diff --git a/changes-set.txt b/changes-set.txt index 9fb320ad95..ffdfbba1cc 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -92,6 +92,15 @@ make a github issue.) DONE: Date Old New Notes +28-Dec-24 fovrnd fovcdmd +28-Dec-24 fovrnda fovcdmda +28-Dec-24 fovrn fovcdm +28-Dec-24 foelrni foelcdmi +28-Dec-24 fornex focdmex +28-Dec-24 fafvelrn fafvelcdm +28-Dec-24 fafv2elrn fafv2elcdm +28-Dec-24 frnvafv2v fcdmafv2v +28-Dec-24 cncffvrn cncfcdm 26-Dec-24 syl5bb bitrid compare to bitri or bitrd 23-Dec-24 sn-iotaex iotaex moved from SN's mathbox to main set.mm 23-Dec-24 sn-iotassuni iotassuni moved from SN's mathbox to main set.mm @@ -99,6 +108,11 @@ Date Old New Notes 23-Dec-24 sn-iotanul iotanul2 moved from SN's mathbox to main set.mm 23-Dec-24 sn-iotauni iotauni2 moved from SN's mathbox to main set.mm 23-Dec-24 iotavallem iotaval2 moved from SN's mathbox to main set.mm +22-Dec-24 frnsuppeq fsuppeq +22-Dec-24 frnsuppeqg fsuppeqg +22-Dec-24 frnfsuppbi ffsuppbi +22-Dec-24 fvmptelrn fvmptelcdm +22-Dec-24 frnssb fcdmssb 19-Dec-24 3albii [same] moved from PM's mathbox to main set.mm 19-Dec-24 ssrel3 [same] moved from PM's mathbox to main set.mm 17-Dec-24 ffvelrnd ffvelcdmd diff --git a/set.mm b/set.mm index 1a86cfc666..c2a005a6ff 100644 --- a/set.mm +++ b/set.mm @@ -64977,7 +64977,7 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ). $( A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.) $) - foelrni $p |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y ) + foelcdmi $p |- ( ( F : A -onto-> B /\ Y e. B ) -> E. x e. A ( F ` x ) = Y ) $= ( wfo wcel cv cfv wceq wrex crn forn eleq2d wfn fofn fvelrnb syl bitr3d wb biimpa ) BCDFZECGZAHDIEJABKZUBEDLZGZUCUDUBUECEBCDMNUBDBOUFUDTBCDPABEDQ @@ -66816,10 +66816,10 @@ in the range of the function (the implication "to the right" is always ${ $d A x $. $d C x $. - fvmptelrn.1 $e |- ( ph -> ( x e. A |-> B ) : A --> C ) $. + fvmptelcdm.1 $e |- ( ph -> ( x e. A |-> B ) : A --> C ) $. $( The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) $) - fvmptelrn $p |- ( ( ph /\ x e. A ) -> B e. C ) $= + fvmptelcdm $p |- ( ( ph /\ x e. A ) -> B e. C ) $= ( wcel cmpt wf wral eqid fmpt sylibr r19.21bi ) ADEGZBCACEBCDHZIOBCJFBCED PPKLMN $. $} @@ -66906,7 +66906,7 @@ in the range of the function (the implication "to the right" is always $d A k $. $d F k $. $d V k $. $( A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.) $) - frnssb $p |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) + fcdmssb $p |- ( ( V C_ W /\ A. k e. A ( F ` k ) e. V ) -> ( F : A --> W <-> F : A --> V ) ) $= ( wss cv cfv wcel wa wf wfn simpr ffn anim12ci ffnfv sylibr simpl anim1ci wral fss syl impbida ) DEFZBGCHDIBATZJZAECKZADCKZUFUGJCALZUEJUHUFUEUGUIUD @@ -72596,25 +72596,25 @@ result of an operator (deduction version). (Contributed by Paul $( An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) $) - fovrn $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> + fovcdm $p |- ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C ) $= ( cxp wf wcel co wa cop opelxpi cfv df-ov ffvelcdm eqeltrid sylan2 3impb ) DEGZCFHZADIZBEIZABFJZCIZUBUCKUAABLZTIZUEABDEMUAUGKUDUFFNCABFOTCUFFPQRS $. ${ - fovrnd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $. + fovcdmd.1 $e |- ( ph -> F : ( R X. S ) --> C ) $. $( An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) $) - fovrnda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $= - ( wcel co cxp wf fovrn syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFGM - NO $. + fovcdmda $p |- ( ( ph /\ ( A e. R /\ B e. S ) ) -> ( A F B ) e. C ) $= + ( wcel co cxp wf fovcdm syl3an1 3expb ) ABEIZCFIZBCGJDIZAEFKDGLPQRHBCDEFG + MNO $. - fovrnd.2 $e |- ( ph -> A e. R ) $. - fovrnd.3 $e |- ( ph -> B e. S ) $. + fovcdmd.2 $e |- ( ph -> A e. R ) $. + fovcdmd.3 $e |- ( ph -> B e. S ) $. $( An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) $) - fovrnd $p |- ( ph -> ( A F B ) e. C ) $= - ( cxp wf wcel co fovrn syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $. + fovcdmd $p |- ( ph -> ( A F B ) e. C ) $= + ( cxp wf wcel co fovcdm syl3anc ) AEFKDGLBEMCFMBCGNDMHIJBCDEFGOP $. $} ${ @@ -76804,7 +76804,7 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this $( If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.) $) - fornex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $= + focdmex $p |- ( A e. C -> ( F : A -onto-> B -> B e. _V ) ) $= ( wfo wcel cvv cdm crn wfun fofun funrnex syl5com fdmd eleq1d 3imtr3d com12 fof forn ) ABDEZACFZBGFZTDHZCFZDIZGFZUAUBTDJUDUFABDKCDLMTUCACTABDABDRNOTUEB GABDSOPQ $. @@ -76814,13 +76814,13 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this by NM, 4-Sep-2004.) $) f1dmex $p |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V ) $= ( wf1 wcel cvv crn ccnv wfo wss f1f frnd ssexg sylan wf1o f1cnv f1ofo syl - ex fornex syl6ci imp ) ABDEZBCFZAGFZUDUEDHZGFZUGADIZJZUFUDUEUHUDUGBKUEUHUDA - BDABDLMUGBCNOTUDUGAUIPUJABDQUGAUIRSUGAGUIUAUBUC $. + ex focdmex syl6ci imp ) ABDEZBCFZAGFZUDUEDHZGFZUGADIZJZUFUDUEUHUDUGBKUEUHUD + ABDABDLMUGBCNOTUDUGAUIPUJABDQUGAUIRSUGAGUIUAUBUC $. $( The range of a 1-1 onto function is a set iff its domain is a set. (Contributed by AV, 21-Mar-2019.) $) f1ovv $p |- ( F : A -1-1-onto-> B -> ( A e. _V <-> B e. _V ) ) $= - ( wf1o cvv wcel wfo f1ofo fornex syl5com wf1 wi f1of1 f1dmex ex syl impbid + ( wf1o cvv wcel wfo f1ofo focdmex syl5com wf1 wi f1of1 f1dmex ex syl impbid ) ABCDZAEFZBEFZRABCGSTABCHABECIJRABCKZTSLABCMUATSABECNOPQ $. ${ @@ -78156,8 +78156,8 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) 27-Jan-2020.) $) ovmpoelrn $p |- ( ( A. x e. A A. y e. B C e. M /\ X e. A /\ Y e. B ) -> ( X O Y ) e. M ) $= - ( wcel wral cxp wf co fmpo fovrn syl3an1b ) EFKBDLACLCDMFGNHCKIDKHIGOFKAB - CDEFGJPHIFCDGQR $. + ( wcel wral cxp wf co fmpo fovcdm syl3an1b ) EFKBDLACLCDMFGNHCKIDKHIGOFKA + BCDEFGJPHIFCDGQR $. $} ${ @@ -78767,7 +78767,7 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) (Contributed by NM, 29-Mar-2008.) $) curry1f $p |- ( ( F : ( A X. B ) --> D /\ C e. A ) -> G : B --> D ) $= - ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry1 sylan fovrn fmpt3d + ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry1 sylan fovcdm fmpt3d 3expa ) ABIZDEJZCAKZLHBCHMZENZDFUEEUDOUFFHBUHPQUDDERHABCEFGSTUEUFUGBKUHDK CUGDABEUAUCUB $. $} @@ -78806,9 +78806,9 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) (Contributed by NM, 16-Dec-2008.) $) curry2f $p |- ( ( F : ( A X. B ) --> D /\ C e. B ) -> G : A --> D ) $= - ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry2 sylan 3com23 fmpt3d - fovrn 3expa ) ABIZDEJZCBKZLHAHMZCENZDFUFEUEOUGFHAUIPQUEDERHABCEFGSTUFUGUH - AKZUIDKZUFUJUGUKUHCDABEUCUAUDUB $. + ( vx cxp wf wcel wa cv co wfn cmpt wceq ffn curry2 sylan fovcdm 3com23 + 3expa fmpt3d ) ABIZDEJZCBKZLHAHMZCENZDFUFEUEOUGFHAUIPQUEDERHABCEFGSTUFUGU + HAKZUIDKZUFUJUGUKUHCDABEUAUBUCUD $. $( The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) $) @@ -79661,8 +79661,8 @@ any sets (which usually are functions) and any element (even not $( Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.) $) - frnsuppeq $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S - -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $= + fsuppeq $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S + -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $= ( wcel wa wf csupp co ccnv csn cdif cima wceq cvv wi cin eqtrd fex imp wfun expcom adantr suppimacnv syl2anc ffun inpreima syl wss cdm cnvimass fimacnv simplr fdm eqtr4d sseqtrid sseqin2 sylib invdif imaeq2i eqtr3di adantl ex ) @@ -79671,16 +79671,16 @@ any sets (which usually are functions) and any element (even not VQVIBUCVTWBPCABUHAVPBUIUJVIVQWAUKWBVQPVIBULZVQWABVPUMVIWCCWACABUPCABUNUQURV QWAUSUTTVSVMVKAVLVAVBVCVDTVE $. - $( Version of ~ frnsuppeq avoiding ~ ax-rep by assuming ` F ` is a set rather + $( Version of ~ fsuppeq avoiding ~ ax-rep by assuming ` F ` is a set rather than its domain ` I ` . (Contributed by SN, 30-Jul-2024.) $) - frnsuppeqg $p |- ( ( F e. V /\ Z e. W ) -> ( F : I --> S - -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $= + fsuppeqg $p |- ( ( F e. V /\ Z e. W ) -> ( F : I --> S + -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) ) $= ( wcel wa wf csupp co ccnv csn cdif cima wceq cvv suppimacnv cin wfun sylib ffun inpreima syl wss cdm cnvimass fdm fimacnv eqtr4d sseqtrid eqtrd invdif sseqin2 imaeq2i eqtr3di sylan9eq ex ) BDGFEGHZCABIZBFJKZBLZAFMZNZOZPUSUTVAV BQVCNZOZVEBDEFRUTVBAVFSZOZVGVEUTVIVBAOZVGSZVGUTBTVIVKPCABUBAVFBUCUDUTVGVJUE VKVGPUTBUFZVGVJBVFUGUTVLCVJCABUHCABUIUJUKVGVJUNUAULVHVDVBAVCUMUOUPUQUR $. - $( $j usage 'frnsuppeqg' avoids 'ax-rep'; $) + $( $j usage 'fsuppeqg' avoids 'ax-rep'; $) ${ $d F i $. $d Z i $. @@ -87997,8 +87997,8 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so wi adantl reeanv sylibr adantr ecexg elisset 3syl biantrud 2rexbidv mpbid 19.42v bicomi rexbii rexcom4 bitri sylib eceq1 eqeq2d anbi1d oveq1 eceq1d cvv anbi12d anbi2d oveq2 cbvrex2vw anbi12i bitr4i wbr wer wss sseldd erth - simprll simprrl wf fovrnd 3imtr3d eqeq2 biimprcd syl6 impd eqeq1 bi2anan9 - cxp imbi12d syl5ibrcom anassrs rexlimdvva syl5bir alrimivv eu4 sylanbrc + simprll simprrl cxp wf fovcdmd 3imtr3d eqeq2 biimprcd syl6 eqeq1 bi2anan9 + impd imbi12d syl5ibrcom anassrs rexlimdvva syl5bir alrimivv eu4 sylanbrc wb ) AQMUPZRNUPZUQZUQZQUCURZIUSZUTZRUBURZJUSZUTZUQZBURZYJYMHVAZKUSZUTZUQZ UBFVBZUCEVBZBVCZUUCYPUOURZYSUTZUQZUBFVBUCEVBZUQZBUOVDZVKZUOVEBVEUUCBVFYIY PYTBVCZUQZUBFVBZUCEVBZUUDYIYPUBFVBUCEVBZUUOYIYLUCEVBZYOUBFVBZUQZUUPYHUUSA @@ -88020,9 +88020,9 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so ZUVFUVRJWTZUQUVJUWBKWTUXOUXRUNUXLUXSUXMUXTUXNUXLUVCUVOILALIXAUXKUGVOUXLEL UVCAELXBUXKUJVOAUXDUXEUXJXEZXCXDUXLUVFUVRJOAOJXAUXKUHVOUXLFOUVFAFOXBUXKUK VOAUXFUXHUXIXFZXCXDWNUXLUVJUWBKPAPKXAUXKUIVOUXLGPUVJAGPXBUXKULVOUXLUVCUVF - GEFHAEFXPGHXGUXKUMVOUYAUYBXHXCXDXIUWDUXQUXRUUEUWCUVKXJXKXLXMUVMUWEUXPUUJU + GEFHAEFXGGHXHUXKUMVOUYAUYBXIXCXDXJUWDUXQUXRUUEUWCUVKXKXLXMXPUVMUWEUXPUUJU XQUVIUWEUXPYEUVLUVIUWAUXOUWDUVEUVQUXMUVHUVTUXNQUVDUVPXNRUVGUVSXNXOWJVOUVL - UUJUXQYEUVIYQUVKUUEXNVLXQXRXMXSXTYAXTYAVOYBUUCUUHBUOUUJUUAUUGUCUBEFUUJYTU + UUJUXQYEUVIYQUVKUUEXNVLXQXRXPXSXTYAXTYAVOYBUUCUUHBUOUUJUUAUUGUCUBEFUUJYTU UFYPYQUUEYSXNWOVTYCYD $. eropr.12 $e |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. B @@ -88076,15 +88076,15 @@ general ordinal versions of these theorems (in this case ~ oa0r ) so 30-Dec-2014.) $) eroprf $p |- ( ph -> .+^ : ( J X. K ) --> L ) $= ( cxp wf cv cec wceq wa co wrex cio cmpo wcel wral cab wi ad2antrr adantr - cqs fovrnda ecelqsg syl2anc eleqtrrdi eleq1a syl rexlimdvva abssdv eroveu - adantld weu iotacl sseldd ralrimivva eqid fmpo sylib erovlem feq1d mpbird - ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZVHZU - FHVJUGGVJZDVKZVLZVDZAXLRVMZCQVNBPVNXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVOZRXLXQX - KDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVPYAXHINVSZRYANUCVM - ZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRVTIXGNUCWAWBVBWCXHRXFWDWE - WIWFWGXQXKDWJXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURWH - XKDWKWEWLWMBCPQXLRXMXMWNWOWPAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIUJUKU - LUMUNUOUPUQURUSWQWRWS $. + cqs ecelqsg syl2anc eleqtrrdi eleq1a syl adantld rexlimdvva abssdv eroveu + fovcdmda iotacl sseldd ralrimivva eqid fmpo sylib erovlem feq1d mpbird + weu ) APQVCZRKVDWTRBCPQBVEZUGVEZLVFVGCVEZUFVEZMVFVGVHZDVEZXBXDJVIZNVFZVGZ + VHZUFHVJUGGVJZDVKZVLZVDZAXLRVMZCQVNBPVNXNAXOBCPQAXAPVMXCQVMVHZVHZXKDVOZRX + LXQXKDRXQXJXFRVMZUGUFGHXQXBGVMXDHVMVHZVHZXIXSXEYAXHRVMXIXSVPYAXHINVSZRYAN + UCVMZXGIVMXHYBVMAYCXPXTUJVQXQXBXDIGHJAGHVCIJVDXPUQVRWIIXGNUCVTWAVBWBXHRXF + WCWDWEWFWGXQXKDWSXLXRVMADEFGHIJLMNOPQSTXAXCUCUDUEUFUGUHUIUJUKULUMUNUOUPUQ + URWHXKDWJWDWKWLBCPQXLRXMXMWMWNWOAWTRKXMABCDEFGHIJKLMNOPQSTUCUDUEUFUGUHUIU + JUKULUMUNUOUPUQURUSWPWQWR $. $} ${ @@ -88711,10 +88711,10 @@ the first case of his notation (simple exponentiation) and subscript it fsetprcnex $p |- ( ( ( A e. V /\ A =/= (/) ) /\ B e/ _V ) -> { f | f : A --> B } e/ _V ) $= ( va vg vm vn wcel c0 wne wa cvv wnel cv wf cab wi wex cfv n0 cmpt cbvabv - feq1 fveq1 cbvmptv fsetfocdm fornex syl5com nelcon3d expcom exlimiv sylbi - wfo impcom imp ) ADIZAJKZLBMNZABCOZPZCQZMNZURUQUSVCRZUREOZAIZESUQVDRZEAUA - VFVGEUQVFVDUQVFLZVBMBMVHVBBFVBVEFOZTZUBZUNVBMIBMIABVKGHVBDVEVAABGOZPCGABU - TVLUDUCFHVBVJVEHOZTVEVIVMUEUFUGVBBMVKUHUIUJUKULUMUOUP $. + wfo fveq1 cbvmptv fsetfocdm focdmex syl5com nelcon3d expcom exlimiv sylbi + feq1 impcom imp ) ADIZAJKZLBMNZABCOZPZCQZMNZURUQUSVCRZUREOZAIZESUQVDRZEAU + AVFVGEUQVFVDUQVFLZVBMBMVHVBBFVBVEFOZTZUBZUDVBMIBMIABVKGHVBDVEVAABGOZPCGAB + UTVLUNUCFHVBVJVEHOZTVEVIVMUEUFUGVBBMVKUHUIUJUKULUMUOUP $. $} ${ @@ -89936,8 +89936,8 @@ the first case of his notation (simple exponentiation) and subscript it $( The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) $) f1oeng $p |- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B ) $= - ( wcel wf1o cvv cen wbr wfo fornex f1ofo impel f1oen2g 3com23 mpd3an3 ) A - CEZABDFZBGEZABHIZQABDJSRABCDKABDLMQSRTABDCGNOP $. + ( wcel wf1o cvv cen wbr wfo focdmex f1ofo impel f1oen2g 3com23 mpd3an3 ) + ACEZABDFZBGEZABHIZQABDJSRABCDKABDLMQSRTABDCGNOP $. $( The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.) $) @@ -92002,32 +92002,33 @@ excluded middle (LEM), and in ILE the LEM is not accepted as necessarily ( ( A ^m B ) ^m C ) ~~ ( A ^m ( B X. C ) ) ) $= ( vx vy wcel cmap co cv cfv cmpt cvv wf wral wa eqid wceq vf w3a cxp cmpo vg ovexd elmapi ffvelcdmda an32s ralrimiva fmpo sylib simp1 xpexg 3adant1 - syl wb elmapg syl2anc syl5ibr adantl fovrn 3expa sylanl1 3adant3 ad2antrr - fmpttd mpbird ex ovex simp3 sylancr sylibrd wfn elmapfn ad2antll adantlrl - fnov 3adant2 simp1l2 simp1l1 fex2 syl3anc fvmpt2 fveq1d sylancl mpoeq3dva - simp2 eqtrd eqtr4d nfcv nfmpt1 nfmpt nfeq2 fveq1 a1d ralrimi jctil eqeq2d - mpoeq123 syl5ibrcom ad2antrl feqmptd simprl sylan mpteq2dva nfmpo2 nfmpo1 - nfv fvex ovmpt4g mp3an3 oveq eqeq1d expcomd ralrimd mpteq12 syl6an impbid - eqidd en3d ) ADIZBEIZCFIZUBZUAUEABJKZCJKZABCUCZJKZGHBCGLZHLZUALZMZMZUDZHC - GBYJYKUELZKZNZNZOOYEYFCJUFYEAYHJUFYLYGIZYOYIIZYEYHAYOPZYTYNAIZHCQZGBQUUBY - TUUDGBYTYJBIZRUUCHCYTYKCIZUUEUUCYTUUFRZBAYJYMUUGYMYFIBAYMPZYTCYFYKYLYLYFC - UGZUHYMABUGUPZUHUIUJUJGHBCYNAYOYOSZUKULYEYBYHOIZUUAUUBUQYBYCYDUMYCYDUULYB - BCEFUNUOAYHYODOURUSUTYEYPYIIZCYFYSPZYSYGIZYEUUMUUNYEUUMRZHCYRYFUUPUUFRZYR - YFIZBAYRPZUUQGBYQAUUPYHAYPPZUUFUUEYQAIZUUMUUTYEYPAYHUGVAUUTUUEUUFUVAUUTUU - EUUFUVAYJYKABCYPVBVCUIVDVGZYEUURUUSUQZUUMUUFYBYCUVCYDABYRDEURVEVFVHVGVIYE - YFOIYDUUOUUNUQABJVJYBYCYDVKYFCYSOFURVLVMYEYTUUMRZYLYSTZYPYOTZUQYEUVDRZUVE - UVFUVGUVFUVEYPGHBCYJYKYSMZMZUDZTUVGYPGHBCYQUDZUVJUVGYPYHVNZYPUVKTUUMUVLYE - YTYPAYHVOVPGHBCYPVRULUVGGHBCUVIYQUVGUUEUUFUBZUVIYJYRMZYQUVMYJUVHYRUVMUUFY - ROIZUVHYRTUVGUUEUUFVKUVMUUSYCYBUVOUVGUUFUUSUUEYEUUMUUFUUSYTUVBVQVSYBYCYDU - VDUUEUUFVTYBYCYDUVDUUEUUFWABAYREDWBWCHCYROYSYSSWDUSWEUVMUUEYQOIUVNYQTUVGU - UEUUFWHYJYKYPVJGBYQOYRYRSWDWFWIWGWJUVEYOUVJYPUVEBBTZCCTZYNUVITZHCQZRZGBQY - OUVJTBSUVEUVTGBGYLYSGHCYRGCWKGBYQWLWMWNUVEUVTUUEUVEUVSUVQUVEUVRHCHYLYSHCY - RWLWNUVEUVRUUFUVEYJYMUVHYKYLYSWOWEWPWQCSZWRWPWQGHBCYNBCUVIWTVLWSXAUVGUVEU - VFYLHCGBYNNZNZTUVGYLHCYMNUWCUVGHCYFYLYTCYFYLPYEUUMUUIXBXCUVGHCYMUWBUVGUUF - RGBAYMUVGYTUUFUUHYEYTUUMXDUUJXEXCXFWIUVFYSUWCYLUVFUVQYRUWBTZHCQYSUWCTUWAU - VFUWDHCHYPYOGHBCYNXGWNUVFUVPUUFYQYNTZGBQUWDUVFBXTUVFUUFUWEGBGYPYOGHBCYNXH - WNUUFGXIUVFUUEUUFUWEUUEUUFRUWEUVFYJYKYOKZYNTZUUEUUFYNOIUWGYJYMXJGHBCYNYOO - UUKXKXLUVFYQUWFYNYJYKYPYOXMXNUTXOXPGBYQBYNXQXRWQHCYRCUWBXQVLWSXAXSVIYA $. + syl wb elmapg syl2anc syl5ibr adantl fovcdm 3expa sylanl1 fmpttd ad2antrr + 3adant3 mpbird ex ovex sylancr sylibrd wfn elmapfn ad2antll fnov adantlrl + simp3 3adant2 simp1l2 simp1l1 fex2 syl3anc fvmpt2 simp2 sylancl mpoeq3dva + fveq1d eqtrd eqtr4d nfcv nfmpt1 nfmpt nfeq2 fveq1 ralrimi mpoeq123 eqeq2d + a1d jctil syl5ibrcom ad2antrl feqmptd simprl sylan mpteq2dva nfmpo2 eqidd + nfmpo1 nfv fvex ovmpt4g mp3an3 oveq eqeq1d expcomd ralrimd mpteq12 syl6an + impbid en3d ) ADIZBEIZCFIZUBZUAUEABJKZCJKZABCUCZJKZGHBCGLZHLZUALZMZMZUDZH + CGBYJYKUELZKZNZNZOOYEYFCJUFYEAYHJUFYLYGIZYOYIIZYEYHAYOPZYTYNAIZHCQZGBQUUB + YTUUDGBYTYJBIZRUUCHCYTYKCIZUUEUUCYTUUFRZBAYJYMUUGYMYFIBAYMPZYTCYFYKYLYLYF + CUGZUHYMABUGUPZUHUIUJUJGHBCYNAYOYOSZUKULYEYBYHOIZUUAUUBUQYBYCYDUMYCYDUULY + BBCEFUNUOAYHYODOURUSUTYEYPYIIZCYFYSPZYSYGIZYEUUMUUNYEUUMRZHCYRYFUUPUUFRZY + RYFIZBAYRPZUUQGBYQAUUPYHAYPPZUUFUUEYQAIZUUMUUTYEYPAYHUGVAUUTUUEUUFUVAUUTU + UEUUFUVAYJYKABCYPVBVCUIVDVEZYEUURUUSUQZUUMUUFYBYCUVCYDABYRDEURVGVFVHVEVIY + EYFOIYDUUOUUNUQABJVJYBYCYDVRYFCYSOFURVKVLYEYTUUMRZYLYSTZYPYOTZUQYEUVDRZUV + EUVFUVGUVFUVEYPGHBCYJYKYSMZMZUDZTUVGYPGHBCYQUDZUVJUVGYPYHVMZYPUVKTUUMUVLY + EYTYPAYHVNVOGHBCYPVPULUVGGHBCUVIYQUVGUUEUUFUBZUVIYJYRMZYQUVMYJUVHYRUVMUUF + YROIZUVHYRTUVGUUEUUFVRUVMUUSYCYBUVOUVGUUFUUSUUEYEUUMUUFUUSYTUVBVQVSYBYCYD + UVDUUEUUFVTYBYCYDUVDUUEUUFWABAYREDWBWCHCYROYSYSSWDUSWHUVMUUEYQOIUVNYQTUVG + UUEUUFWEYJYKYPVJGBYQOYRYRSWDWFWIWGWJUVEYOUVJYPUVEBBTZCCTZYNUVITZHCQZRZGBQ + YOUVJTBSUVEUVTGBGYLYSGHCYRGCWKGBYQWLWMWNUVEUVTUUEUVEUVSUVQUVEUVRHCHYLYSHC + YRWLWNUVEUVRUUFUVEYJYMUVHYKYLYSWOWHWSWPCSZWTWSWPGHBCYNBCUVIWQVKWRXAUVGUVE + UVFYLHCGBYNNZNZTUVGYLHCYMNUWCUVGHCYFYLYTCYFYLPYEUUMUUIXBXCUVGHCYMUWBUVGUU + FRGBAYMUVGYTUUFUUHYEYTUUMXDUUJXEXCXFWIUVFYSUWCYLUVFUVQYRUWBTZHCQYSUWCTUWA + UVFUWDHCHYPYOGHBCYNXGWNUVFUVPUUFYQYNTZGBQUWDUVFBXHUVFUUFUWEGBGYPYOGHBCYNX + IWNUUFGXJUVFUUEUUFUWEUUEUUFRUWEUVFYJYKYOKZYNTZUUEUUFYNOIUWGYJYMXKGHBCYNYO + OUUKXLXMUVFYQUWFYNYJYKYPYOXNXOUTXPXQGBYQBYNXRXSWPHCYRCUWBXRVKWRXAXTVIYA + $. $} ${ @@ -95429,13 +95430,13 @@ of Infinity (see comments in ~ fin2inf ). (Contributed by NM, $( Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.) $) - frnfsuppbi $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S + ffsuppbi $p |- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F finSupp Z <-> ( `' F " ( S \ { Z } ) ) e. Fin ) ) ) $= ( wcel wa wf cfsupp wbr ccnv csn cdif cima cfn wb csupp cvv imp wfun adantl - co ffun wi fex expcom adantr simplr funisfsupp syl3anc wceq frnsuppeq bitrd - eleq1d ex ) CDGZFEGZHZCABIZBFJKZBLAFMNOZPGZQUSUTHZVABFRUCZPGZVCVDBUAZBSGZUR - VAVFQUTVGUSCABUDUBUSUTVHUQUTVHUEURUTUQVHCADBUFUGUHTUQURUTUIBSEFUJUKVDVEVBPU - SUTVEVBULABCDEFUMTUOUNUP $. + co ffun wi fex expcom adantr simplr funisfsupp syl3anc fsuppeq eleq1d bitrd + wceq ex ) CDGZFEGZHZCABIZBFJKZBLAFMNOZPGZQUSUTHZVABFRUCZPGZVCVDBUAZBSGZURVA + VFQUTVGUSCABUDUBUSUTVHUQUTVHUEURUTUQVHCADBUFUGUHTUQURUTUIBSEFUJUKVDVEVBPUSU + TVEVBUOABCDEFULTUMUNUP $. ${ $d A k $. $d F k $. $d Z k $. $d ph k $. @@ -105438,58 +105439,58 @@ x finSupp (/) } |-> ( ( _I |` _om ) o. ( y o. `' ( Y o. `' X ) ) ) ) $. c0 df1o2 mpbird wa wf wral seqomsuc ad2antrl vex fvex reseq1 fveq2d fveq1 oveq12d cbvmptv suceq adantr oveq2d simpr reseq2 fveq12d mpteq12dv eqtrid simpl nfcv cbvmpo ovex mptex ovmpoa cxp ad2antrr f1of f1f ad2antll elmapi - mp2an adantl sssucid fssres elmapg ffvelcdmd sucid ffvelcdm fovrnd fmpt3d - sylancl wfn a1i fvreseq syl21anc eqeq12d ralsn fveq1d fvmpt eqtrd opelxpd - ffnd df-ov f1fveq syl12anc bitrdi bicomi anbi12d eqid opth simplrr anbi1d - 3bitrd eqfnfv cun df-suc raleqi ralunb 3bitr4d biimpd ralrimivva sylanbrc - bitri dff13 expr expcom finds2 vtoclga impcom ) EUFOACEPUGZCEIQZUMZACUAUH - ZPUGZCUVGIQZUMZUIAUVFUIUAEUFUVGERZUVJUVFAUVKUVJUVHCUVEUMZUVFUVKUVIUVERUVJ - UVLSUVGEIUJUVHCUVIUVEUKUNUVKUVHUVDRUVLUVFSUVGECPULUVHUVDCUVEUOUNUPUQUVJCV - TPUGZCVTDURZUSZUMZCUBUHZPUGZCUVQIQZUMZCUVQUTZPUGZCUWAIQZUMZAUAUBUVGVTRZUV - JUVHCUVOUMZUVPUWEUVIUVORUVJUWFSUWEUVIVTIQZUVOUVGVTIUJUVOTOUWGUVORUVNVAGFT - TBCGUHZUTZPUGZBUHZUWHVBZFUHZQZUWHUWKQZHUGZVCZVDZIUVOTNVEVFVGUVHCUVIUVOUKU - 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xpnz biimpi 3syl eldifd 3expib + sylbi ralrimivv fmpo sylib dcomex ccom cvv 2ndcof 3ad2ant1 adantl mp3an23 + axdc3 fex2 syl fvco3 mpan2 fveq2 3ad2ant2 eqtrd op2ndg nfv eqeq12d exbidv + opeq1 weq opeq2 eqeq2d imp df-ov simplr ffvelcdm eleq1 syl6bi suceq sneqd + opelxp2 oveq1 xpeq12d oveq2 xpeq2d ovmpo eleq12d elxp exlimiv com3l com12 + fveq2d op2nd eqtrdi oveq2d ex fveq1 sylan9eqr nfra1 nfan spcegv 3ad2antr2 + nf3an eqtr4di mpan9 2fveq3 rspcv eleq2 velsn biimpac adantrr eximi expcom + ad2antlr sylsyld cbvexvw syl8ib 3adant2 finds2 rspccv 3impia simp21 simp3 + impancom ralrimiv rspa 3ad2antl3 3adant1 simpl simprr impel eleq2d sylan2 + eqtr3id simpll biimprcd exp4c impcom sylbid 3imp syl121anc 3expia ralrimi + exlimivv 3jca feq1 eqeq1d ralbidv 3anbi123d spcedv exlimdv adantr mpd ) C + BLZMBUFZBUGZNUHZUKZGUIZOMUWRUAUJZUIZNUXCPZNCULZQZEUJZUMZUXCPZUXHUXCPZHPZL + ZEMUNZUOZUARZMBDUJZUIZNUXQPZCQZUXIUXQPZUXHUXHUXQPZGVDZLZEMUNZUOZDRZUWQUXF + 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WUDWUBVVPVYKVVFWUBVVPOZVYKOZVVFWUDWUIVVCVXCVVEVYOWUIVVCVYDTPZVXCWUIVVCUXJ + TPZWUJVYKVVCWUKQZWUHVVMUXDUXIMLWULUXHVBMUWRUXITUXCXCUVPWRWUIUXJVYDTWUBVVP + VYKUVRYOXGVYCVXCUEWAUDWAYPYQWUIVVDVVNUXHGWUIVVDVVOTPZVVNWUIVVDUXKTPZWUMVY + KVVDWUNQWUHMUWRUXHTUXCXCWRWUIUXKVVOTWUBVVPVYKXRYOXGUXHVVNEWAKWAYPYQYRYJUV + SUVTWRUWAUWGWIYMXPUWBYSYLUWCUWDUWEUWFUWHUXQVUSQZUXRVUTUXTVVBUYEVVGMBUXQVU + SUWIWUOUXSVVACNUXQVUSYTUWJWUOUYDVVFEMWUOUYAVVCUYCVVEUXIUXQVUSYTWUOUYBVVDU + XHGUXHUXQVUSYTYRYJUWKUWLUWMYSUWNUWOUWP $. $} ${ @@ -113798,14 +113799,14 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario 15-May-2015.) $) ttukeylem1 $p |- ( ph -> ( C e. A <-> ( ~P C i^i Fin ) C_ A ) ) $= ( cvv wcel cpw cfn cin wss cdom cun ccrd ssexg wb wi elex a1i wa wbr cuni - cdif ssun1 undif1 sseqtrri cfv wfo fvex wf1o f1ofo syl fornex mpsyl unexg - id syl2anc sylancr uniexb sylibr syl2anr infpwfidom reldom brrelex1i 3syl - ex cv wal wceq eleq1 pweq ineq1d sseq1d bibi12d spcgv syl5com pm5.21ndd ) - AEJKZECKZELZMNZCOZWCWBUAAECUBUCAWFWBAWFUDWEJKZEWEPUEWBWFWFCJKZWGAWFUTACUF - ZJKZWHAWIWIDUGZDQZOWLJKZWJWIWIDQWLWIDUHWIDUIUJAWKJKZDCKWMWKRUKZJKAWOWKFUL - ZWNWKRUMAWOWKFUNWPGWOWKFUOUPWOWKJFUQURHWKDJCUSVAWIWLJSVBCVCVDWECJSVEEVFEW - EPVGVHVIVJABVKZCKZWQLZMNZCOZTZBVLWBWCWFTZIXBXCBEJWQEVMZWRWCXAWFWQECVNXDWT - WECXDWSWDMWQEVOVPVQVRVSVTWA $. + id cdif ssun1 undif1 sseqtrri cfv wfo fvex wf1o f1ofo focdmex mpsyl unexg + syl2anc sylancr uniexb sylibr syl2anr infpwfidom reldom brrelex1i 3syl ex + syl cv wal wceq eleq1 pweq ineq1d sseq1d bibi12d spcgv syl5com pm5.21ndd + ) AEJKZECKZELZMNZCOZWCWBUAAECUBUCAWFWBAWFUDWEJKZEWEPUEWBWFWFCJKZWGAWFUGAC + UFZJKZWHAWIWIDUHZDQZOWLJKZWJWIWIDQWLWIDUIWIDUJUKAWKJKZDCKWMWKRULZJKAWOWKF + UMZWNWKRUNAWOWKFUOWPGWOWKFUPVJWOWKJFUQURHWKDJCUSUTWIWLJSVACVBVCWECJSVDEVE + EWEPVFVGVHVIABVKZCKZWQLZMNZCOZTZBVLWBWCWFTZIXBXCBEJWQEVMZWRWCXAWFWQECVNXD + WTWECXDWSWDMWQEVOVPVQVRVSVTWA $. $( Lemma for ~ ttukey . A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) $) @@ -114126,12 +114127,12 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario ( (/) ~< B /\ B ~<_ A ) ) $= ( c0 wne cv wfo wa csdm wbr cdom wceq eqeq1d wb cvv wcel mpcom 0sdomg syl adantl wex cdm fof fdmd crn dm0rn0 forn bitrid necon3bid biimpac vex dmex - bitr3d eqeltrrdi fornex mpbird ex fodomg jca2 exlimdv sdomdomtr brrelex2i - imp reldom mpbid fodomr jca impbii ) ADEZABCFZGZCUAZHDBIJZBAKJZHZVIVLVOVI - VKVOCVIVKVMVNVIVKVMVIVKHVMBDEZVKVIVPVKADBDVKVJUBZDLZADLBDLZVKVQADVKABVJAB - VJUCUDZMVRVJUEZDLVKVSVJUFVKWABDABVJUGMUHUMUIUJVKVMVPNZVIVKBOPZWBAOPZVKWCV - KAVQOVTVJCUKULUNZABOVJUOQBORSTUPUQWDVKVNWEABVJOURQUSUTVCVOVIVLVODAIJZVIDB - AVAVOWDWFVINVNWDVMBAKVDVBTAORSVEABCVFVGVH $. + bitr3d eqeltrrdi focdmex mpbird ex fodomg exlimdv imp sdomdomtr brrelex2i + jca2 reldom mpbid fodomr jca impbii ) ADEZABCFZGZCUAZHDBIJZBAKJZHZVIVLVOV + IVKVOCVIVKVMVNVIVKVMVIVKHVMBDEZVKVIVPVKADBDVKVJUBZDLZADLBDLZVKVQADVKABVJA + BVJUCUDZMVRVJUEZDLVKVSVJUFVKWABDABVJUGMUHUMUIUJVKVMVPNZVIVKBOPZWBAOPZVKWC + VKAVQOVTVJCUKULUNZABOVJUOQBORSTUPUQWDVKVNWEABVJOURQVCUSUTVOVIVLVODAIJZVID + BAVAVOWDWFVINVNWDVMBAKVDVBTAORSVEABCVFVGVH $. $} $( When assuming AC, weak and usual dominance coincide. It is not known if @@ -134768,7 +134769,7 @@ Nonnegative integers (as a subset of complex numbers) Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.) $) frnnn0supp $p |- ( ( I e. V /\ F : I --> NN0 ) -> ( F supp 0 ) = ( `' F " NN ) ) $= - ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex frnsuppeq + ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeq wi mpan2 imp dfn2 imaeq2i eqtr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $. @@ -134777,16 +134778,16 @@ Nonnegative integers (as a subset of complex numbers) frnnn0fsupp $p |- ( ( I e. V /\ F : I --> NN0 ) -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) ) $= ( wcel cn0 wf wa cc0 cfsupp wbr ccnv csn cdif cima cfn cn wb cvv wi c0ex - frnfsuppbi mpan2 imp dfn2 imaeq2i eleq1i bitr4di ) BCDZBEAFZGAHIJZAKZEHLMZN - ZODZUKPNZODUHUIUJUNQZUHHRDUIUPSTEABCRHUAUBUCUOUMOPULUKUDUEUFUG $. + ffsuppbi mpan2 imp dfn2 imaeq2i eleq1i bitr4di ) BCDZBEAFZGAHIJZAKZEHLMZNZO + DZUKPNZODUHUIUJUNQZUHHRDUIUPSTEABCRHUAUBUCUOUMOPULUKUDUEUFUG $. $( Version of ~ frnnn0supp avoiding ~ ax-rep by assuming ` F ` is a set rather than its domain ` I ` . (Contributed by SN, 5-Aug-2024.) $) frnnn0suppg $p |- ( ( F e. V /\ F : I --> NN0 ) -> ( F supp 0 ) = ( `' F " NN ) ) $= - ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq wi c0ex frnsuppeqg - cvv mpan2 imp dfn2 imaeq2i eqtr4di ) ACDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKP - ZUFHTDUGULQREABCTHSUAUBOUJUIUCUDUE $. + ( wcel cn0 wf wa cc0 csupp co ccnv csn cdif cima cn wceq cvv c0ex fsuppeqg + wi mpan2 imp dfn2 imaeq2i eqtr4di ) ACDZBEAFZGAHIJZAKZEHLMZNZUIONUFUGUHUKPZ + UFHQDUGULTREABCQHSUAUBOUJUIUCUDUE $. $( $j usage 'frnnn0suppg' avoids 'ax-rep'; $) $( Version of ~ frnnn0fsupp avoiding ~ ax-rep by assuming ` F ` is a set @@ -138958,8 +138959,8 @@ Rational numbers (as a subset of complex numbers) (Proof modification is discouraged.) (New usage is discouraged.) $) cnexALT $p |- CC e. _V $= ( vx vy cr cxp cvv wcel cc cv ci cmul co caddc cmpo wfo reexALT xpex wf1o - eqid cnref1o f1ofo ax-mp fornex mp2 ) CCDZEFUDGABCCAHIBHJKLKMZNZGEFCCOOPU - DGUEQUFABUEUERSUDGUETUAUDGEUEUBUC $. + eqid cnref1o f1ofo ax-mp focdmex mp2 ) CCDZEFUDGABCCAHIBHJKLKMZNZGEFCCOOP + UDGUEQUFABUEUERSUDGUETUAUDGEUEUBUC $. $} $( The set of extended reals exists. (Contributed by NM, 24-Dec-2006.) $) @@ -149264,38 +149265,38 @@ equinumerous to omega (the set of finite ordinal numbers). (Contributed E. g ( g : Z --> A /\ ( g ` M ) = C /\ A. k e. Z ( g ` ( k + 1 ) ) e. ( k F ( g ` k ) ) ) ) $= ( wcel com cfv vf vm cxp cpw c0 csn cdif wf wa cv wceq csuc co wral w3a - wex c1 caddc wf1o cuz om2uzf1oi f1oeq3 ax-mp mpbir f1of ffvelcdmi fovrn - wb syl3an2 3expb ralrimivva fmpo sylib sylan2 ccnv ccom f1ocnv mp2b fco - axdc4 mpan2 3ad2ant1 cz eleqtrri fvco3 mp2an om2uz0i wi peano1 f1ocnvfv - uzid fveq2i eqtri simp2 eqtrid adantl suceq fveq2 oveq12d eleq12d rspcv - fveq2d id syl peano2uzs sylancr om2uzsuci f1ocnvfv2 oveq1d eqtrd peano2 - mpan eqtr2d ffvelcdm oveq2 ovex syl2anc eqcomd sylibd impancom ralrimiv - mpd ovmpo 3adant2 vex cmpt crdg cres wfun rdgfun omex resfunexg eqeltri - cvv cnvex coex feq1 fveq1 eqeq1d oveq2d ralbidv 3anbi123d spcev syl3anc - exlimiv ) DCRZLCUCCUDUEUFUGZHUHZUISCUAUJZUHZUEUUITZDUKZUBUJZULZUUITZUUM - UUMUUITZJUMZRZUBSUNZUOZUAUPZLCEUJZUHZKUVBTZDUKZFUJZUQURUMZUVBTZUVFUVFUV - BTZHUMZRZFLUNZUOZEUPZUUHUUFSCUCUUGJUHZUVAUUHGUJZITZAUJZHUMZUUGRZACUNGSU - NUVOUUHUVTGASCUUHUVPSRZUVRCRZUVTUWAUUHUVQLRUWBUVTSLUVPISLIUSZSLIUHUWCSK - UTTZIUSZBKIMPVALUWDUKUWCUWEVHNLUWDSIVBVCVDZSLIVEVCVFUVQUVRUUGLCHVGVIVJV - KGASCUVSUUGJQVLVMCDUAUBJOVTVNUUTUVNUAUUTLCUUIIVOZVPZUHZKUWHTZDUKZUVGUWH - TZUVFUVFUWHTZHUMZRZFLUNZUVNUUJUULUWIUUSUUJLSUWGUHZUWIUWCLSUWGUSUWQUWFSL - IVQLSUWGVEVRZLSCUUIUWGVSWAWBUUTUWJUUKDUWJKUWGTZUUITZUUKUWQKLRUWJUWTUKUW - RKUWDLKWCRKUWDRMKWKVCNWDLSKUUIUWGWEWFUWSUEUUIUEITKUKZUWSUEUKZBKIMPWGUWC - UESRUXAUXBWHUWFWISLUEKIWJWFVCWLWMUUJUULUUSWNWOUUJUUSUWPUULUUJUUSUIUWOFL - UUJUVFLRZUUSUWOUUJUXCUIZUUSUVFUWGTZULZUUITZUXEUXEUUITZJUMZRZUWOUXDUXESR - ZUUSUXJWHUXCUXKUUJLSUVFUWGUWRVFZWPZUURUXJUBUXESUUMUXEUKZUUOUXGUUQUXIUXN - UUNUXFUUIUUMUXEWQXBUXNUUMUXEUUPUXHJUXNXCUUMUXEUUIWRWSWTXAXDUXDUXGUWLUXI - UWNUXCUXGUWLUKUUJUXCUWLUVGUWGTZUUITZUXGUXCUWQUVGLRUWLUXPUKUWRKUVFLNXELS - UVGUUIUWGWEXFUXCUXOUXFUUIUXCUXFITZUVGUKZUXOUXFUKZUXCUXQUXEITZUQURUMZUVG - UXCUXKUXQUYAUKUXLBUXEKIMPXGXDUXCUXTUVFUQURUWCUXCUXTUVFUKUWFSLUVFIXHXLZX - IXJUXCUWCUXFSRZUXRUXSWHUWFUXCUXKUYCUXLUXEXKXDSLUXFUVGIWJXFYBXBXMWPUXDUX - IUXTUXHHUMZUWNUXDUXKUXHCRZUXIUYDUKUXMUXCUUJUXKUYEUXLSCUXEUUIXNVNGAUXEUX - HSCUVSUYDJUXTUVRHUMUVPUXEUKUVQUXTUVRHUVPUXEIWRXIUVRUXHUXTHXOQUXTUXHHXPY - CXQUXCUYDUWNUKUUJUXCUXTUVFUXHUWMHUYBUXCUWMUXHUWQUXCUWMUXHUKUWRLSUVFUUIU - WGWEXLXRWSWPXJWTXSXTYAYDUVMUWIUWKUWPUOEUWHUUIUWGUAYEIIBYNBUJUQURUMYFZKY - GZSYHZYNPUYGYISYNRUYHYNRKUYFYJYKUYGSYNYLWFYMYOYPUVBUWHUKZUVCUWIUVEUWKUV - LUWPLCUVBUWHYQUYIUVDUWJDKUVBUWHYRYSUYIUVKUWOFLUYIUVHUWLUVJUWNUVGUVBUWHY - RUYIUVIUWMUVFHUVFUVBUWHYRYTWTUUAUUBUUCUUDUUEXD $. + wex c1 caddc wf1o om2uzf1oi wb f1oeq3 ax-mp mpbir f1of ffvelcdmi fovcdm + cuz syl3an2 3expb ralrimivva fmpo sylib axdc4 sylan2 ccnv ccom mp2b fco + f1ocnv mpan2 3ad2ant1 uzid eleqtrri fvco3 mp2an om2uz0i peano1 f1ocnvfv + cz wi fveq2i eqtri simp2 eqtrid adantl suceq fveq2d fveq2 oveq12d rspcv + id eleq12d syl peano2uzs sylancr om2uzsuci f1ocnvfv2 mpan oveq1d peano2 + eqtrd eqtr2d ffvelcdm oveq2 ovex ovmpo syl2anc eqcomd impancom ralrimiv + mpd sylibd 3adant2 vex cvv cmpt crdg cres wfun rdgfun resfunexg eqeltri + omex cnvex coex feq1 fveq1 eqeq1d oveq2d ralbidv spcev syl3anc exlimiv + 3anbi123d ) DCRZLCUCCUDUEUFUGZHUHZUISCUAUJZUHZUEUUITZDUKZUBUJZULZUUITZU + UMUUMUUITZJUMZRZUBSUNZUOZUAUPZLCEUJZUHZKUVBTZDUKZFUJZUQURUMZUVBTZUVFUVF + UVBTZHUMZRZFLUNZUOZEUPZUUHUUFSCUCUUGJUHZUVAUUHGUJZITZAUJZHUMZUUGRZACUNG + SUNUVOUUHUVTGASCUUHUVPSRZUVRCRZUVTUWAUUHUVQLRUWBUVTSLUVPISLIUSZSLIUHUWC + SKVHTZIUSZBKIMPUTLUWDUKUWCUWEVANLUWDSIVBVCVDZSLIVEVCVFUVQUVRUUGLCHVGVIV + JVKGASCUVSUUGJQVLVMCDUAUBJOVNVOUUTUVNUAUUTLCUUIIVPZVQZUHZKUWHTZDUKZUVGU + WHTZUVFUVFUWHTZHUMZRZFLUNZUVNUUJUULUWIUUSUUJLSUWGUHZUWIUWCLSUWGUSUWQUWF + SLIVTLSUWGVEVRZLSCUUIUWGVSWAWBUUTUWJUUKDUWJKUWGTZUUITZUUKUWQKLRUWJUWTUK + UWRKUWDLKWJRKUWDRMKWCVCNWDLSKUUIUWGWEWFUWSUEUUIUEITKUKZUWSUEUKZBKIMPWGU + WCUESRUXAUXBWKUWFWHSLUEKIWIWFVCWLWMUUJUULUUSWNWOUUJUUSUWPUULUUJUUSUIUWO + FLUUJUVFLRZUUSUWOUUJUXCUIZUUSUVFUWGTZULZUUITZUXEUXEUUITZJUMZRZUWOUXDUXE + SRZUUSUXJWKUXCUXKUUJLSUVFUWGUWRVFZWPZUURUXJUBUXESUUMUXEUKZUUOUXGUUQUXIU + XNUUNUXFUUIUUMUXEWQWRUXNUUMUXEUUPUXHJUXNXBUUMUXEUUIWSWTXCXAXDUXDUXGUWLU + XIUWNUXCUXGUWLUKUUJUXCUWLUVGUWGTZUUITZUXGUXCUWQUVGLRUWLUXPUKUWRKUVFLNXE + LSUVGUUIUWGWEXFUXCUXOUXFUUIUXCUXFITZUVGUKZUXOUXFUKZUXCUXQUXEITZUQURUMZU + VGUXCUXKUXQUYAUKUXLBUXEKIMPXGXDUXCUXTUVFUQURUWCUXCUXTUVFUKUWFSLUVFIXHXI + ZXJXLUXCUWCUXFSRZUXRUXSWKUWFUXCUXKUYCUXLUXEXKXDSLUXFUVGIWIXFYBWRXMWPUXD + UXIUXTUXHHUMZUWNUXDUXKUXHCRZUXIUYDUKUXMUXCUUJUXKUYEUXLSCUXEUUIXNVOGAUXE + UXHSCUVSUYDJUXTUVRHUMUVPUXEUKUVQUXTUVRHUVPUXEIWSXJUVRUXHUXTHXOQUXTUXHHX + PXQXRUXCUYDUWNUKUUJUXCUXTUVFUXHUWMHUYBUXCUWMUXHUWQUXCUWMUXHUKUWRLSUVFUU + IUWGWEXIXSWTWPXLXCYCXTYAYDUVMUWIUWKUWPUOEUWHUUIUWGUAYEIIBYFBUJUQURUMYGZ + KYHZSYIZYFPUYGYJSYFRUYHYFRKUYFYKYNUYGSYFYLWFYMYOYPUVBUWHUKZUVCUWIUVEUWK + UVLUWPLCUVBUWHYQUYIUVDUWJDKUVBUWHYRYSUYIUVKUWOFLUYIUVHUWLUVJUWNUVGUVBUW + HYRUYIUVIUWMUVFHUVFUVBUWHYRYTXCUUAUUEUUBUUCUUDXD $. $} $( A version of ~ axdc4 that works on an upper set of integers instead of @@ -154456,13 +154457,6 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ ZGZCHBDEZPRSCRABQIZPSABQJPTSABQKLMNO $. $} - $( The codomain of an onto function is a set if its domain is a set. - (Contributed by AV, 4-May-2021.) $) - focdmex $p |- ( ( A e. V /\ F : A -onto-> B ) -> B e. _V ) $= - ( wcel wfo wa crn cvv wf fof anim2i ancomd rnexg 3syl wb forn eleq1d adantl - fex mpbid ) ADEZABCFZGZCHZIEZBIEZUDABCJZUBGCIEUFUDUBUHUCUHUBABCKLMABDCTCINO - UCUFUGPUBUCUEBIABCQRSUA $. - ${ $d A f g $. $d B f g $. $d V f $. $( The size of two sets is equal if there is a bijection mapping one of the @@ -154480,8 +154474,8 @@ notation is used by Donald Knuth for iterated exponentiation (_Science_ XBXCWGBAWFUAZIZXBXCWFUHZWGXEUIWGXFWEABWFUJQABWFUKULXCXEWTXBXCWFRGZXDRGXEW TNWGABWFVEWEXGABWFUMABDWFUNUOWFRUPWSXEEXDRBAWRXDUQURUSWTWKUTVAVBVCVDVFVGS VHULVIWQWOOZWEWLXHWEOZWGWKCXIWGWKXIWGOZWITWJXIWITMZWGXHWEXKWQWEXKNWOWEWQX - KADVJVKQVLQXJBRGZWJTMZXIWEABWFVMXLWGXHWEVNABWFVOABWFDVPVQWOXLXMNWQWEWGXLW - OXMBRVJVKVRVSVTSWASWB $. + KADVJVKQVLQXJBRGZWJTMZXIWEABWFVMZXLWGXHWEVNABWFVOWEXNXLABDWFVPVLVQWOXLXMN + WQWEWGXLWOXMBRVJVKVRVSVTSWASWB $. $} ${ @@ -170596,8 +170590,8 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= $( Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.) $) rlimcn2 $p |- ( ph -> ( z e. A |-> ( B F C ) ) ~~>r ( R F S ) ) $= - ( cv wcel wa cc cxp wf adantr fovrnd rlimcn3 ) ABCDEFGHIJKLMNOPQACUDFUEZU - FGHUGLMKALMUHUGKUIUMUBUJPQUKAIJUGLMKUBRSUKTUAUCUL $. + ( cv wcel wa cc cxp wf adantr fovcdmd rlimcn3 ) ABCDEFGHIJKLMNOPQACUDFUEZ + UFGHUGLMKALMUHUGKUIUMUBUJPQUKAIJUGLMKUBRSUKTUAUCUL $. $} ${ @@ -170936,8 +170930,8 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= $( Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.) $) rlimmptrcl $p |- ( ( ph /\ k e. A ) -> B e. CC ) $= - ( cc cmpt cdm wf crli wbr rlimf syl eqid dmmptd feq2d mpbid fvmptelrn ) A - EBCIAEBCJZKZIUBLZBIUBLAUBDMNUDHDUBOPAUCBIUBAEUBBCFUBQGRSTUA $. + ( cc cmpt cdm wf crli wbr rlimf syl eqid dmmptd feq2d mpbid fvmptelcdm ) + AEBCIAEBCJZKZIUBLZBIUBLAUBDMNUDHDUBOPAUCBIUBAEUBBCFUBQGRSTUA $. $( Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.) $) @@ -171148,8 +171142,8 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= (Contributed by Mario Carneiro, 26-May-2016.) $) lo1mptrcl $p |- ( ( ph /\ x e. A ) -> B e. RR ) $= ( cr cmpt cdm wf clo1 wcel lo1f syl wral wceq ralrimiva dmmptg feq2d - mpbid fvmptelrn ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEGUCNOAUDCHUCADEMZBCPUDC - QAUFBCFRBCDESOTUAUB $. + mpbid fvmptelcdm ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEGUCNOAUDCHUCADEMZBCPUD + CQAUFBCFRBCDESOTUAUB $. $} ${ @@ -171158,8 +171152,8 @@ seq M ( + , ( ( ZZ>= ` M ) X. { 0 } ) ) ~~> 0 ) $= Mario Carneiro, 26-May-2016.) $) o1mptrcl $p |- ( ( ph /\ x e. A ) -> B e. CC ) $= ( cc cmpt cdm wf co1 wcel o1f syl wral wceq ralrimiva dmmptg feq2d - mpbid fvmptelrn ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEGUCNOAUDCHUCADEMZBCPUDC - QAUFBCFRBCDESOTUAUB $. + mpbid fvmptelcdm ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEGUCNOAUDCHUCADEMZBCPUD + CQAUFBCFRBCDESOTUAUB $. $} o1add2.2 $e |- ( ( ph /\ x e. A ) -> C e. V ) $. @@ -205808,15 +205802,15 @@ topology is based on the order and not the extended metric (which would csn cmulr ctp csca cvsca cmpo cip cun cts ctopn cqtop cple ccom ccnv c1st cds cn wceq c2nd caddc cmin cfz wral w3a cxp cmap crab cxrs cgsu cmpt crn cxr clt cinf cvv c2 cdc imasval imasvalstr baseid snsstp1 ssun1 sstri wfo - wcel fvex eqeltrdi fornex sylc strfv3 eqcomd ) ADUILZBAXBBMUILBNZMUJLUAFU - BFUAOZELUBOZELZNZXDXECUJLZPELNUKQQZNZMULLUAFUBFXGXDXECULLZPELNUKQQZNZUMZM - UNLCUNLZNMUOLUBFUAUCXOUILZXFUKXDXECUOLZPELUPQZNMUQLUAFUBFXGXDXECUQLZPNUKQ - QZNUMZURZMUSLCUTLZEVAPZNMVBLECVBLZVCEVDVCZNMVFLUCUDBBUEVGUFRUGOZLVELELUCO - VHUEOZYGLVILELUDOVHUHOZYGLVILELYIRVJPYGLVELELVHUHRYHRVKPVLPVMVNUGFFVORYHV - LPVPPVQVRCVFLZUFOVCVSPVTWAQWBWCWDUPZNUMZURZDUIWERRWFWGNAUCUDBYKXHXICXLXQX - KXRDUFUGUHUEYJEXOXSXTYCXPYFYEYDFGUBUAHIXHSXKSXOSXPSXQSXSSYCSYJSYESAXITAXL - TAXRTAXTTAYDTAYKTAYFTJKWHBYKXIXOXRXLYMXTYFYDYMSWIWJXCUKZYBYMYNXNYBXCXJXMW - KXNYAWLWMYBYLWLWMAFWEWOFBEWNBWEWOAFCUILWEICUIWPWQJFBWEEWRWSXBSWTXA $. + wcel fvex eqeltrdi focdmex sylc strfv3 eqcomd ) ADUILZBAXBBMUILBNZMUJLUAF + UBFUAOZELUBOZELZNZXDXECUJLZPELNUKQQZNZMULLUAFUBFXGXDXECULLZPELNUKQQZNZUMZ + MUNLCUNLZNMUOLUBFUAUCXOUILZXFUKXDXECUOLZPELUPQZNMUQLUAFUBFXGXDXECUQLZPNUK + QQZNUMZURZMUSLCUTLZEVAPZNMVBLECVBLZVCEVDVCZNMVFLUCUDBBUEVGUFRUGOZLVELELUC + OVHUEOZYGLVILELUDOVHUHOZYGLVILELYIRVJPYGLVELELVHUHRYHRVKPVLPVMVNUGFFVORYH + VLPVPPVQVRCVFLZUFOVCVSPVTWAQWBWCWDUPZNUMZURZDUIWERRWFWGNAUCUDBYKXHXICXLXQ + XKXRDUFUGUHUEYJEXOXSXTYCXPYFYEYDFGUBUAHIXHSXKSXOSXPSXQSXSSYCSYJSYESAXITAX + LTAXRTAXTTAYDTAYKTAYFTJKWHBYKXIXOXRXLYMXTYFYDYMSWIWJXCUKZYBYMYNXNYBXCXJXM + WKXNYAWLWMYBYLWLWMAFWEWOFBEWNBWEWOAFCUILWEICUIWPWQJFBWEEWRWSXBSWTXA $. ${ imasds.e $e |- E = ( dist ` R ) $. @@ -205834,18 +205828,18 @@ topology is based on the order and not the extended metric (which would ( vp vq cn c1 cv cfv c1st wceq c2nd caddc co cmin cfz wral w3a cxp cmap crab cxrs ccom cgsu cmpt crn ciun cxr clt cinf cnx cbs cop cplusg cmulr cmpo csn ctp csca cvsca cip cun cts ctopn cple ccnv cds cvv c2 cdc eqid - cqtop eqidd imasval imasvalstr dsid snsstp3 ssun2 sstri wcel wfo fornex - fvex eqeltrdi sylc mpoexga syl2anc strfv3 ) AEBCDDKUDHUEIUFZUGUHUGMUGBU - FUIKUFZXGUGUJUGMUGCUFUIJUFZXGUGUJUGMUGXIUEUKULXGUGUHUGMUGUIJUEXHUEUMULU - NULUOUPINNUQUEXHUNULURULUSUTLHUFVAVBULVCVDVEVFVGVHZVNZVIVJUGDVKVIVLUGUB - NUCNUBUFZMUGUCUFZMUGZVKZXLXMFVLUGZULMUGVKVOVEVEZVKVIVMUGUBNUCNXOXLXMFVM - UGZULMUGVKVOVEVEZVKVPVIVQUGFVQUGZVKVIVRUGUCNUBBXTVJUGZXNVOXLXMFVRUGZULM - UGVNVEZVKVIVSUGUBNUCNXOXLXMFVSUGZULVKVOVEVEZVKVPVTZVIWAUGFWBUGZMWJULZVK - ZVIWCUGMFWCUGZVAMWDVAZVKZVIWEUGXKVKZVPZVTZGWEWFUEUEWGWHVKABCDXKXPXQFXSY - BXRYCGHIJKLMXTYDYEYGYAYKYJYHNOUCUBPQXPWIXRWIXTWIYAWIYBWIYDWIYGWITYJWIAX - QWKAXSWKAYCWKAYEWKAYHWKAXKWKAYKWKRSWLDXKXQXTYCXSYOYEYKYHYOWIWMWNYMVOYNY - OYIYLYMWOYNYFWPWQADWFWRZYPXKWFWRANWFWRNDMWSYPANFVJUGWFQFVJXAXBRNDWFMWTX - CZYQBCDDXJWFWFXDXEUAXF $. + cqtop eqidd imasval imasvalstr dsid snsstp3 ssun2 wcel wfo fvex focdmex + sstri eqeltrdi sylc mpoexga syl2anc strfv3 ) AEBCDDKUDHUEIUFZUGUHUGMUGB + UFUIKUFZXGUGUJUGMUGCUFUIJUFZXGUGUJUGMUGXIUEUKULXGUGUHUGMUGUIJUEXHUEUMUL + UNULUOUPINNUQUEXHUNULURULUSUTLHUFVAVBULVCVDVEVFVGVHZVNZVIVJUGDVKVIVLUGU + BNUCNUBUFZMUGUCUFZMUGZVKZXLXMFVLUGZULMUGVKVOVEVEZVKVIVMUGUBNUCNXOXLXMFV + MUGZULMUGVKVOVEVEZVKVPVIVQUGFVQUGZVKVIVRUGUCNUBBXTVJUGZXNVOXLXMFVRUGZUL + MUGVNVEZVKVIVSUGUBNUCNXOXLXMFVSUGZULVKVOVEVEZVKVPVTZVIWAUGFWBUGZMWJULZV + KZVIWCUGMFWCUGZVAMWDVAZVKZVIWEUGXKVKZVPZVTZGWEWFUEUEWGWHVKABCDXKXPXQFXS + YBXRYCGHIJKLMXTYDYEYGYAYKYJYHNOUCUBPQXPWIXRWIXTWIYAWIYBWIYDWIYGWITYJWIA + XQWKAXSWKAYCWKAYEWKAYHWKAXKWKAYKWKRSWLDXKXQXTYCXSYOYEYKYHYOWIWMWNYMVOYN + YOYIYLYMWOYNYFWPXAADWFWQZYPXKWFWQANWFWQNDMWRYPANFVJUGWFQFVJWSXBRNDWFMWT + XCZYQBCDDXJWFWFXDXEUAXF $. $( The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.) $) @@ -214518,10 +214512,10 @@ is always a subcategory (and it is full, meaning that all morphisms of 11-Jan-2017.) $) homarel $p |- Rel ( X H Y ) $= ( vf co wrel cvv cxp wss cv wcel cbs cfv xpss cpw eqid homarcl homaf - homarcl2 simpld simprd fovrnd elelpwi mpdan sselid ssriv df-rel mpbir ) C - DBGZHUKIIJZKFUKULFLZUKMZANOZUOJZIJZULUMUPIPUNUKUQQZMUMUQMUNCDURUOUOBUNUOA - BEUORZAUMBCDESTUNCUOMZDUOMZUOAUMBCDEUSUAZUBUNUTVAVBUCUDUMUKUQUEUFUGUHUKUI - UJ $. + homarcl2 simpld simprd fovcdmd elelpwi mpdan sselid ssriv df-rel mpbir ) + CDBGZHUKIIJZKFUKULFLZUKMZANOZUOJZIJZULUMUPIPUNUKUQQZMUMUQMUNCDURUOUOBUNUO + ABEUORZAUMBCDESTUNCUOMZDUOMZUOAUMBCDEUSUAZUBUNUTVAVBUCUDUMUKUQUEUFUGUHUKU + IUJ $. $( The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) $) @@ -218266,60 +218260,60 @@ that F(A') is isomorphic to B'.". Therefore, the category of sets and ( cfv co vy vz vg vw vh c1st c2nd cfunc cv cmpt chom ccid cmpo eqid curf1 cop wceq cbs fvexi mptex mpoex op1std syl op2ndd opeq12d eqtr4d wcel cxpc wbr cco ccat wa funcrcl simprd cxp wf xpcbas wrel relfunc 1st2ndbr funcf1 - sylancr adantr simpr fovrnd fmpt3d wfn fnmpoi fneq1d mpbiri oveqd ovmpt4g + sylancr adantr simpr fovcdmd fmpt3d wfn ovex fnmpoi fneq1d mpbiri ovmpt4g cvv mp3an3 sylan9eq ad2antrr simplrl opelxpd simplrr funcf2 xpchom curf11 - df-ov eqtr2di oveq12d feq23d mpbid catidcl op1stg syl2anc eleqtrrd op2ndg - ovex xpcid fveq2d eqtr4di opelxpi funcid eqtr3d curf12 eqtrdi 3eqtr4d w3a - 3ad2ant1 simp21 simp22 simp23 simp3l simp3r xpcco2 catlid xpchom2 catcocl - sylan opeq1d eqtrd funcco oveq123d isfuncd df-br sylib eqeltrd ) AIIUFSZI - UGSZUPZEFUHTZAIUACJUAUIZGUFSZTZUJZUAUBCCUCUUGUBUIZEUKSZTZJDULSZSZUCUIZJUU - GUPZJUUKUPZGUGSZTZTZUJZUMZUPZUUEAUAUBBCDEUUNUCFGHUULIJKLMNOPQRUULUNZUUNUN - ZUOZAUUCUUJUUDUVCAIUVDUQZUUCUUJUQUVGUUJUVCIUACUUICEURPUSZUTZUAUBCCUVBUVIU - 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Therefore, the category of sets and L e. 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Therefore, the category of sets and ( K ( <. X , Y >. ( 2nd ` M ) <. Z , W >. ) L ) ) ) $= ( vf vg co cop cco cfv cv cmpt c2nd wcel eqid ccat adantr simpr catcocl catass oveq1d eqtrd eqtr3d mpteq2dva hof2val ccom oveq12d chomf homfval - wa cxp wfn crn wss wf homffn a1i sylanbrc fovrnd eqeltrrd fmpttd setcco - df-f eqidd wceq oveq2 fmptco 3eqtrd 3eqtr4d ) AURQRJUTZFLRPVAHCVBVCZUTU - TZURVDZQRVAZHXDUTUTZKEGSVAZQXDUTUTZGQVAHXDUTUTZVEURXCFLXFXGPXDUTUTZKSQV - AZPXDUTUTZSPVAZHXDUTZUTZEXIHXDUTZUTZVEZXJXEXGGHVAZMVFVCZUTUTEFXOYAYBUTU - TZKLXGXOYBUTUTZXCSPJUTZVAGHJUTZDVBVCZUTZUTZAURXCXKXSAXFXCVGZWCZXHKXMHXD - UTZUTZEXRUTXKXSYKBCXDEKJXHHGSQUFUGXDVHZACVIVGZYJUCVJZAGBVGZYJULVJASBVGZ - YJUJVJZAQBVGYJUHVJZAEGSJUTVGZYJUPVJAKSQJUTVGYJUNVJZAHBVGZYJUMVJZYKBCXDX - FXEJQRHUFUGYNYPYTARBVGYJUIVJZUUDAYJVKZAXERHJUTVGYJABCXDLFJRPHUFUGYNUCUI - UKUMUOUQVLZVJVLVMYKYMXQEXRYKYMFXLQPVAHXDUTUTZKYLUTXQYKXHUUHKYLYKBCXDXFL - JFHQRPUFUGYNYPYTUUEAPBVGZYJUKVJZUUFALRPJUTVGYJUOVJZUUDAFPHJUTVGZYJUQVJZ - VMVNYKBCXDKXLJFHSQPUFUGYNYPYSYTUUJUUBYKBCXDXFLJQRPUFUGYNYPYTUUEUUJUUFUU - KVLZUUDUUMVMVOVNVPVQABCXDURXJXEJMHQRGTUCUFUGUHUIULUMYNABCXDEKJGSQUFUGYN - UCULUJUHUPUNVLUUGVRAYIUSYEFUSVDZXPUTZEXRUTZVEZURXCXNVEZYHUTUURUUSVSXTAY - CUURYDUUSYHABCXDUSEFJMHSPGTUCUFUGUJUKULUMYNUPUQVRABCXDURKLJMPQRSTUCUFUG - UHUIUJUKYNUNUOVRVTADYGIUUSUUROXCYEYFUBUDYGVHAQRCWAVCZUTXCIABCUUTJQRUUTV - HZUFUGUHUIWBAQRIBBUUTAUUTBBWDZWEZUUTWFIWGUVBIUUTWHUVCABCUUTUVAUFWIWJUEU - VBIUUTWPWKZUHUIWLWMASPUUTUTYEIABCUUTJSPUVAUFUGUJUKWBASPIBBUUTUVDUJUKWLW - MAGHUUTUTYFIABCUUTJGHUVAUFUGULUMWBAGHIBBUUTUVDULUMWLWMAURXCXNYEYKBCXDKX - LJSQPUFUGYNYPYSYTUUJUUBUUNVLZWNAUSYEUUQYFAUUOYEVGZWCZBCXDEUUPJGSHUFUGYN - AYOUVFUCVJZAYQUVFULVJAYRUVFUJVJZAUUCUVFUMVJZAUUAUVFUPVJUVGBCXDUUOFJSPHU - FUGYNUVHUVIAUUIUVFUKVJUVJAUVFVKAUULUVFUQVJVLVLWNWOAURUSXCYEXNUUQXSUUSUU - RUVEAUUSWQAUURWQUUOXNWRUUPXQEXRUUOXNFXPWSVNWTXAXB $. + cxp wfn crn wss homffn a1i df-f sylanbrc fovcdmd eqeltrrd fmpttd setcco + wa wf eqidd wceq oveq2 fmptco 3eqtrd 3eqtr4d ) AURQRJUTZFLRPVAHCVBVCZUT + UTZURVDZQRVAZHXDUTUTZKEGSVAZQXDUTUTZGQVAHXDUTUTZVEURXCFLXFXGPXDUTUTZKSQ + VAZPXDUTUTZSPVAZHXDUTZUTZEXIHXDUTZUTZVEZXJXEXGGHVAZMVFVCZUTUTEFXOYAYBUT + UTZKLXGXOYBUTUTZXCSPJUTZVAGHJUTZDVBVCZUTZUTZAURXCXKXSAXFXCVGZWOZXHKXMHX + DUTZUTZEXRUTXKXSYKBCXDEKJXHHGSQUFUGXDVHZACVIVGZYJUCVJZAGBVGZYJULVJASBVG + ZYJUJVJZAQBVGYJUHVJZAEGSJUTVGZYJUPVJAKSQJUTVGYJUNVJZAHBVGZYJUMVJZYKBCXD + XFXEJQRHUFUGYNYPYTARBVGYJUIVJZUUDAYJVKZAXERHJUTVGYJABCXDLFJRPHUFUGYNUCU + IUKUMUOUQVLZVJVLVMYKYMXQEXRYKYMFXLQPVAHXDUTUTZKYLUTXQYKXHUUHKYLYKBCXDXF + LJFHQRPUFUGYNYPYTUUEAPBVGZYJUKVJZUUFALRPJUTVGYJUOVJZUUDAFPHJUTVGZYJUQVJ + ZVMVNYKBCXDKXLJFHSQPUFUGYNYPYSYTUUJUUBYKBCXDXFLJQRPUFUGYNYPYTUUEUUJUUFU + UKVLZUUDUUMVMVOVNVPVQABCXDURXJXEJMHQRGTUCUFUGUHUIULUMYNABCXDEKJGSQUFUGY + NUCULUJUHUPUNVLUUGVRAYIUSYEFUSVDZXPUTZEXRUTZVEZURXCXNVEZYHUTUURUUSVSXTA + YCUURYDUUSYHABCXDUSEFJMHSPGTUCUFUGUJUKULUMYNUPUQVRABCXDURKLJMPQRSTUCUFU + GUHUIUJUKYNUNUOVRVTADYGIUUSUUROXCYEYFUBUDYGVHAQRCWAVCZUTXCIABCUUTJQRUUT + VHZUFUGUHUIWBAQRIBBUUTAUUTBBWCZWDZUUTWEIWFUVBIUUTWPUVCABCUUTUVAUFWGWHUE + UVBIUUTWIWJZUHUIWKWLASPUUTUTYEIABCUUTJSPUVAUFUGUJUKWBASPIBBUUTUVDUJUKWK + WLAGHUUTUTYFIABCUUTJGHUVAUFUGULUMWBAGHIBBUUTUVDULUMWKWLAURXCXNYEYKBCXDK + XLJSQPUFUGYNYPYSYTUUJUUBUUNVLZWMAUSYEUUQYFAUUOYEVGZWOZBCXDEUUPJGSHUFUGY + NAYOUVFUCVJZAYQUVFULVJAYRUVFUJVJZAUUCUVFUMVJZAUUAUVFUPVJUVGBCXDUUOFJSPH + UFUGYNUVHUVIAUUIUVFUKVJUVJAUVFVKAUULUVFUQVJVLVLWMWNAURUSXCYEXNUUQXSUUSU + URUVEAUUSWQAUURWQUUOXNWRUUPXQEXRUUOXNFXPWSVNWTXAXB $. $} $( Closure of the Hom functor. Note that the codomain is the category @@ -219748,94 +219742,94 @@ that F(A') is isomorphic to B'.". Therefore, the category of sets and 1st2ndbr ffvelcdmd adantr fuccoval wf cbs funcf1 setcbas feq3d eleqtrrd mpbird nat1st2nd natcl elsetchom mpbid setcco eqtrd syl2anc yon11 fvco3 3eqtrd catidcl ccat fveq2d 3eqtr3d wral cxp simpld opelxpd df-ov eqcomi - oveq12i a1i feq23d fovrnd yonedalem21 feq123d sylibr yonedalem3a simprd - syl fmpt fmptcof 3eqtr4d oppcbas fuchom chomf ssexd yoncl simpr fuccocl - crn unssbd fucbas fco yon2 catrid oppchom eleqtrrdi nati catlid 3eqtr2d - yon12 mpteq2dva eqtrid cxpc xpcbas yonedalem1 xpeq2i eqtrdi yonedalem22 - xpchom2 oppccat setccat fuccat hof2val fveq1 coeq1d evlf1 fcompt 2fveq3 - evlf2val ) APFSVKZCROUCVLZPFVLZUEVMVNZVKZVKZVOZCRUVTUWANVMVNZVKZVKZOUCS - VKZVOZUVSUWDOUCUEVPVNZVKZPFUWKVKZVLPFNVPVNZVKZJWDVNZVKVKUWHUWIUWLOUCUWN - VKZVLUWOUWPVKVKAVIUCUDVPVNZVNZOTIVQVKZVKZFLVNZFCVIVRZUWSOVLZPGWDVNZVKZV - KZRFUCUDVMVNZVKZVNZFUWRVNZUWSVLPUXEVKZVKZVNZVNZVSZUFUXAUCLVNZUCUFVRZVNZ - VNZRUCFOVMVNZVKZVNZVNZFCVNZVNZVSZUWEUWJAUXPUFUXAUXBFCUXRUXFVKZUXJUXLVKZ - 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( O Func S ) , x e. B |-> @@ -219846,48 +219840,48 @@ that F(A') is isomorphic to B'.". 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for ~ yonffth . (Contributed by Mario Carneiro, 29-Jan-2017.) $) @@ -226242,35 +226236,35 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is Carneiro, 24-Feb-2015.) $) imasmnd2 $p |- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) ) $= ( vu vv vw cmnd wcel cfv c0g wceq cplusg imasbas eqidd wf cv eqid 3expb - cxp co caovclg imasaddf fovrn syl3an1 w3a wrex crn wfo eleq2d 3anbi123d - forn syl wfn wb fofn fvelrnb bitr3d 3reeanv bitr4di wa 3adant3r3 simpr3 - simpl imasaddval syl3anc simpr1 3adantr1 3eqtr4d oveq1d 3adant3r1 simp1 - wi oveq2d simp2 oveq12d simp3 syl5ibcom 3exp2 imp32 rexlimdv rexlimdvva - eqeq12d sylbid imp ffvelcdmd adantr simpr eqtrd oveq2 rexlimdva mpd3an3 - fof id oveq1 ismndd grpidd jca ) AHUKULLIUMZHUNUMUOAUHUIUJEHUPUMZHYBAEG - HIJKQRTUBUQZAYCURZAEEVCEYCUSUHUTZEULZUIUTZEULZYFYHYCVDZEULAEGYCFHIJKMNO - PTUAQRUBSYCVAZABCNUTMUTJJJFABUTZJULZCUTZJULZYLYNFVDZJULZUCVBVEZVFYFYHEE - EYCVGVHAYGYIUJUTZEULZVIZYJYSYCVDZYFYHYSYCVDZYCVDZUOZAUUAYLIUMZYFUOZYNIU - MZYHUOZDUTZIUMZYSUOZVIZDJVJZCJVJBJVJZUUEAUUAUUGBJVJZUUICJVJZUULDJVJZVIZ - UUOAYFIVKZULZYHUUTULZYSUUTULZVIZUUAUUSAUVAYGUVBYIUVCYTAUUTEYFAJEIVLZUUT - EUOTJEIVOVPZVMZAUUTEYHUVFVMAUUTEYSUVFVMVNAIJVQZUVDUUSVRAUVEUVHTJEIVSVPZ - UVHUVAUUPUVBUUQUVCUURBJYFIVTZCJYHIVTDJYSIVTVNVPWAUUGUUIUULBCDJJJWBWCAUU - NUUEBCJJAYMYOWDWDUUMUUEDJAYMYOUUJJULZUUMUUEWPZWPAYMYOUVKUVLAYMYOUVKVIZW - DZUUFUUHYCVDZUUKYCVDZUUFUUHUUKYCVDZYCVDZUOUUMUUEUVNYPIUMZUUKYCVDZUUFYNU - UJFVDZIUMZYCVDZUVPUVRUVNYPUUJFVDIUMZYLUWAFVDIUMZUVTUWCUDUVNAYQUVKUVTUWD - UOAUVMWGZAYMYOYQUVKUCWEAYMYOUVKWFAEGYCFHIJYPUUJKMNOPTUAQRUBSYKWHWIUVNAY - MUWAJULZUWCUWEUOUWFAYMYOUVKWJAYOUVKUWGYMANMYNUUJJJJFYRVEWKAEGYCFHIJYLUW - AKMNOPTUAQRUBSYKWHWIWLUVNUVOUVSUUKYCAYMYOUVOUVSUOUVKAEGYCFHIJYLYNKMNOPT - UAQRUBSYKWHWEWMUVNUVQUWBUUFYCAYOUVKUVQUWBUOYMAEGYCFHIJYNUUJKMNOPTUAQRUB - SYKWHWNWQWLUUMUVPUUBUVRUUDUUMUVOYJUUKYSYCUUMUUFYFUUHYHYCUUGUUIUULWOZUUG - UUIUULWRZWSUUGUUIUULWTZWSUUMUUFYFUVQUUCYCUWHUUMUUHYHUUKYSYCUWIUWJWSWSXF - XAXBXCXDXEXGXHAJELIAUVEJEIUSTJEIXPVPUEXIZAYGYBYFYCVDZYFUOZAYGUUPUWMAUVA - YGUUPUVGAUVHUVAUUPVRUVIUVJVPWAZAUUGUWMBJAYMWDZYBUUFYCVDZUUFUOUUGUWMUWOU - WPLYLFVDIUMZUUFUWOALJULZYMUWPUWQUOAYMWGAUWRYMUEXJZAYMXKAEGYCFHIJLYLKMNO - PTUAQRUBSYKWHWIUFXLUUGUWPUWLUUFYFUUFYFYBYCXMUUGXQZXFXAXNXGXHZAYGYFYBYCV - DZYFUOZAYGUUPUXCUWNAUUGUXCBJUWOUUFYBYCVDZUUFUOUUGUXCUWOUXDYLLFVDIUMZUUF - AYMUWRUXDUXEUOUWSAEGYCFHIJYLLKMNOPTUAQRUBSYKWHXOUGXLUUGUXDUXBUUFYFUUFYF - YBYCXRUWTXFXAXNXGXHZXSAUHEYCHYBYDYEUWKUXAUXFXTYA $. + cxp co caovclg imasaddf fovcdm syl3an1 w3a wrex crn wfo forn syl eleq2d + 3anbi123d wfn wb fofn fvelrnb bitr3d 3reeanv bitr4di wa simpl 3adant3r3 + simpr3 imasaddval syl3anc simpr1 3adantr1 oveq1d 3adant3r1 oveq2d simp1 + wi 3eqtr4d simp2 oveq12d simp3 eqeq12d syl5ibcom 3exp2 imp32 rexlimdvva + rexlimdv sylbid imp fof ffvelcdmd adantr simpr eqtrd oveq2 id rexlimdva + mpd3an3 oveq1 ismndd grpidd jca ) AHUKULLIUMZHUNUMUOAUHUIUJEHUPUMZHYBAE + GHIJKQRTUBUQZAYCURZAEEVCEYCUSUHUTZEULZUIUTZEULZYFYHYCVDZEULAEGYCFHIJKMN + OPTUAQRUBSYCVAZABCNUTMUTJJJFABUTZJULZCUTZJULZYLYNFVDZJULZUCVBVEZVFYFYHE + EEYCVGVHAYGYIUJUTZEULZVIZYJYSYCVDZYFYHYSYCVDZYCVDZUOZAUUAYLIUMZYFUOZYNI + UMZYHUOZDUTZIUMZYSUOZVIZDJVJZCJVJBJVJZUUEAUUAUUGBJVJZUUICJVJZUULDJVJZVI + ZUUOAYFIVKZULZYHUUTULZYSUUTULZVIZUUAUUSAUVAYGUVBYIUVCYTAUUTEYFAJEIVLZUU + TEUOTJEIVMVNZVOZAUUTEYHUVFVOAUUTEYSUVFVOVPAIJVQZUVDUUSVRAUVEUVHTJEIVSVN + ZUVHUVAUUPUVBUUQUVCUURBJYFIVTZCJYHIVTDJYSIVTVPVNWAUUGUUIUULBCDJJJWBWCAU + UNUUEBCJJAYMYOWDWDUUMUUEDJAYMYOUUJJULZUUMUUEWPZWPAYMYOUVKUVLAYMYOUVKVIZ + WDZUUFUUHYCVDZUUKYCVDZUUFUUHUUKYCVDZYCVDZUOUUMUUEUVNYPIUMZUUKYCVDZUUFYN + UUJFVDZIUMZYCVDZUVPUVRUVNYPUUJFVDIUMZYLUWAFVDIUMZUVTUWCUDUVNAYQUVKUVTUW + DUOAUVMWEZAYMYOYQUVKUCWFAYMYOUVKWGAEGYCFHIJYPUUJKMNOPTUAQRUBSYKWHWIUVNA + YMUWAJULZUWCUWEUOUWFAYMYOUVKWJAYOUVKUWGYMANMYNUUJJJJFYRVEWKAEGYCFHIJYLU + WAKMNOPTUAQRUBSYKWHWIWQUVNUVOUVSUUKYCAYMYOUVOUVSUOUVKAEGYCFHIJYLYNKMNOP + TUAQRUBSYKWHWFWLUVNUVQUWBUUFYCAYOUVKUVQUWBUOYMAEGYCFHIJYNUUJKMNOPTUAQRU + BSYKWHWMWNWQUUMUVPUUBUVRUUDUUMUVOYJUUKYSYCUUMUUFYFUUHYHYCUUGUUIUULWOZUU + GUUIUULWRZWSUUGUUIUULWTZWSUUMUUFYFUVQUUCYCUWHUUMUUHYHUUKYSYCUWIUWJWSWSX + AXBXCXDXFXEXGXHAJELIAUVEJEIUSTJEIXIVNUEXJZAYGYBYFYCVDZYFUOZAYGUUPUWMAUV + AYGUUPUVGAUVHUVAUUPVRUVIUVJVNWAZAUUGUWMBJAYMWDZYBUUFYCVDZUUFUOUUGUWMUWO + UWPLYLFVDIUMZUUFUWOALJULZYMUWPUWQUOAYMWEAUWRYMUEXKZAYMXLAEGYCFHIJLYLKMN + OPTUAQRUBSYKWHWIUFXMUUGUWPUWLUUFYFUUFYFYBYCXNUUGXOZXAXBXPXGXHZAYGYFYBYC + VDZYFUOZAYGUUPUXCUWNAUUGUXCBJUWOUUFYBYCVDZUUFUOUUGUXCUWOUXDYLLFVDIUMZUU + FAYMUWRUXDUXEUOUWSAEGYCFHIJYLLKMNOPTUAQRUBSYKWHXQUGXMUUGUXDUXBUUFYFUUFY + FYBYCXRUWTXAXBXPXGXHZXSAUHEYCHYBYDYEUWKUXAUXFXTYA $. $} ${ @@ -229770,8 +229764,8 @@ with the same (relevant) properties is also a group. (Contributed by $( Closure of group subtraction. (Contributed by NM, 31-Mar-2014.) $) grpsubcl $p |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) e. B ) $= - ( cgrp wcel cxp wf co grpsubf fovrn syl3an1 ) BHIAAJACKDAIEAIDECLAIABCFGM - DEAAACNO $. + ( cgrp wcel cxp wf co grpsubf fovcdm syl3an1 ) BHIAAJACKDAIEAIDECLAIABCFG + MDEAAACNO $. $( Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.) $) @@ -230303,47 +230297,47 @@ with the same (relevant) properties is also a group. (Contributed by imasgrp2 $p |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) ) $= ( vu vv vw cgrp wcel cfv c0g wceq cplusg imasbas eqidd cxp wf cv co w3a wa wb oveqd fveq2d eqeq12d 3ad2ant1 sylibd eqid adantr caovclg eqeltrrd - 3expb imasaddf fovrn syl3an1 wrex crn wfo forn syl eleq2d 3anbi123d wfn - fofn fvelrnb bitr3d 3reeanv bitr4di 3eqtr3d 3adant3r3 simpr3 imasaddval - wi simpl syl3anc simpr1 3adantr1 3eqtr4d eqtr4d oveq1d 3adant3r1 oveq2d - simp1 simp2 oveq12d simp3 syl5ibcom 3exp2 imp32 rexlimdv rexlimdvva imp - sylbid fof ffvelcdmd simpr 3eqtr2d oveq2 rexlimdva oveq1 eqeq1d syl2anc - id rspcev rexbidv isgrpde grpidd2 jca ) AHUMUNMIUOZHUPUOUQAUJUKULEHURUO - ZHYNAEGHIKLRSUAUCUSZAYOUTZAEEVAEYOVBUJVCZEUNZUKVCZEUNZYRYTYOVDZEUNAEGYO - GURUOZHIKLNOPQUAAPVCZKUNQVCZKUNVFZOVCZKUNNVCZKUNVFZVEUUDIUOUUGIUOUQUUEI - UOUUHIUOUQVFUUDUUEFVDZIUOZUUGUUHFVDZIUOZUQZUUDUUEUUCVDZIUOZUUGUUHUUCVDZ - IUOZUQZUBAUUFUUNUUSVGUUIAUUKUUPUUMUURAUUJUUOIAFUUCUUDUUETVHVIAUULUUQIAF - UUCUUGUUHTVHZVIVJVKVLZRSUCUUCVMZYOVMZAUUIVFUULUUQKAUULUUQUQUUIUUTVNABCU - UGUUHKKKFABVCZKUNZCVCZKUNZUVDUVFFVDZKUNZUDVQVOZVPVRYRYTEEEYOVSVTAYSUUAU - 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YNYPYQUYOVUHVVAYLYM $. + imasaddf fovcdm syl3an1 wrex crn wfo forn syl eleq2d 3anbi123d wfn fofn + 3expb fvelrnb bitr3d 3reeanv bitr4di 3eqtr3d simpl 3adant3r3 imasaddval + wi simpr3 syl3anc simpr1 3adantr1 3eqtr4d eqtr4d oveq1d 3adant3r1 simp1 + oveq2d simp2 simp3 syl5ibcom 3exp2 imp32 rexlimdv rexlimdvva sylbid imp + oveq12d fof ffvelcdmd simpr 3eqtr2d oveq2 rexlimdva oveq1 eqeq1d rspcev + id syl2anc rexbidv isgrpde grpidd2 jca ) AHUMUNMIUOZHUPUOUQAUJUKULEHURU + OZHYNAEGHIKLRSUAUCUSZAYOUTZAEEVAEYOVBUJVCZEUNZUKVCZEUNZYRYTYOVDZEUNAEGY + OGURUOZHIKLNOPQUAAPVCZKUNQVCZKUNVFZOVCZKUNNVCZKUNVFZVEUUDIUOUUGIUOUQUUE + IUOUUHIUOUQVFUUDUUEFVDZIUOZUUGUUHFVDZIUOZUQZUUDUUEUUCVDZIUOZUUGUUHUUCVD + ZIUOZUQZUBAUUFUUNUUSVGUUIAUUKUUPUUMUURAUUJUUOIAFUUCUUDUUETVHVIAUULUUQIA + FUUCUUGUUHTVHZVIVJVKVLZRSUCUUCVMZYOVMZAUUIVFUULUUQKAUULUUQUQUUIUUTVNABC + UUGUUHKKKFABVCZKUNZCVCZKUNZUVDUVFFVDZKUNZUDWIVOZVPVQYRYTEEEYOVRVSAYSUUA + ULVCZEUNZVEZUUBUVKYOVDZYRYTUVKYOVDZYOVDZUQZAUVMUVDIUOZYRUQZUVFIUOZYTUQZ + DVCZIUOZUVKUQZVEZDKVTZCKVTBKVTZUVQAUVMUVSBKVTZUWACKVTZUWDDKVTZVEZUWGAYR + IWAZUNZYTUWLUNZUVKUWLUNZVEZUVMUWKAUWMYSUWNUUAUWOUVLAUWLEYRAKEIWBZUWLEUQ + UAKEIWCWDZWEZAUWLEYTUWRWEAUWLEUVKUWRWEWFAIKWGZUWPUWKVGAUWQUWTUAKEIWHWDZ + UWTUWMUWHUWNUWIUWOUWJBKYRIWJZCKYTIWJDKUVKIWJWFWDWKUVSUWAUWDBCDKKKWLWMAU + WFUVQBCKKAUVEUVGVFVFUWEUVQDKAUVEUVGUWBKUNZUWEUVQWRZWRAUVEUVGUXCUXDAUVEU + VGUXCVEZVFZUVRUVTYOVDZUWCYOVDZUVRUVTUWCYOVDZYOVDZUQUWEUVQUXFUVHIUOZUWCY + OVDZUVRUVFUWBFVDZIUOZYOVDZUXHUXJUXFUVHUWBUUCVDZIUOZUVDUXMUUCVDZIUOZUXLU + XOUXFUVHUWBFVDZIUOUVDUXMFVDZIUOUXQUXSUEUXFUXTUXPIUXFFUUCUVHUWBAFUUCUQZU + XETVNZVHVIUXFUYAUXRIUXFFUUCUVDUXMUYCVHVIWNUXFAUVIUXCUXLUXQUQAUXEWOZAUVE + UVGUVIUXCUDWPAUVEUVGUXCWSAEGYOUUCHIKUVHUWBLNOPQUAUVARSUCUVBUVCWQWTUXFAU + VEUXMKUNZUXOUXSUQUYDAUVEUVGUXCXAAUVGUXCUYEUVEAONUVFUWBKKKFUVJVOXBAEGYOU + UCHIKUVDUXMLNOPQUAUVARSUCUVBUVCWQWTXCUXFUXGUXKUWCYOUXFUXGUVDUVFUUCVDZIU + OZUXKAUVEUVGUXGUYGUQUXCAEGYOUUCHIKUVDUVFLNOPQUAUVARSUCUVBUVCWQWPUXFUVHU + YFIUXFFUUCUVDUVFUYCVHVIXDXEUXFUXIUXNUVRYOUXFUXIUVFUWBUUCVDZIUOZUXNAUVGU + XCUXIUYIUQUVEAEGYOUUCHIKUVFUWBLNOPQUAUVARSUCUVBUVCWQXFUXFUXMUYHIUXFFUUC + UVFUWBUYCVHVIXDXHXCUWEUXHUVNUXJUVPUWEUXGUUBUWCUVKYOUWEUVRYRUVTYTYOUVSUW + AUWDXGZUVSUWAUWDXIZXRUVSUWAUWDXJZXRUWEUVRYRUXIUVOYOUYJUWEUVTYTUWCUVKYOU + YKUYLXRXRVJXKXLXMXNXOXPXQAKEMIAUWQKEIVBZUAKEIXSWDZUFXTZAYSYNYRYOVDZYRUQ + ZAYSUWHUYQAUWMYSUWHUWSAUWTUWMUWHVGUXAUXBWDWKZAUVSUYQBKAUVEVFZYNUVRYOVDZ + UVRUQUVSUYQUYSUYTMUVDUUCVDZIUOZMUVDFVDZIUOUVRUYSAMKUNZUVEUYTVUBUQAUVEWO + ZAVUDUVEUFVNAUVEYAZAEGYOUUCHIKMUVDLNOPQUAUVARSUCUVBUVCWQWTUYSVUCVUAIUYS + FUUCMUVDAUYBUVETVNZVHVIUGYBUVSUYTUYPUVRYRUVRYRYNYOYCUVSYHVJXKYDXPXQZAYS + YTYRYOVDZYNUQZUKEVTZAYSUWHVUKUYRAUVSVUKBKUYSYTUVRYOVDZYNUQZUKEVTZUVSVUK + UYSJIUOZEUNVUOUVRYOVDZYNUQZVUNUYSKEJIAUYMUVEUYNVNUHXTUYSVUPJUVDUUCVDZIU + OZJUVDFVDZIUOYNUYSAJKUNUVEVUPVUSUQVUEUHVUFAEGYOUUCHIKJUVDLNOPQUAUVARSUC + UVBUVCWQWTUYSVUTVURIUYSFUUCJUVDVUGVHVIUIYBVUMVUQUKVUOEYTVUOUQVULVUPYNYT + VUOUVRYOYEYFYGYIUVSVUMVUJUKEUVSVULVUIYNUVRYRYTYOYCYFYJXKYDXPXQYKZAUJEYO + HYNYPYQUYOVUHVVAYLYM $. $} ${ @@ -230474,15 +230468,15 @@ with the same (relevant) properties is also a group. (Contributed by mhmid $p |- ( ph -> ( F ` .0. ) = ( 0g ` H ) ) $= ( cfv va vi c0g eqid wfo wf fof syl cmnd wcel mndidcl ffvelcdmd cv wa wceq co simplll syl3an1 ad3antrrr simplr mhmlem mndlid syl2anc fveq2d - eqtr3d simpr oveq2d 3eqtr3d wrex foelrni sylan mndrid oveq1d ismgmid2 - r19.29a ) AUAJEKFTZHHUCTZNVQUDPAIJKFAIJFUEZIJFUFQIJFUGUHAGUIUJZKIUJZR - IGKMSUKZUHULAUAUMZJUJZUNZUBUMZFTZWBUOZVPWBEUPZWBUOUBIWDWEIUJZUNZWGUNZ - VPWFEUPZWFWHWBWKKWEDUPZFTWLWFWKBCKWEDEFIWKABUMZIUJCUMZIUJWNWODUPFTWNF - TWOFTEUPUOAWCWIWGUQLURZWKVSVTAVSWCWIWGRUSZWAUHZWDWIWGUTZVAWKWMWEFWKVS - WIWMWEUOWQWSIDGWEKMOSVBVCVDVEWKWFWBVPEWJWGVFZVGWTVHAVRWCWGUBIVIQUBIJF - WBVJVKZVOWDWGWBVPEUPZWBUOUBIWKWFVPEUPZWFXBWBWKWEKDUPZFTXCWFWKBCWEKDEF - IWPWSWRVAWKXDWEFWKVSWIXDWEUOWQWSIDGWEKMOSVLVCVDVEWKWFWBVPEWTVMWTVHXAV - OVN $. + eqtr3d simpr oveq2d 3eqtr3d wrex foelcdmi sylan r19.29a mndrid oveq1d + ismgmid2 ) AUAJEKFTZHHUCTZNVQUDPAIJKFAIJFUEZIJFUFQIJFUGUHAGUIUJZKIUJZ + RIGKMSUKZUHULAUAUMZJUJZUNZUBUMZFTZWBUOZVPWBEUPZWBUOUBIWDWEIUJZUNZWGUN + ZVPWFEUPZWFWHWBWKKWEDUPZFTWLWFWKBCKWEDEFIWKABUMZIUJCUMZIUJWNWODUPFTWN + FTWOFTEUPUOAWCWIWGUQLURZWKVSVTAVSWCWIWGRUSZWAUHZWDWIWGUTZVAWKWMWEFWKV + SWIWMWEUOWQWSIDGWEKMOSVBVCVDVEWKWFWBVPEWJWGVFZVGWTVHAVRWCWGUBIVIQUBIJ + FWBVJVKZVLWDWGWBVPEUPZWBUOUBIWKWFVPEUPZWFXBWBWKWEKDUPZFTXCWFWKBCWEKDE + FIWPWSWRVAWKXDWEFWKVSWIXDWEUOWQWSIDGWEKMOSVMVCVDVEWKWFWBVPEWTVNWTVHXA + VLVO $. $} $( The image of a monoid ` G ` under a monoid homomorphism ` F ` is a @@ -230490,40 +230484,40 @@ with the same (relevant) properties is also a group. (Contributed by mhmmnd $p |- ( ph -> H e. Mnd ) $= ( co wcel wa va vb vc vd vi vj vk wceq wral wrex cmnd cfv simpllr simpr cv oveq12d simp-5l syl3an1 simp-4r simplr mhmlem wf wfo fof syl ad5antr - mndcl syl3anc ffvelcdmd eqeltrrd foelrni syl2an ad2antrr r19.29a simpll - simpl simplrl simplrr w3a simp-6r mndass syl13anc simp-7l 3eqtr3d simp1 - fveq2d simp2 oveq1d oveq2d simp-5r 3ad2antr3 ad4antr 3adantr3 ralrimiva - simp3 sylan jca ralrimivva c0g mndidcl simplll ad3antrrr mndlid syl2anc - eqid eqtr3d mndrid oveq1 eqeq1d ovanraleqv rspcev ismnd sylanbrc ) AUAU - OZUBUOZERZJSZXPUCUOZERZXNXOXRERZERZUHZUCJUIZTZUBJUIUAJUIUDUOZXNERZXNUHZ - XNYEERXNUHTUAJUIZUDJUJZHUKSAYDUAUBJJAXNJSZXOJSZTZTZXQYCYMUEUOZFULZXNUHZ - XQUEIYMYNISZTZYPTZUFUOZFULZXOUHZXQUFIYSYTISZTZUUBTZYOUUAERZXPJUUEYOXNUU - AXOEYRYPUUCUUBUMUUDUUBUNUPUUEYNYTDRZFULZUUFJUUEBCYNYTDEFIUUEABUOZISZCUO - ZISZUUIUUKDRFULUUIFULUUKFULERUHZAYLYQYPUUCUUBUQKURYMYQYPUUCUUBUSZYSUUCU - UBUTZVAUUEIJUUGFAIJFVBZYLYQYPUUCUUBAIJFVCZUUPPIJFVDVEZVFUUEGUKSZYQUUCUU - GISZAUUSYLYQYPUUCUUBQVFUUNUUOIDGYNYTLNVGZVHVIVJVJYMUUBUFIUJZYQYPAUUQYKU - VBYLPYJYKUNUFIJFXOVKVLZVMVNAUUQYJYPUEIUJZYLPYJYKVPUEIJFXNVKZVLZVNYMYBUC - JYMXRJSZTAYJYKUVGYBAYLUVGVOAYJYKUVGVQAYJYKUVGVRYMUVGUNAYJYKUVGVSZTZYPYB - UEIUVIYQTZYPTZUUBYBUFIUVKUUCTZUUBTZUGUOZFULZXRUHZYBUGIUVMUVNISZTZUVPTZU - UFUVOERZYOUUAUVOERZERZXSYAUVSUUHUVOERZYOYTUVNDRZFULZERZUVTUWBUVSUUGUVND - RZFULYNUWDDRZFULUWCUWFUVSUWGUWHFUVSUUSYQUUCUVQUWGUWHUHUVJUUSYPUUCUUBUVQ - UVPAUUSUVHYQQVMVFZUVIYQYPUUCUUBUVQUVPVTZUVKUUCUUBUVQUVPUSZUVMUVQUVPUTZI - DGYNYTUVNLNWAWBWFUVSBCUUGUVNDEFIUVSAUUJUULUUMAUVHYQYPUUCUUBUVQUVPWCZKUR - ZUVSUUSYQUUCUUTUWIUWJUWKUVAVHUWLVAUVSBCYNUWDDEFIUWNUWJUVSUUSUUCUVQUWDIS - UWIUWKUWLIDGYTUVNLNVGVHVAWDUVSUUHUUFUVOEUVSAYQUUCUUHUUFUHUWMUWJUWKAYQUU - CVSZBCYNYTDEFIUWOAUUJUULUUMAYQUUCWEKURAYQUUCWGAYQUUCWOVAVHWHUVSUWEUWAYO - EUVSBCYTUVNDEFIUWNUWKUWLVAWIWDUVSUUFXPUVOXREUVSYOXNUUAXOEUVJYPUUCUUBUVQ - UVPWJZUVLUUBUVQUVPUMZUPUVRUVPUNZUPUVSYOXNUWAXTEUWPUVSUUAXOUVOXREUWQUWRU - PUPWDUVIUVPUGIUJZYQYPUUCUUBAYJUVGUWSYKAUUQUVGUWSPUGIJFXRVKWPWKWLVNUVIUV - BYQYPAYJYKUVBUVGUVCWMVMVNAYJYKUVDUVGUVFWMVNWBWNWQWRAGWSULZFULZJSUXAXNER - ZXNUHZXNUXAERZXNUHZTZUAJUIZYIAIJUWTFUURAUUSUWTISZQIGUWTLUWTXEZWTZVEVIAU - XFUAJAYJTZYPUXFUEIUXKYQTZYPTZUXCUXEUXMUXAYOERZYOUXBXNUXMUWTYNDRZFULUXNY - OUXMBCUWTYNDEFIUXMAUUJUULUUMAYJYQYPXAKURZUXMUUSUXHAUUSYJYQYPQXBZUXJVEZU - XKYQYPUTZVAUXMUXOYNFUXMUUSYQUXOYNUHUXQUXSIDGYNUWTLNUXIXCXDWFXFUXMYOXNUX - AEUXLYPUNZWIUXTWDUXMYOUXAERZYOUXDXNUXMYNUWTDRZFULUYAYOUXMBCYNUWTDEFIUXP - UXSUXRVAUXMUYBYNFUXMUUSYQUYBYNUHUXQUXSIDGYNUWTLNUXIXGXDWFXFUXMYOXNUXAEU - XTWHUXTWDWQAUUQYJUVDPUVEWPVNWNYHUXGUDUXAJYGUXCUAXNYEXNEJUXAYEUXAUHYFUXB - XNYEUXAXNEXHXIXJXKXDJEUDHUAUBUCMOXLXM $. + mndcl syl3anc ffvelcdmd eqeltrrd foelcdmi syl2an ad2antrr r19.29a simpl + simpll simplrl simplrr w3a simp-6r mndass syl13anc fveq2d simp-7l simp1 + 3eqtr3d simp2 simp3 oveq1d simp-5r 3ad2antr3 ad4antr 3adantr3 ralrimiva + oveq2d jca ralrimivva c0g eqid mndidcl simplll ad3antrrr mndlid syl2anc + sylan eqtr3d mndrid oveq1 eqeq1d ovanraleqv rspcev ismnd sylanbrc ) AUA + UOZUBUOZERZJSZXPUCUOZERZXNXOXRERZERZUHZUCJUIZTZUBJUIUAJUIUDUOZXNERZXNUH + ZXNYEERXNUHTUAJUIZUDJUJZHUKSAYDUAUBJJAXNJSZXOJSZTZTZXQYCYMUEUOZFULZXNUH + ZXQUEIYMYNISZTZYPTZUFUOZFULZXOUHZXQUFIYSYTISZTZUUBTZYOUUAERZXPJUUEYOXNU + UAXOEYRYPUUCUUBUMUUDUUBUNUPUUEYNYTDRZFULZUUFJUUEBCYNYTDEFIUUEABUOZISZCU + OZISZUUIUUKDRFULUUIFULUUKFULERUHZAYLYQYPUUCUUBUQKURYMYQYPUUCUUBUSZYSUUC + UUBUTZVAUUEIJUUGFAIJFVBZYLYQYPUUCUUBAIJFVCZUUPPIJFVDVEZVFUUEGUKSZYQUUCU + UGISZAUUSYLYQYPUUCUUBQVFUUNUUOIDGYNYTLNVGZVHVIVJVJYMUUBUFIUJZYQYPAUUQYK + UVBYLPYJYKUNUFIJFXOVKVLZVMVNAUUQYJYPUEIUJZYLPYJYKVOUEIJFXNVKZVLZVNYMYBU + CJYMXRJSZTAYJYKUVGYBAYLUVGVPAYJYKUVGVQAYJYKUVGVRYMUVGUNAYJYKUVGVSZTZYPY + BUEIUVIYQTZYPTZUUBYBUFIUVKUUCTZUUBTZUGUOZFULZXRUHZYBUGIUVMUVNISZTZUVPTZ + UUFUVOERZYOUUAUVOERZERZXSYAUVSUUHUVOERZYOYTUVNDRZFULZERZUVTUWBUVSUUGUVN + DRZFULYNUWDDRZFULUWCUWFUVSUWGUWHFUVSUUSYQUUCUVQUWGUWHUHUVJUUSYPUUCUUBUV + QUVPAUUSUVHYQQVMVFZUVIYQYPUUCUUBUVQUVPVTZUVKUUCUUBUVQUVPUSZUVMUVQUVPUTZ + IDGYNYTUVNLNWAWBWCUVSBCUUGUVNDEFIUVSAUUJUULUUMAUVHYQYPUUCUUBUVQUVPWDZKU + RZUVSUUSYQUUCUUTUWIUWJUWKUVAVHUWLVAUVSBCYNUWDDEFIUWNUWJUVSUUSUUCUVQUWDI + SUWIUWKUWLIDGYTUVNLNVGVHVAWFUVSUUHUUFUVOEUVSAYQUUCUUHUUFUHUWMUWJUWKAYQU + UCVSZBCYNYTDEFIUWOAUUJUULUUMAYQUUCWEKURAYQUUCWGAYQUUCWHVAVHWIUVSUWEUWAY + OEUVSBCYTUVNDEFIUWNUWKUWLVAWOWFUVSUUFXPUVOXREUVSYOXNUUAXOEUVJYPUUCUUBUV + QUVPWJZUVLUUBUVQUVPUMZUPUVRUVPUNZUPUVSYOXNUWAXTEUWPUVSUUAXOUVOXREUWQUWR + UPUPWFUVIUVPUGIUJZYQYPUUCUUBAYJUVGUWSYKAUUQUVGUWSPUGIJFXRVKXEWKWLVNUVIU + VBYQYPAYJYKUVBUVGUVCWMVMVNAYJYKUVDUVGUVFWMVNWBWNWPWQAGWRULZFULZJSUXAXNE + RZXNUHZXNUXAERZXNUHZTZUAJUIZYIAIJUWTFUURAUUSUWTISZQIGUWTLUWTWSZWTZVEVIA + UXFUAJAYJTZYPUXFUEIUXKYQTZYPTZUXCUXEUXMUXAYOERZYOUXBXNUXMUWTYNDRZFULUXN + YOUXMBCUWTYNDEFIUXMAUUJUULUUMAYJYQYPXAKURZUXMUUSUXHAUUSYJYQYPQXBZUXJVEZ + UXKYQYPUTZVAUXMUXOYNFUXMUUSYQUXOYNUHUXQUXSIDGYNUWTLNUXIXCXDWCXFUXMYOXNU + XAEUXLYPUNZWOUXTWFUXMYOUXAERZYOUXDXNUXMYNUWTDRZFULUYAYOUXMBCYNUWTDEFIUX + PUXSUXRVAUXMUYBYNFUXMUUSYQUYBYNUHUXQUXSIDGYNUWTLNUXIXGXDWCXFUXMYOXNUXAE + UXTWIUXTWFWPAUUQYJUVDPUVEXEVNWNYHUXGUDUXAJYGUXCUAXNYEXNEJUXAYEUXAUHYFUX + BXNYEUXAXNEXHXIXJXKXDJEUDHUAUBUCMOXLXM $. $( The function fulfilling the conditions of ~ mhmmnd is a monoid homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.) $) @@ -230546,15 +230540,15 @@ since the target of the homomorphism (operator ` O ` in our model) need ( wcel cfv wceq vf va vi cmnd cv co wrex wral cgrp grpmndd mhmmnd cminusg c0g wfo fof syl ad3antrrr simplr eqid grpinvcl syl2anc ffvelcdmd 3adant1r wa sylan simpr mhmlem ad4ant13 grplinv fveq2d mhmid eqtrd oveq2d 3eqtr3rd - wf oveq1 eqeq1d rspcev foelrni r19.29a ralrimiva isgrp sylanbrc ) AHUDRUA - UEZUBUEZEUFZHUMSZTZUAJUGZUBJUHHUIRABCDEFGHIJKLMNOPAGQUJZUKAWIUBJAWEJRZVDZ - UCUEZFSZWETZWIUCIWLWMIRZVDZWOVDZWMGULSZSZFSZJRXAWEEUFZWGTZWIWRIJWTFAIJFVO - ZWKWPWOAIJFUNZXDPIJFUOUPUQWRGUIRZWPWTIRZAXFWKWPWOQUQZWLWPWOURZIGWSWMLWSUS - ZUTZVAVBWRWTWMDUFZFSZXAWNEUFZWGXBAWPXMXNTWKWOAWPVDBCWTWMDEFIABUEZIRCUEZIR - XOXPDUFFSXOFSXPFSEUFTWPKVCAXFWPXGQXKVEAWPVFVGVHWRXMGUMSZFSZWGWRXFWPXMXRTX - HXIXFWPVDXLXQFIDGWSWMXQLNXQUSZXJVIVJVAAXRWGTWKWPWOABCDEFGHIJXQKLMNOPWJXSV - KUQVLWRWNWEXAEWQWOVFVMVNWHXCUAXAJWDXATWFXBWGWDXAWEEVPVQVRVAAXEWKWOUCIUGPU - CIJFWEVSVEVTWAJEUAHWGUBMOWGUSWBWC $. + wf oveq1 eqeq1d rspcev foelcdmi r19.29a ralrimiva isgrp sylanbrc ) AHUDRU + AUEZUBUEZEUFZHUMSZTZUAJUGZUBJUHHUIRABCDEFGHIJKLMNOPAGQUJZUKAWIUBJAWEJRZVD + ZUCUEZFSZWETZWIUCIWLWMIRZVDZWOVDZWMGULSZSZFSZJRXAWEEUFZWGTZWIWRIJWTFAIJFV + OZWKWPWOAIJFUNZXDPIJFUOUPUQWRGUIRZWPWTIRZAXFWKWPWOQUQZWLWPWOURZIGWSWMLWSU + SZUTZVAVBWRWTWMDUFZFSZXAWNEUFZWGXBAWPXMXNTWKWOAWPVDBCWTWMDEFIABUEZIRCUEZI + RXOXPDUFFSXOFSXPFSEUFTWPKVCAXFWPXGQXKVEAWPVFVGVHWRXMGUMSZFSZWGWRXFWPXMXRT + XHXIXFWPVDXLXQFIDGWSWMXQLNXQUSZXJVIVJVAAXRWGTWKWPWOABCDEFGHIJXQKLMNOPWJXS + VKUQVLWRWNWEXAEWQWOVFVMVNWHXCUAXAJWDXATWFXBWGWDXAWEEVPVQVRVAAXEWKWOUCIUGP + UCIJFWEVSVEVTWAJEUAHWGUBMOWGUSWBWC $. $} @@ -233820,25 +233814,25 @@ permutation associated with the composition of these two elements (in ( ( .(+) |` ( X X. Z ) ) e. ( G GrpAct Z ) <-> A. x e. X A. y e. Z ( x .(+) y ) e. Z ) ) $= ( vz vu vv co wcel wa cv wral wceq ovres ad2antrr syl2anc cga cres adantl - wss cxp wf gaf fovrnda eqeltrrd ralrimivva cvv c0g cfv cplusg gagrp gaset - cgrp adantr simpr ssexd jca wfn ffnd simplr xpss2 fnssres eleq1d ralbidva - ralbiia sylibr ffnov sylanbrc eqid grpidcl simpll sselda gagrpid syl2an2r + wss cxp wf gaf fovcdmda eqeltrrd ralrimivva cgrp cvv c0g cfv cplusg gagrp + gaset adantr simpr ssexd jca wfn simplr xpss2 syl fnssres eleq1d ralbidva + ffnd ralbiia sylibr ffnov sylanbrc grpidcl simpll sselda gagrpid syl2an2r sylan eqtrd simprl simprr gaass syl13anc simpllr ovrspc2v syl21anc eqtr4d - syl grpcl syl3anc oveq2d 3eqtr4d ralrimiva isga impbida ) CDFUALMZGFUDZNZ - CEGUEZUBZDGUALMZAOZBOZCLZGMZBGPZAEPZWSXBNZXFABEGXIXCEMZXDGMNZNXCXDXALZXEG - XKXLXEQXIXCXDEGCRZUCXIXCXDGEGXAXBWTGXAUFZWSXADEGHUGUCUHUIUJWSXHNZDUQMZGUK - MZNXNDULUMZIOZXALZXSQZJOZKOZDUNUMZLZXSXALZYBYCXSXALZXALZQZKEPJEPZNZIGPZNX - BXOXPXQWQXPWRXHCDFUOSZWSXQXHWSGFUKWQFUKMWRCDFUPURWQWRUSUTURVAXOXNYLXOXAWT - VBZXLGMZBGPZAEPZXNXOCEFUEZVBWTYRUDZYNXOYRFCWQYRFCUFWRXHCDEFHUGSVCXOWRYSWQ - WRXHVDZGFEVEWIYRWTCVFTXOXHYQWSXHUSYPXGAEXJYOXFBGXKXLXEGXMVGVHVIVJABEGGXAV - KVLXOYKIGXOXSGMZNZYAYJUUBXTXRXSCLZXSXOXREMZUUAXTUUCQXOXPUUDYMEDXRHXRVMZVN - WIXRXSEGCRVSXOWQUUAXSFMZUUCXSQWQWRXHVOZXOGFXSYTVPZXSCDFXRUUEVQVRVTUUBYIJK - EEUUBYBEMZYCEMZNZNZYEXSCLZYBYCXSCLZXALZYFYHUULUUMYBUUNCLZUUOUULWQUUIUUJUU - FUUMUUPQXOWQUUAUUKUUGSUUBUUIUUJWAZUUBUUIUUJWBZUUBUUFUUKUUHURYBYCXSYDCDEFH - YDVMZWCWDUULUUIUUNGMZUUOUUPQUUQUULUUJUUAXHUUTUURXOUUAUUKVDZWSXHUUAUUKWEAB - EGGCYCXSWFWGYBUUNEGCRTWHUULYEEMZUUAYFUUMQUULXPUUIUUJUVBXOXPUUAUUKYMSUUQUU - REYDDYBYCHUUSWJWKUVAYEXSEGCRTUULYGUUNYBXAUULUUJUUAYGUUNQUURUVAYCXSEGCRTWL - WMUJVAWNVAIJKYDXADEGXRHUUSUUEWOVLWP $. + eqid grpcl syl3anc oveq2d 3eqtr4d ralrimiva isga impbida ) CDFUALMZGFUDZN + ZCEGUEZUBZDGUALMZAOZBOZCLZGMZBGPZAEPZWSXBNZXFABEGXIXCEMZXDGMNZNXCXDXALZXE + GXKXLXEQXIXCXDEGCRZUCXIXCXDGEGXAXBWTGXAUFZWSXADEGHUGUCUHUIUJWSXHNZDUKMZGU + LMZNXNDUMUNZIOZXALZXSQZJOZKOZDUOUNZLZXSXALZYBYCXSXALZXALZQZKEPJEPZNZIGPZN + XBXOXPXQWQXPWRXHCDFUPSZWSXQXHWSGFULWQFULMWRCDFUQURWQWRUSUTURVAXOXNYLXOXAW + TVBZXLGMZBGPZAEPZXNXOCEFUEZVBWTYRUDZYNXOYRFCWQYRFCUFWRXHCDEFHUGSVIXOWRYSW + QWRXHVCZGFEVDVEYRWTCVFTXOXHYQWSXHUSYPXGAEXJYOXFBGXKXLXEGXMVGVHVJVKABEGGXA + VLVMXOYKIGXOXSGMZNZYAYJUUBXTXRXSCLZXSXOXREMZUUAXTUUCQXOXPUUDYMEDXRHXRWIZV + NVEXRXSEGCRVSXOWQUUAXSFMZUUCXSQWQWRXHVOZXOGFXSYTVPZXSCDFXRUUEVQVRVTUUBYIJ + KEEUUBYBEMZYCEMZNZNZYEXSCLZYBYCXSCLZXALZYFYHUULUUMYBUUNCLZUUOUULWQUUIUUJU + UFUUMUUPQXOWQUUAUUKUUGSUUBUUIUUJWAZUUBUUIUUJWBZUUBUUFUUKUUHURYBYCXSYDCDEF + HYDWIZWCWDUULUUIUUNGMZUUOUUPQUUQUULUUJUUAXHUUTUURXOUUAUUKVCZWSXHUUAUUKWEA + BEGGCYCXSWFWGYBUUNEGCRTWHUULYEEMZUUAYFUUMQUULXPUUIUUJUVBXOXPUUAUUKYMSUUQU + UREYDDYBYCHUUSWJWKUVAYEXSEGCRTUULYGUUNYBXAUULUUJUUAYGUUNQUURUVAYCXSEGCRTW + LWMUJVAWNVAIJKYDXADEGXRHUUSUUEWOVMWP $. $} ${ @@ -233852,24 +233846,24 @@ permutation associated with the composition of these two elements (in ( vx vy vz co wcel cfv wa cxp wceq eqid adantr syl ovres syl2anc cga cgrp csubg cvv cbs cres wf c0g cv cplusg wral gaset subggrp anim12ci gaf simpr wss subgss xpss1 fssresd subgbas xpeq1d feq2d mpbid subg0cl subg0 gagrpid - oveq1d adantlr 3eqtr3d sselda simprl ad2antrr simprr sseldd fovrnd oveq2d - eqimss2 anim12dan ad3antlr simplll gaass syl13anc 3eqtr4d 3expb ressplusg - subgcl sylan oveqd 3eqtr2rd syldan ralrimivva jca ralrimiva isga sylanbrc - ) ACEUAJKZBCUCLZKZMZDUBKZEUDKZMDUELZENZEABENZUFZUGZDUHLZGUIZXFJZXIOZHUIZI - UIZDUJLZJZXIXFJZXLXMXIXFJZXFJZOZIXCUKHXCUKZMZGEUKZMXFDEUAJKWQXBWSXAACEULB - CDFUMUNWTXGYBWTXEEXFUGXGWTCUELZENZEXEAWQYDEAUGZWSACYCEYCPZUOQZWTBYCUQZXEY - DUQWTWSYHWQWSUPZYCBCYFURZRBYCEUSRUTWTXEXDEXFWTBXCEWTWSBXCOZYIBCDFVARZVBVC - VDWTYAGEWTXIEKZMZXKXTYNCUHLZXIXFJZYOXIAJZXJXIYNYOBKZYMYPYQOYNWSYRWTWSYMYI - QZBCYOYOPZVERWTYMUPZYOXIBEASTYNYOXHXIXFYNWSYOXHOYSBCDYOFYTVFRVHWQYMYQXIOW - SXIACEYOYTVGVIVJYNXSHIXCXCYNXLXCKZXMXCKZMXLBKZXMBKZMZXSYNUUBUUDUUCUUEYNXC - BXLWTXCBUQZYMWTYKUUGYLXCBVRRQZVKYNXCBXMUUHVKVSYNUUFMZXRXLXMCUJLZJZXIAJZUU - KXIXFJZXPUUIXLXMXIAJZXFJZXLUUNAJZXRUULUUIUUDUUNEKUUOUUPOYNUUDUUEVLZUUIXMX - IEYCEAWTYEYMUUFYGVMUUIBYCXMWSYHWQYMUUFYJVTZYNUUDUUEVNZVOZYNYMUUFUUAQZVPXL - UUNBEASTUUIXQUUNXLXFUUIUUEYMXQUUNOUUSUVAXMXIBEASTVQUUIWQXLYCKXMYCKYMUULUU - POWQWSYMUUFWAUUIBYCXLUURUUQVOUUTUVAXLXMXIUUJACYCEYFUUJPZWBWCWDUUIUUKBKZYM - UUMUULOYNWSUUFUVCYSWSUUDUUEUVCUUJBCXLXMUVBWGWEWHUVAUUKXIBEASTUUIUUKXOXIXF - UUIUUJXNXLXMWSUUJXNOWQYMUUFBUUJCDWRFUVBWFVTWIVHWJWKWLWMWNWMGHIXNXFDXCEXHX - CPXNPXHPWOWP $. + oveq1d adantlr 3eqtr3d eqimss2 sselda anim12dan ad2antrr ad3antlr fovcdmd + simprl simprr sseldd oveq2d simplll gaass syl13anc subgcl 3expb ressplusg + 3eqtr4d sylan oveqd 3eqtr2rd syldan ralrimivva ralrimiva isga sylanbrc + jca ) ACEUAJKZBCUCLZKZMZDUBKZEUDKZMDUELZENZEABENZUFZUGZDUHLZGUIZXFJZXIOZH + UIZIUIZDUJLZJZXIXFJZXLXMXIXFJZXFJZOZIXCUKHXCUKZMZGEUKZMXFDEUAJKWQXBWSXAAC + EULBCDFUMUNWTXGYBWTXEEXFUGXGWTCUELZENZEXEAWQYDEAUGZWSACYCEYCPZUOQZWTBYCUQ + ZXEYDUQWTWSYHWQWSUPZYCBCYFURZRBYCEUSRUTWTXEXDEXFWTBXCEWTWSBXCOZYIBCDFVARZ + VBVCVDWTYAGEWTXIEKZMZXKXTYNCUHLZXIXFJZYOXIAJZXJXIYNYOBKZYMYPYQOYNWSYRWTWS + YMYIQZBCYOYOPZVERWTYMUPZYOXIBEASTYNYOXHXIXFYNWSYOXHOYSBCDYOFYTVFRVHWQYMYQ + XIOWSXIACEYOYTVGVIVJYNXSHIXCXCYNXLXCKZXMXCKZMXLBKZXMBKZMZXSYNUUBUUDUUCUUE + YNXCBXLWTXCBUQZYMWTYKUUGYLXCBVKRQZVLYNXCBXMUUHVLVMYNUUFMZXRXLXMCUJLZJZXIA + JZUUKXIXFJZXPUUIXLXMXIAJZXFJZXLUUNAJZXRUULUUIUUDUUNEKUUOUUPOYNUUDUUEVQZUU + IXMXIEYCEAWTYEYMUUFYGVNUUIBYCXMWSYHWQYMUUFYJVOZYNUUDUUEVRZVSZYNYMUUFUUAQZ + VPXLUUNBEASTUUIXQUUNXLXFUUIUUEYMXQUUNOUUSUVAXMXIBEASTVTUUIWQXLYCKXMYCKYMU + ULUUPOWQWSYMUUFWAUUIBYCXLUURUUQVSUUTUVAXLXMXIUUJACYCEYFUUJPZWBWCWGUUIUUKB + KZYMUUMUULOYNWSUUFUVCYSWSUUDUUEUVCUUJBCXLXMUVBWDWEWHUVAUUKXIBEASTUUIUUKXO + XIXFUUIUUJXNXLXMWSUUJXNOWQYMUUFBUUJCDWRFUVBWFVOWIVHWJWKWLWPWMWPGHIXNXFDXC + EXHXCPXNPXHPWNWO $. $} ${ @@ -233911,12 +233905,12 @@ permutation associated with the composition of these two elements (in -> ( ( A .(+) B ) = C <-> ( ( N ` A ) .(+) C ) = B ) ) $= ( cga co wcel w3a wceq cfv adantr eqid syl2anc syl13anc cplusg cgrp gagrp wa c0g simpr1 grprinv oveq1d simpl grpinvcl simpr3 gagrpid 3eqtr3d eqeq2d - gaass wb simpr2 cxp wf gaf fovrnd galcan bitr3d eqcom bitrdi ) DEHKLMZAGM - ZBHMZCHMZNZUDZABDLZCOZBAFPZCDLZOZVOBOVKVLAVODLZOZVMVPVKVQCVLVKAVNEUAPZLZC - DLZEUEPZCDLZVQCVKVTWBCDVKEUBMZVGVTWBOVFWDVJDEHUCQZVFVGVHVIUFZGVSEFAWBIVSR - ZWBRZJUGSUHVKVFVGVNGMZVIWAVQOVFVJUIZWFVKWDVGWIWEWFGEFAIJUJSZVFVGVHVIUKZAV - NCVSDEGHIWGUOTVKVFVIWCCOWJWLCDEHWBWHULSUMUNVKVFVGVHVOHMVRVPUPWJWFVFVGVHVI - UQVKVNCHGHDVFGHURHDUSVJDEGHIUTQWKWLVAABVODEGHIVBTVCBVOVDVE $. + gaass wb simpr2 cxp wf gaf fovcdmd galcan bitr3d eqcom bitrdi ) DEHKLMZAG + MZBHMZCHMZNZUDZABDLZCOZBAFPZCDLZOZVOBOVKVLAVODLZOZVMVPVKVQCVLVKAVNEUAPZLZ + CDLZEUEPZCDLZVQCVKVTWBCDVKEUBMZVGVTWBOVFWDVJDEHUCQZVFVGVHVIUFZGVSEFAWBIVS + RZWBRZJUGSUHVKVFVGVNGMZVIWAVQOVFVJUIZWFVKWDVGWIWEWFGEFAIJUJSZVFVGVHVIUKZA + VNCVSDEGHIWGUOTVKVFVIWCCOWJWLCDEHWBWHULSUMUNVKVFVGVHVOHMVRVPUPWJWFVFVGVHV + IUQVKVNCHGHDVFGHURHDUSVJDEGHIUTQWKWLVAABVODEGHIVBTVCBVOVDVE $. $} ${ @@ -233928,13 +233922,13 @@ permutation associated with the composition of these two elements (in Mario Carneiro, 13-Jan-2015.) $) gapm $p |- ( ( .(+) e. ( G GrpAct Y ) /\ A e. X ) -> F : Y -1-1-onto-> Y ) $= - ( vy co wcel wa cv cfv ad2antrr simplr simpr fovrnd wceq cga cminusg cgrp - cxp wf gagrp eqid grpinvcl syl2anc wb simpll simprl simprr gacan syl13anc - gaf bicomd eqcom 3bitr4g f1o2d ) CEGUAKLZBFLZMZAJGGBANZCKZBEUBOZOZJNZCKZD - IVCVDGLZMBVDGFGCVAFGUDGCUEZVBVJCEFGHUPZPVAVBVJQVCVJRSVCVHGLZMZVGVHGFGCVAV - KVBVMVLPVNEUCLZVBVGFLVAVOVBVMCEGUFPVAVBVMQFEVFBHVFUGZUHUIVCVMRSVCVJVMMZMZ - VIVDTZVEVHTZVDVITVHVETVRVTVSVRVAVBVJVMVTVSUJVAVBVQUKVAVBVQQVCVJVMULVCVJVM - UMBVDVHCEVFFGHVPUNUOUQVDVIURVHVEURUSUT $. + ( vy co wcel wa cv cfv ad2antrr simplr simpr fovcdmd wceq cga cminusg cxp + wf gaf cgrp gagrp grpinvcl syl2anc wb simpll simprl simprr gacan syl13anc + eqid bicomd eqcom 3bitr4g f1o2d ) CEGUAKLZBFLZMZAJGGBANZCKZBEUBOZOZJNZCKZ + DIVCVDGLZMBVDGFGCVAFGUCGCUDZVBVJCEFGHUEZPVAVBVJQVCVJRSVCVHGLZMZVGVHGFGCVA + VKVBVMVLPVNEUFLZVBVGFLVAVOVBVMCEGUGPVAVBVMQFEVFBHVFUPZUHUIVCVMRSVCVJVMMZM + ZVIVDTZVEVHTZVDVITVHVETVRVTVSVRVAVBVJVMVTVSUJVAVBVQUKVAVBVQQVCVJVMULVCVJV + MUMBVDVHCEVFFGHVPUNUOUQVDVIURVHVEURUSUT $. $} ${ @@ -234020,14 +234014,14 @@ permutation associated with the composition of these two elements (in ( co wcel wa cfv wceq wb syl cga wbr cminusg cplusg w3a wss csubg gastacl cgrp adantr subgrcl subgss eqid eqgval syl2anc df-3an bitrdi simpr simpll biantrurd simprl grpinvcl simprr simplr gaass syl13anc eqeq1d grpcl oveq1 - syl3anc cv elrab2 baib cxp wf gaf fovrnd gacan 3bitr4d 3bitr2d ) EGJUANOZ - BJOZPZCIOZDIOZPZPZCDFUBZWFCGUCQZQZDGUDQZNZHOZPZWMCBENDBENZRZWGWHWDWEWMUEZ - WNWGGUIOZHIUFZWHWQSWGHGUGQOZWRWCWTWFABEGHIJKLUHUJZHGUKTZWGWTWSXAIHGKULTCD - WKFHGWIUIIKWIUMZWKUMZMUNUOWDWEWMUPUQWGWFWMWCWFURUTWGWLBENZBRZWJWOENZBRZWM - WPWGXEXGBWGWAWJIOZWEWBXEXGRWAWBWFUSZWGWRWDXIXBWCWDWEVAZIGWICKXCVBUOZWCWDW - EVCZWAWBWFVDZWJDBWKEGIJKXDVEVFVGWGWLIOZWMXFSWGWRXIWEXOXBXLXMIWKGWJDKXDVHV - JWMXOXFAVKZBENZBRXFAWLIHXPWLRXQXEBXPWLBEVIVGLVLVMTWGWAWDWBWOJOWPXHSXJXKXN - WGDBJIJEWGWAIJVNJEVOXJEGIJKVPTXMXNVQCBWOEGWIIJKXCVRVFVSVT $. + syl3anc cv elrab2 baib cxp wf gaf fovcdmd gacan 3bitr4d 3bitr2d ) EGJUANO + ZBJOZPZCIOZDIOZPZPZCDFUBZWFCGUCQZQZDGUDQZNZHOZPZWMCBENDBENZRZWGWHWDWEWMUE + ZWNWGGUIOZHIUFZWHWQSWGHGUGQOZWRWCWTWFABEGHIJKLUHUJZHGUKTZWGWTWSXAIHGKULTC + DWKFHGWIUIIKWIUMZWKUMZMUNUOWDWEWMUPUQWGWFWMWCWFURUTWGWLBENZBRZWJWOENZBRZW + MWPWGXEXGBWGWAWJIOZWEWBXEXGRWAWBWFUSZWGWRWDXIXBWCWDWEVAZIGWICKXCVBUOZWCWD + WEVCZWAWBWFVDZWJDBWKEGIJKXDVEVFVGWGWLIOZWMXFSWGWRXIWEXOXBXLXMIWKGWJDKXDVH + VJWMXOXFAVKZBENZBRXFAWLIHXPWLRXQXEBXPWLBEVIVGLVLVMTWGWAWDWBWOJOWPXHSXJXKX + NWGDBJIJEWGWAIJVNJEVOXJEGIJKVPTXMXNVQCBWOEGWIIJKXCVRVFVSVT $. orbsta.f $e |- F = ran ( k e. X |-> <. [ k ] .~ , ( k .(+) A ) >. ) $. $( Existence and uniqueness for the function of ~ orbsta . (Contributed by @@ -234061,31 +234055,31 @@ permutation associated with the composition of these two elements (in orbsta $p |- ( ( .(+) e. ( G GrpAct Y ) /\ A e. Y ) -> F : ( X /. .~ ) -1-1-onto-> [ A ] O ) $= ( wceq va vb vh vz vw cga co wcel wa cqs cec wf1 wfo wf1o wf cv wral wfun - cfv orbstafun cvv wbr wrex simpr adantr cxp gaf fovrnd eqid eqeq1d rspcev - wi oveq1 sylancl gaorb syl3anbrc wb ovex elecg sylancr mpbird wer gastacl - csubg eqger syl cbs fvexi a1i qliftf mpbid fveqeq2 imbi12d ralbidv eqeq2d - eqeq1 fveq2 eqeq2 orbstaval adantrr adantrl eqeq12d gastacos erth 3bitr2d - simprl biimpd anassrs ectocld ralrimiva sylanbrc w3a bitrdi biimpa simp3d - dff13 vex cqg ecelqsi eqcomd rspceeqv syl2an2 rexbidv syl5ibcom rexlimdva - ovexi imp syldan dffo3 df-f1o ) EJNUFUGUHZDNUHZUIZMFUJZDLUKZIULZYNYOIUMZY - NYOIUNYMYNYOIUOZUAUPZIUSUBUPZIUSZTZYSYTTZVLZUBYNUQZUAYNUQYPYMIURYRCDEFHIJ - KMNOPQRUTYMHHUPZDEUGZFIVAMYORYMUUFMUHZUIZUUGYOUHZDUUGLVBZUUIYLUUGNUHUCUPZ - DEUGZUUGTZUCMVCZUUKYMYLUUHYKYLVDZVEZUUIUUFDNMNEYMMNVFNEUOZUUHYKUURYLEJMNO - VGVEVEYMUUHVDZUUQVHUUIUUHUUGUUGTZUUOUUSUUGVIUUNUUTUCUUFMUULUUFTUUMUUGUUGU - ULUUFDEVMVJVKVNABDUUGELGUCMNSVOVPUUIUUGVAUHYLUUJUUKVQUUFDEVRUUQUUGDLVANVS - VTWAYMKJWDUSUHMFWBZCDEJKMNOPWCFJMKOQWEWFZMVAUHYMMJWGOWHWIWJWKZYMUUEUAYNUD - UPZFUKZIUSZUUATZUVEYTTZVLZUBYNUQUUEYMUDYSMFYNYNVIZUVEYSTZUVIUUDUBYNUVKUVG - UUBUVHUUCUVEYSUUAIWLUVEYSYTWPWMWNYMUVDMUHZUIZUVIUBYNUVFUEUPZFUKZIUSZTZUVE - UVOTZVLZUVIUVMUEYTMFYNUVJUVOYTTZUVQUVGUVRUVHUVTUVPUUAUVFUVOYTIWQWOUVOYTUV - EWRWMYMUVLUVNMUHZUVSYMUVLUWAUIZUIZUVQUVRUWCUVQUVDDEUGZUVNDEUGZTUVDUVNFVBU - VRUWCUVFUWDUVPUWEYMUVLUVFUWDTUWACDUVDEFHIJKMNOPQRWSWTYMUWAUVPUWETUVLCDUVN - EFHIJKMNOPQRWSZXAXBCDUVDUVNEFJKMNOPQXCUWCUVDUVNFMYMUVAUWBUVBVEYMUVLUWAXFX - DXEXGXHXIXJXIXJUAUBYNYOIXPXKYMYRUULUVDIUSZTZUDYNVCZUCYOUQYQUVCYMUWIUCYOYM - UULYOUHZUWEUULTZUEMVCZUWIYMUWJUIYLUULNUHZUWLYMUWJYLUWMUWLXLZYMUWJDUULLVBZ - UWNYMUULVAUHYLUWJUWOVQUCXQUUPUULDLVANVSVTABDUULELGUEMNSVOXMXNXOYMUWLUWIYM - UWKUWIUEMYMUWAUIZUWEUWGTZUDYNVCZUWKUWIUWAUVOYNUHYMUWEUVPTUWRMUVNFFJKXRQYF - XSUWPUVPUWEUWFXTUDUVOYNUWGUVPUWEUVDUVOIWQYAYBUWKUWQUWHUDYNUWEUULUWGWPYCYD - YEYGYHXJUDUCYNYOIYIXKYNYOIYJXK $. + cfv orbstafun cvv wbr wrex simpr adantr cxp gaf fovcdmd eqid oveq1 eqeq1d + wi rspcev sylancl gaorb syl3anbrc ovex elecg sylancr mpbird csubg gastacl + wb wer eqger syl cbs fvexi a1i qliftf mpbid fveqeq2 eqeq1 imbi12d ralbidv + fveq2 eqeq2d eqeq2 orbstaval adantrr adantrl eqeq12d gastacos simprl erth + 3bitr2d biimpd anassrs ectocld ralrimiva dff13 sylanbrc w3a bitrdi biimpa + vex simp3d cqg ovexi ecelqsi rspceeqv syl2an2 rexbidv syl5ibcom rexlimdva + eqcomd imp syldan dffo3 df-f1o ) EJNUFUGUHZDNUHZUIZMFUJZDLUKZIULZYNYOIUMZ + YNYOIUNYMYNYOIUOZUAUPZIUSUBUPZIUSZTZYSYTTZVLZUBYNUQZUAYNUQYPYMIURYRCDEFHI + JKMNOPQRUTYMHHUPZDEUGZFIVAMYORYMUUFMUHZUIZUUGYOUHZDUUGLVBZUUIYLUUGNUHUCUP + ZDEUGZUUGTZUCMVCZUUKYMYLUUHYKYLVDZVEZUUIUUFDNMNEYMMNVFNEUOZUUHYKUURYLEJMN + OVGVEVEYMUUHVDZUUQVHUUIUUHUUGUUGTZUUOUUSUUGVIUUNUUTUCUUFMUULUUFTUUMUUGUUG + UULUUFDEVJVKVMVNABDUUGELGUCMNSVOVPUUIUUGVAUHYLUUJUUKWCUUFDEVQUUQUUGDLVANV + RVSVTYMKJWAUSUHMFWDZCDEJKMNOPWBFJMKOQWEWFZMVAUHYMMJWGOWHWIWJWKZYMUUEUAYNU + DUPZFUKZIUSZUUATZUVEYTTZVLZUBYNUQUUEYMUDYSMFYNYNVIZUVEYSTZUVIUUDUBYNUVKUV + GUUBUVHUUCUVEYSUUAIWLUVEYSYTWMWNWOYMUVDMUHZUIZUVIUBYNUVFUEUPZFUKZIUSZTZUV + EUVOTZVLZUVIUVMUEYTMFYNUVJUVOYTTZUVQUVGUVRUVHUVTUVPUUAUVFUVOYTIWPWQUVOYTU + VEWRWNYMUVLUVNMUHZUVSYMUVLUWAUIZUIZUVQUVRUWCUVQUVDDEUGZUVNDEUGZTUVDUVNFVB + UVRUWCUVFUWDUVPUWEYMUVLUVFUWDTUWACDUVDEFHIJKMNOPQRWSWTYMUWAUVPUWETUVLCDUV + NEFHIJKMNOPQRWSZXAXBCDUVDUVNEFJKMNOPQXCUWCUVDUVNFMYMUVAUWBUVBVEYMUVLUWAXD + XEXFXGXHXIXJXIXJUAUBYNYOIXKXLYMYRUULUVDIUSZTZUDYNVCZUCYOUQYQUVCYMUWIUCYOY + MUULYOUHZUWEUULTZUEMVCZUWIYMUWJUIYLUULNUHZUWLYMUWJYLUWMUWLXMZYMUWJDUULLVB + ZUWNYMUULVAUHYLUWJUWOWCUCXPUUPUULDLVANVRVSABDUULELGUEMNSVOXNXOXQYMUWLUWIY + MUWKUWIUEMYMUWAUIZUWEUWGTZUDYNVCZUWKUWIUWAUVOYNUHYMUWEUVPTUWRMUVNFFJKXRQX + SXTUWPUVPUWEUWFYFUDUVOYNUWGUVPUWEUVDUVOIWPYAYBUWKUWQUWHUDYNUWEUULUWGWMYCY + DYEYGYHXJUDUCYNYOIYIXLYNYOIYJXL $. $} ${ @@ -234979,9 +234973,9 @@ operation is a permutation group (group consisting of permutations), see $( For a permutation of a set, each element of the set is replaced by an(other) element of the set. (Contributed by AV, 2-Jan-2019.) $) symgmov2 $p |- ( Q e. P -> A. n e. N E. k e. N ( Q ` k ) = n ) $= - ( wcel wf1o wfo cv cfv wceq wrex wral csymg eqid symgbasf1o f1ofo foelrni - ralrimiva 3syl ) BAGEEBHEEBIZCJBKDJZLCEMZDENEABEOKZUEPFQEEBRUBUDDECEEBUCS - TUA $. + ( wcel wf1o wfo cv cfv wceq wrex wral csymg symgbasf1o foelcdmi ralrimiva + eqid f1ofo 3syl ) BAGEEBHEEBIZCJBKDJZLCEMZDENEABEOKZUESFPEEBTUBUDDECEEBUC + QRUA $. $} $( The base set of the symmetric group on the empty set is the singleton @@ -235298,18 +235292,18 @@ operation is a permutation group (group consisting of permutations), see vw cbs symggrp syl cv wf1o gapm wb elsymgbas mpbird fmptd df-3an sylan2br gaass anassrs mpteq2dva oveq1 mpteq2dv simprl simprr grpcl syl3anc mptexd w3a fvmptd3 ccom wf ffvelcdmd symgov syl2anc cxp gaf ad2antrr simpr oveq2 - fovrnd cbvmptv eqtrdi fmptco eqtrd 3eqtr4d isghmd ) CEHUBLMZUAUGEUCNZFUCN - ZEFDGFUHNZIWLOZWJOZWKOZCEHUDZWIHPMZFUEMCEHUFZHFPJUIUJWIAGBHAUKZBUKZCLZQZW - LDWIWSGMZRZXBWLMZHHXBULZBWSCXBEGHIXBOUMXDWQXEXFUNWIWQXCWRSHWLXBFPJWMUOUJU - PKUQZWIUAUKZGMZUGUKZGMZRZRZBHXHXJWJLZWTCLZQZBHXHXJWTCLZCLZQZXNDNXHDNZXJDN - ZWKLZXMBHXOXRWIXLWTHMZXOXRTZXLYCRWIXIXKYCVJYDXIXKYCURXHXJWTWJCEGHIWNUTUSV - AVBXMAXNXBXPGDPKWSXNTBHXAXOWSXNWTCVCVDXMEUEMZXIXKXNGMWIYEXLWPSWIXIXKVEZWI - XIXKVFZGWJEXHXJIWNVGVHXMBHXOPWIWQXLWRSZVIVKXMYBXTYAVLZXSXMXTWLMYAWLMYBYIT - XMGWLXHDWIGWLDVMXLXGSZYFVNXMGWLXJDYJYGVNHWLWKFXTYAJWMWOVOVPXMBAHHXQXHWSCL - ZXRYAXTXMYCRXJWTHGHCWIGHVQHCVMXLYCCEGHIVRVSXMXKYCYGSXMYCVTWBXMAXJXBBHXQQG - DPKWSXJTBHXAXQWSXJWTCVCVDYGXMBHXQPYHVIVKXMXTBHXHWTCLZQZAHYKQXMAXHXBYMGDPK - WSXHTBHXAYLWSXHWTCVCVDYFXMBHYLPYHVIVKBAHYLYKWTWSXHCWAWCWDWSXQXHCWAWEWFWGW - H $. + fovcdmd cbvmptv eqtrdi fmptco eqtrd 3eqtr4d isghmd ) CEHUBLMZUAUGEUCNZFUC + NZEFDGFUHNZIWLOZWJOZWKOZCEHUDZWIHPMZFUEMCEHUFZHFPJUIUJWIAGBHAUKZBUKZCLZQZ + WLDWIWSGMZRZXBWLMZHHXBULZBWSCXBEGHIXBOUMXDWQXEXFUNWIWQXCWRSHWLXBFPJWMUOUJ + UPKUQZWIUAUKZGMZUGUKZGMZRZRZBHXHXJWJLZWTCLZQZBHXHXJWTCLZCLZQZXNDNXHDNZXJD + NZWKLZXMBHXOXRWIXLWTHMZXOXRTZXLYCRWIXIXKYCVJYDXIXKYCURXHXJWTWJCEGHIWNUTUS + VAVBXMAXNXBXPGDPKWSXNTBHXAXOWSXNWTCVCVDXMEUEMZXIXKXNGMWIYEXLWPSWIXIXKVEZW + IXIXKVFZGWJEXHXJIWNVGVHXMBHXOPWIWQXLWRSZVIVKXMYBXTYAVLZXSXMXTWLMYAWLMYBYI + TXMGWLXHDWIGWLDVMXLXGSZYFVNXMGWLXJDYJYGVNHWLWKFXTYAJWMWOVOVPXMBAHHXQXHWSC + LZXRYAXTXMYCRXJWTHGHCWIGHVQHCVMXLYCCEGHIVRVSXMXKYCYGSXMYCVTWBXMAXJXBBHXQQ + GDPKWSXJTBHXAXQWSXJWTCVCVDYGXMBHXQPYHVIVKXMXTBHXHWTCLZQZAHYKQXMAXHXBYMGDP + KWSXHTBHXAYLWSXHWTCVCVDYFXMBHYLPYHVIVKBAHYLYKWTWSXHCWAWCWDWSXQXHCWAWEWFWG + WH $. $} ${ @@ -238620,42 +238614,42 @@ P pGrp ( G |`s S ) ) $= cfn sylanbrc fmpo adantr grpidcl simpr weq simpl oveq1d mpteq12dv rneqd mptex ovmpoa cid ssrab3 sselid sselda grplid mpteq2dva mptresid eqtr4di rnex rnresi eqtrdi wrex cab ovex oveq2 rnmpt abbidv ad2antrr jca grpass - 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X ( g .(+) x ) = y ) } $. @@ -239059,40 +239053,40 @@ P pGrp ( G |`s S ) ) $= ecss hashnncl mpbird pceq0 cuni csn c1o cen crab cmul ssrab2 ssfi sylancl oveq2 dvdsmul1 cqg orbsta2 syl21anc breqtrrd simprd breq2 biimpcd reximdv eqeq12d weq eleqtrd elsn sylc pcprmpw2 eqcomd c1 zcnd exp0d hash1 eqtr4di - df1o2 snfi eqeltri hashen bitrd en1b bitrdi cxp wf ad2antrr simprl fovrnd - gaf oveq1 eqeq1d rspcev gaorb syl3anbrc ovex simprr eqtr4d expr ralrimdva - sylbid syl5 sylbird ralbidv elrab2 baib sylibrd sylow2alem1 snssd eqsstrd - id adantl ex syld con1d ectocld impr sylan2b fsumdvds ) AMHUFZNUGZUHZDUIZ - UJUKZDFAUWKULUMUWMULUMAMUGZUWKAMULUMZUWPULUMSMUSUNAMHAGKMUOUPUMZMHUQZPBCG - HIKLMUAOURUTZVAZVBUWKUWLVCUTAFVDUMZFVEUMZAUXBLUJUKZFUBUIVFUPZVGZUBVHVIZAF - KVJVNZUXBUXGVKZQAKVLUMZLULUMZUXHUXIVOAUWRUXJPGKMVMUTRFUBKLOVPVQVRZVSZFVTU - TZAUWNUWMUMZVKZUWOUXPUWNULUMZUWOVHUMUXOAUWNUWKUMZUXQUWNUWKUWLWAAUXRVKZMUW - NAUWQUXRSWBUXSUWNMAUWKUWPUWNUXAWCWDVBWEUWNWFUTWGUXOAUXRUWNUWLUMZWHZVKFUWO - WIVNZUWNUWKUWLWJAUXRUYAUYBUCUIZHWKZNWLZWHZFUYDUJUKZWIVNZWSUYAUYBWSAUCUWNM - 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If ` G ` is a finite ` P ` -group that acts on the finite set ` Y ` , then the set @@ -239433,47 +239427,47 @@ P pGrp ( G |`s S ) ) $= weq vex ovmpoa cid cres ad2antrr slwsubg adantl sselda grplid grpsubid1 mptex eqtrd mpteq2dva mptresid eqtr4di rnresi eqtrdi wrex oveq2 abrexco cab simprr simplr rnmpt rexeqdv abbidv grpcl grpsubsub4 syl13anc grpass - adantlr grpaddsubass eqeq2d rexbidva 3eqtr4a 3eqtr4g fovrnd jca cbvmptv - 3eqtr4rd ralrimiva isga sylanbrc ) AHUGRZEHUHSZUIRZUJJUUGUKUUGGULZHUMUN - ZUAUOZGSZUUKTZUBUOZUCUOZFSZUUKGSZUUNUUOUUKGSZGSZTZUCJUQUBJUQZUJZUAUUGUQ - ZUJGHUUGUPSRAUUFUUHLEHUHURUSAUUIUVCADCUOZBUOZDUOZFSZUVEISZUTZVAZUUGRZCU - UGUQBJUQUUIAUVKBCJUUGAUVEJRZUVDUUGRZUJZUJZUVKUVJHVBUNZRZUVJVCUNZEEJVCUN - VDSVESZTZUVOUVDUVPRZUVLUVQUVOUWAUVDVCUNZUVSTZAUVMUWAUWCUJZUVLAUVMUWDAUU - FJVFRZEVGRZUVMUWDVJLMNEHUVDJKVKVHVIVLZVMZAUVLUVMVNZDUVEFUVDUVIHIJKOPUVI - VOZVPVQZUVOUWBUVRUVSUVOUWBUVRTZUVDUVJVRVSZUVOUWAUVLUWMUWHUWIDUVEFUVDUVI - HIJKOPUWJVTVQUVOUVDVFRUVJVFRUWLUWMVJUVOJUVDAUWEUVNMWAZUVOUWAUVDJWBUWHJU - 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BUURDUUKUYTUTZVAZVUAUYBUXTUWQUURVUHTUWRUXSUXTYDZAUWQUYAYEZBCUUOUUKJUUGU - VJVUHGBUCXBZUXGUJZUVIVUGVULDUVDUVHUUKUYTVUKUXGWOVULUVGUYSUVEUUOIVULUVEU - UOUVFFVUKUXGWPZWQVUMWRWSWTQVUGDUUKUYTUXJXMXAXDVQDUFUUKUYTVUGVUGVOYFXSYG - YHUYBUYQVUFUEUYBUYPVUEDUUKUYBUXLUJZUYIVUDUYLVUNUYHUUOISZUUNISZUYIVUDVUN - UUFUYHJRZUXTUXSVUPUYITUYBUUFUXLUWRUUFUYAUXDWAZWAZVUNUUFUUPJRZUXOVUQVUSU - YBVUTUXLUYBUUFUXSUXTVUTVURUWRUXSUXTVNZVUIJFHUUNUUOKOYIVHZWAUWRUXLUXOUYA - UXRYMZJFHUUPUVFKOYIVHUYBUXTUXLVUIWAZUYBUXSUXLVVAWAZJFHIUYHUUOUUNKOPYJYK - VUNVUOVUCUUNIVUNVUOUUNUYSFSZUUOISZVUCVUNUYHVVFUUOIVUNUUFUXSUXTUXOUYHVVF - TVUSVVEVVDVVCJFHUUNUUOUVFKOYLYKWQVUNUUFUXSUYSJRZUXTVVGVUCTVUSVVEVUNUUFU - XTUXOVVHVUSVVDVVCJFHUUOUVFKOYIVHVVDJFHIUUNUYSUUOKOPYNYKXNWQWIYOYPYHYQUD - UEUURUYEUYFUYFVOYFDUEUUKUYIUYJUYJVOYFYRUYBUXSUURUUGRUUSUYGTVVAUYBUUOUUK - UUGJUUGGAUUIUWQUYAUWPXGVUIVUJYSBCUUNUURJUUGUVJUYGGBUBXBZUVDUURTZUJZUVIU - YFVVKUVIDUURUUNUVFFSZUUNISZUTUYFVVKDUVDUVHUURVVMVVIVVJWOVVKUVGVVLUVEUUN - IVVKUVEUUNUVFFVVIVVJWPZWQVVNWRWSDUDUURVVMUYEDUDXBVVLUYDUUNIUVFUYCUUNFYA - WQUUAXSWTQUYFUDUURUYEUUOUUKGURXMXAXDVQUYBVUTUWQUUQUYKTVVBVUJBCUUPUUKJUU - GUVJUYKGUVEUUPTZUXGUJZUVIUYJVVPDUVDUVHUUKUYIVVOUXGWOVVPUVGUYHUVEUUPIVVP - UVEUUPUVFFVVOUXGWPZWQVVQWRWSWTQUYJDUUKUYIUXJXMXAXDVQUUBWJYTUUCYTUAUBUCF - GHJUUGUUJKOUXEUUDUUE $. + adantlr grpaddsubass eqeq2d rexbidva 3eqtr4a 3eqtr4g fovcdmd jca isga + cbvmptv 3eqtr4rd ralrimiva sylanbrc ) AHUGRZEHUHSZUIRZUJJUUGUKUUGGULZHU + MUNZUAUOZGSZUUKTZUBUOZUCUOZFSZUUKGSZUUNUUOUUKGSZGSZTZUCJUQUBJUQZUJZUAUU + GUQZUJGHUUGUPSRAUUFUUHLEHUHURUSAUUIUVCADCUOZBUOZDUOZFSZUVEISZUTZVAZUUGR + ZCUUGUQBJUQUUIAUVKBCJUUGAUVEJRZUVDUUGRZUJZUJZUVKUVJHVBUNZRZUVJVCUNZEEJV + CUNVDSVESZTZUVOUVDUVPRZUVLUVQUVOUWAUVDVCUNZUVSTZAUVMUWAUWCUJZUVLAUVMUWD + AUUFJVFRZEVGRZUVMUWDVJLMNEHUVDJKVKVHVIVLZVMZAUVLUVMVNZDUVEFUVDUVIHIJKOP + UVIVOZVPVQZUVOUWBUVRUVSUVOUWBUVRTZUVDUVJVRVSZUVOUWAUVLUWMUWHUWIDUVEFUVD + UVIHIJKOPUWJVTVQUVOUVDVFRUVJVFRUWLUWMVJUVOJUVDAUWEUVNMWAZUVOUWAUVDJWBUW + HJUVDHKWCWDWEUVOJUVJUWNUVOUVQUVJJWBUWKJUVJHKWCWDWEUVDUVJWFVQWGUVOUWAUWC + UWGWHWIAUVKUVQUVTUJVJZUVNAUUFUWEUWFUWOLMNEHUVJJKVKVHWAWLWJBCJUUGUVJUUGG + QWKWMZAUVBUAUUGAUUKUUGRZUJZUUMUVAUWRUULDUUKUUJUVFFSZUUJISZUTZVAZUUKUWRU + UJJRZUWQUULUXBTUWRUUFUXCAUUFUWQLWAZJHUUJKUUJVOZWNWDAUWQWOBCUUJUUKJUUGUV + JUXBGUVEUUJTZCUAXBZUJZUVIUXAUXHDUVDUVHUUKUWTUXFUXGWOUXHUVGUWSUVEUUJIUXH + UVEUUJUVFFUXFUXGWPZWQUXIWRWSWTQUXADUUKUWTUAXCZXMXAXDVQUWRUXBXEUUKXFZVAU + UKUWRUXAUXKUWRUXADUUKUVFUTUXKUWRDUUKUWTUVFUWRUVFUUKRZUJZUWTUVFUUJISZUVF + UXMUWSUVFUUJIUXMUUFUVFJRZUWSUVFTAUUFUWQUXLLXGZUWRUUKJUVFUWRUUKUVPRZUUKJ + WBUWQUXQAEHUUKXHXIJUUKHKWCWDXJZJFHUVFUUJKOUXEXKVQWQUXMUUFUXOUXNUVFTUXPU + XRJHIUVFUUJKUXEPXLVQXNXODUUKXPXQWTUUKXRXSXNUWRUUTUBUCJJUWRUUNJRZUUOJRZU + JZUJZUDUURUUNUDUOZFSZUUNISZUTZVAZDUUKUUPUVFFSZUUPISZUTZVAZUUSUUQUYBUEUO + ZUYETZUDUURXTZUEYCZUYLUYITZDUUKXTZUEYCZUYGUYKUYBUYMUDUFUOUUOUVFFSZUUOIS + ZTDUUKXTUFYCZXTZUEYCUYLUUNUYTFSZUUNISZTZDUUKXTZUEYCUYOUYRUEUDUFDUUKUYTU + YEVUDUYSUUOIURUYCUYTTUYDVUCUUNIUYCUYTUUNFYAWQYBUYBUYNVUBUEUYBUYMUDUURVU + AUYBUURDUUKUYTUTZVAZVUAUYBUXTUWQUURVUHTUWRUXSUXTYDZAUWQUYAYEZBCUUOUUKJU + UGUVJVUHGBUCXBZUXGUJZUVIVUGVULDUVDUVHUUKUYTVUKUXGWOVULUVGUYSUVEUUOIVULU + VEUUOUVFFVUKUXGWPZWQVUMWRWSWTQVUGDUUKUYTUXJXMXAXDVQDUFUUKUYTVUGVUGVOYFX + SYGYHUYBUYQVUFUEUYBUYPVUEDUUKUYBUXLUJZUYIVUDUYLVUNUYHUUOISZUUNISZUYIVUD + VUNUUFUYHJRZUXTUXSVUPUYITUYBUUFUXLUWRUUFUYAUXDWAZWAZVUNUUFUUPJRZUXOVUQV + USUYBVUTUXLUYBUUFUXSUXTVUTVURUWRUXSUXTVNZVUIJFHUUNUUOKOYIVHZWAUWRUXLUXO + UYAUXRYMZJFHUUPUVFKOYIVHUYBUXTUXLVUIWAZUYBUXSUXLVVAWAZJFHIUYHUUOUUNKOPY + JYKVUNVUOVUCUUNIVUNVUOUUNUYSFSZUUOISZVUCVUNUYHVVFUUOIVUNUUFUXSUXTUXOUYH + VVFTVUSVVEVVDVVCJFHUUNUUOUVFKOYLYKWQVUNUUFUXSUYSJRZUXTVVGVUCTVUSVVEVUNU + UFUXTUXOVVHVUSVVDVVCJFHUUOUVFKOYIVHVVDJFHIUUNUYSUUOKOPYNYKXNWQWIYOYPYHY + QUDUEUURUYEUYFUYFVOYFDUEUUKUYIUYJUYJVOYFYRUYBUXSUURUUGRUUSUYGTVVAUYBUUO + UUKUUGJUUGGAUUIUWQUYAUWPXGVUIVUJYSBCUUNUURJUUGUVJUYGGBUBXBZUVDUURTZUJZU + VIUYFVVKUVIDUURUUNUVFFSZUUNISZUTUYFVVKDUVDUVHUURVVMVVIVVJWOVVKUVGVVLUVE + UUNIVVKUVEUUNUVFFVVIVVJWPZWQVVNWRWSDUDUURVVMUYEDUDXBVVLUYDUUNIUVFUYCUUN + FYAWQUUBXSWTQUYFUDUURUYEUUOUUKGURXMXAXDVQUYBVUTUWQUUQUYKTVVBVUJBCUUPUUK + JUUGUVJUYKGUVEUUPTZUXGUJZUVIUYJVVPDUVDUVHUUKUYIVVOUXGWOVVPUVGUYHUVEUUPI + VVPUVEUUPUVFFVVOUXGWPZWQVVQWRWSWTQUYJDUUKUYIUXJXMXAXDVQUUCWJYTUUDYTUAUB + UCFGHJUUGUUJKOUXEUUAUUE $. sylow3lem2.k $e |- ( ph -> K e. ( P pSyl G ) ) $. sylow3lem2.h $e |- H = { u e. X | ( u .(+) K ) = K } $. @@ -241965,73 +241959,73 @@ an extension of the previous (inserting an element and its inverse at wb cs2 cmpo efgtf ffn 3syl adantr caddc cpfx syl ccatass syl3anc oveq1d efgtval eqtrd oveq2d cuz cn0 lencl nn0uz uzaddcl ccatlen nn0cnd addcomd eleqtrdi eqeltrd c0 vex erref ex pm4.71d w3a df-3an anidm anbi1i 3bitri - bitr4di iserd ovelrn ad2antlr simprl simprr cop csubstr pfxcl ffvelcdmi - fovrnd efgmf swrdcl splval syl13anc 3eqtr4rd elfznn0 cz elfzuz3 eluzadd - s2cld nn0zd 3eltr3d elfzuzb sylanbrc wrd0 ccatrid eluzfz2 ccatpfx pfxid - fveq2d 3eqtr2rd pfxlen eqtr2d nn0addcld addid1d eqtr2id splval2 3eqtr4d - hash0 oveq2i fnovrn efgi2 elec breq2 bitrid syl5ibrcom rexlimdvva ssrdv - sylbid ralrimiva fvexi erex mpisyl ereq1 eceq2 sseq2d ralbidv mpbir2and - anbi12d elabg intss1 eqsstrid ) GUAUQZHUAUQZURZJUAUIVGZUSZUJVGZLUTZVAZU - XQUXOVBZVCZUJUAVDZURZUIVHZVEZSUJBCDEJLQRTUAUIUBUCUDUEVFUXNSUYDUQZUYESVC - 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freeGrp ` I ) $. frgpup.x $e |- X = ( Base ` G ) $. @@ -243179,15 +243173,15 @@ an extension of the previous (inserting an element and its inverse at ghmcmn $p |- ( ph -> H e. CMnd ) $= ( wcel co cfv vi vj va vb cmnd wceq wral ccmn cmnmnd syl mhmmnd simp-6l cv wa simp-4r simplr cmncom syl3anc fveq2d syl3an1 mhmlem 3eqtr3d simpr - simpllr oveq12d wrex wfo foelrni sylan r19.29a adantr anasss ralrimivva - ad5ant13 iscmn sylanbrc ) AHUERUAUMZUBUMZESZVRVQESZUFZUBJUGUAJUGHUHRABC - DEFGHIJOKLMNPAGUHRZGUERQGUIUJUKAWAUAUBJJAVQJRZVRJRZWAAWCUNZWDUNZUCUMZFT - ZVQUFZWAUCIWFWGIRZUNZWIUNZUDUMZFTZVRUFZWAUDIWLWMIRZUNZWOUNZWHWNESZWNWHE - SZVSVTWRWGWMDSZFTWMWGDSZFTWSWTWRXAXBFWRWBWJWPXAXBUFWRAWBAWCWDWJWIWPWOUL - ZQUJWFWJWIWPWOUOZWLWPWOUPZIDGWGWMKMUQURUSWRBCWGWMDEFIWRABUMZIRCUMZIRXFX - GDSFTXFFTXGFTESUFXCOUTZXDXEVAWRBCWMWGDEFIXHXEXDVAVBWRWHVQWNVREWKWIWPWOV - DZWQWOVCZVEWRWNVRWHVQEXJXIVEVBAWDWOUDIVFZWCWJWIAIJFVGZWDXKPUDIJFVRVHVIV - NVJWEWIUCIVFZWDAXLWCXMPUCIJFVQVHVIVKVJVLVMUAUBJEHLNVOVP $. + simpllr oveq12d wrex foelcdmi ad5ant13 r19.29a adantr anasss ralrimivva + wfo sylan iscmn sylanbrc ) AHUERUAUMZUBUMZESZVRVQESZUFZUBJUGUAJUGHUHRAB + CDEFGHIJOKLMNPAGUHRZGUERQGUIUJUKAWAUAUBJJAVQJRZVRJRZWAAWCUNZWDUNZUCUMZF + TZVQUFZWAUCIWFWGIRZUNZWIUNZUDUMZFTZVRUFZWAUDIWLWMIRZUNZWOUNZWHWNESZWNWH + ESZVSVTWRWGWMDSZFTWMWGDSZFTWSWTWRXAXBFWRWBWJWPXAXBUFWRAWBAWCWDWJWIWPWOU + LZQUJWFWJWIWPWOUOZWLWPWOUPZIDGWGWMKMUQURUSWRBCWGWMDEFIWRABUMZIRCUMZIRXF + XGDSFTXFFTXGFTESUFXCOUTZXDXEVAWRBCWMWGDEFIXHXEXDVAVBWRWHVQWNVREWKWIWPWO + VDZWQWOVCZVEWRWNVRWHVQEXJXIVEVBAWDWOUDIVFZWCWJWIAIJFVMZWDXKPUDIJFVRVGVN + VHVIWEWIUCIVFZWDAXLWCXMPUCIJFVQVGVNVJVIVKVLUAUBJEHLNVOVP $. $} ghmabl.3 $e |- ( ph -> G e. Abel ) $. @@ -246333,14 +246327,14 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements gsumxp2 $p |- ( ph -> ( G gsum ( k e. C |-> ( G gsum ( j e. A |-> ( j F k ) ) ) ) ) = ( G gsum ( j e. A |-> ( G gsum ( k e. C |-> ( j F k ) ) ) ) ) ) $= - ( co wcel cv cmpt cgsu csupp fovrnda fsuppimpd wa wbr wn wceq cop cfv cxp - simpl opelxpi ad2antlr simpr eldifd cvv ssidd xpexd c0g fvexi a1i suppssr - cdif syl2an2r ex df-br notbii df-ov eqeq1i 3imtr4g impr gsumcom3 eqcomd ) - AHEBHFDEUAZFUAZGSZUBUCSUBUCSHFDHEBVSUBUCSUBUCSABCDGKUDSZEFHIJVSKLMNOPAVQV - RCBDGQUEAGKRUFAVQBTVRDTUGZVQVRVTUHZUIZVSKUJZAWAUGZVQVRUKZVTTZUIZWFGULZKUJ - ZWCWDWEWHWJWEAWHWFBDUMZVTVFTWJAWAUNWEWHUGWFWKVTWAWFWKTAWHVQVRBDUOUPWEWHUQ - URAWKCUSGUSVTWFKQAVTUTABDIJOPVAKUSTAKHVBMVCVDVEVGVHWBWGVQVRVTVIVJVSWIKVQV - RGVKVLVMVNVOVP $. + ( co wcel cv cmpt cgsu csupp fovcdmda fsuppimpd wbr wceq cop cfv cxp cdif + wa wn simpl opelxpi ad2antlr simpr eldifd cvv ssidd xpexd c0g a1i suppssr + fvexi syl2an2r ex df-br notbii df-ov eqeq1i 3imtr4g impr gsumcom3 eqcomd + ) AHEBHFDEUAZFUAZGSZUBUCSUBUCSHFDHEBVSUBUCSUBUCSABCDGKUDSZEFHIJVSKLMNOPAV + QVRCBDGQUEAGKRUFAVQBTVRDTUMZVQVRVTUGZUNZVSKUHZAWAUMZVQVRUIZVTTZUNZWFGUJZK + UHZWCWDWEWHWJWEAWHWFBDUKZVTULTWJAWAUOWEWHUMWFWKVTWAWFWKTAWHVQVRBDUPUQWEWH + URUSAWKCUTGUTVTWFKQAVTVAABDIJOPVBKUTTAKHVCMVFVDVEVGVHWBWGVQVRVTVIVJVSWIKV + QVRGVKVLVMVNVOVP $. $} ${ @@ -252083,72 +252077,72 @@ the additive structure must be abelian (see ~ ringcom ), care must be imasring $p |- ( ph -> ( U e. Ring /\ ( F ` .1. ) = ( 1r ` U ) ) ) $= ( vu vv vw vx vy vz crg wcel cfv cur wceq cplusg cmulr imasbas eqidd cgrp c0g a1i ringgrp syl eqid imasgrp simpld cxp wf cv co wa cbs adantr simprl - eleqtrd simprr ringcl syl3anc eleqtrrd caovclg imasmulf fovrn syl3an1 w3a - wrex crn wfo forn eleq2d 3anbi123d wb fofn fvelrnb bitr3d 3reeanv bitr4di + eleqtrd simprr ringcl syl3anc eleqtrrd caovclg fovcdm syl3an1 w3a crn wfo + imasmulf wrex forn eleq2d 3anbi123d wfn wb fvelrnb bitr3d 3reeanv bitr4di wi simp2 3ad2ant1 3adant3r3 simp3 simpr3 ringass syl13anc fveq2d 3adantr3 - wfn simpl imasmulval simpr1 3eqtr4d oveq1d 3adant3r1 oveq2d simp1 oveq12d - 3adantr1 eqeq12d syl5ibcom 3exp2 imp32 rexlimdv rexlimdvva sylbid ringacl - imp ringdi 3adantr2 imasaddval 3adant3r2 ringdir ringidcl ffvelcdmd simpr - fof biimpa ringlidm syl2an2r eqtrd oveq2 rexlimdva mpd3an3 ringridm oveq1 - id isringd wral jcad sylbird ralrimiv isringid mpbi2and eqcomd jca ) AFUI - 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W /\ X e. V ) /\ ( N e. Fin /\ M e. Fin ) /\ ( I e. N /\ J e. M ) ) -> ( I X J ) e. B ) $= ( wcel wa cfn w3a cxp cmap syl cbs cfv eleq2i biimpi adantl 3ad2ant1 wceq - co wf xpfi anim12i 3adant3 frlmfibas eleqtrrd elmapi simp3l simp3r fovrnd - simpl ) BINZJHNZOZGPNFPNOZDGNZEFNZOZQZDEAGFJVGJAGFRZSUHZNVHAJUIVGJCUAUBZV - IVBVCJVJNZVFVAVKUTVAVKHVJJMUCUDUEUFVGUTVHPNZOZVIVJUGVBVCVMVFVBUTVCVLUTVAU - SGFUJUKULBCVHAIKLUMTUNJAVHUOTVBVCVDVEUPVBVCVDVEUQUR $. + co wf simpl xpfi anim12i 3adant3 frlmfibas eleqtrrd elmapi simp3l fovcdmd + simp3r ) BINZJHNZOZGPNFPNOZDGNZEFNZOZQZDEAGFJVGJAGFRZSUHZNVHAJUIVGJCUAUBZ + VIVBVCJVJNZVFVAVKUTVAVKHVJJMUCUDUEUFVGUTVHPNZOZVIVJUGVBVCVMVFVBUTVCVLUTVA + UJGFUKULUMBCVHAIKLUNTUOJAVHUPTVBVCVDVEUQVBVCVDVEUSUR $. $} ${ @@ -269394,38 +269388,38 @@ is used for the (special) unit vectors forming the standard basis of free ( vb va wcel cfv co cmpt eqid wceq crg w3a cv cmulr cgsu cof cbs frlmbasf wf 3adant1 feqmptd wa c0g cmnd simpl1 ringmnd syl simpl2 simpr ffvelcdmda csn cxp uvcff 3adant3 frlmvscafval adantr syl2anc fconstmpt offval2 eqtrd - a1i clmod csca frlmlmod frlmsca fveq2d eleqtrd lmodvscl syl3anc fvmptelrn - 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O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) $) mamucl $p |- ( ph -> ( X F Y ) e. ( B ^m ( M X. 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N ) X. { X } ) oF .x. Y ) F Z ) = ( ( ( M X. O ) X. { X } ) oF .x. ( Y F Z ) ) ) $= ( vi vk vj cxp csn cof co wceq wral wcel cmpt cgsu cplusg cfv cfn c0g - cv wa eqid crg adantr ad2antrr cmap elmapi syl simplrl fovrnd simplrr - wf simpr ringcl syl3anc ovexd fvexd fsuppmptdm gsummulc2 df-ov simprl - cvv cop opelxpi sylan xpfi syl2anc wfn ffn 3syl eqcomi a1i ofc1 mpdan - eqtrid oveq1d ringass syl13anc mpteq2dva oveq2d simprr mamufv 3eqtr4d - eqtrd fconst6g wb cbs elmapg sylancr mpbird ringvcl adantl ralrimivva - fvexi mamucl eqfnov2 ) AFGUEZIUFZUEZJDUGZUHZKEUHZFHUEZXPUEZJKEUHZXRUH - ZUIZUBURZUCURZXTUHZYFYGYDUHZUIZUCHUJUBFUJZAYJUBUCFHAYFFUKZYGHUKZUSZUS - ZCUDGYFUDURZXSUHZYPYGKUHZDUHZULZUMUHZIYFYGYCUHZDUHZYHYIYOCUDGIYFYPJUH - ZYRDUHZDUHZULZUMUHICUDGUUEULZUMUHZDUHUUAUUCYOGBCUNUOZCDUDUPUUEICUQUOZ - LUUKUTUUJUTRACVAUKZYNMVBZAGUPUKZYNPVBZAIBUKZYNSVBZYOYPGUKZUSZUULUUDBU - KZYRBUKZUUEBUKAUULYNUURMVCZUUSYFYPBFGJAXOBJVJZYNUURAJBXOVDUHZUKZUVCTJ - BXOVEZVFVCAYLYMUURVGYOUURVKZVHZUUSYPYGBGHKAGHUEZBKVJZYNUURAKBUVIVDUHU - KZUVJUAKBUVIVEVFVCUVGAYLYMUURVIVHZBCDUUDYRLRVLVMYOUDGUUHVTVTUUEUUKUUH - UTUUOUUSUUDYRDVNYOCUQVOVPVQYOYTUUGCUMYOUDGYSUUFUUSYSIUUDDUHZYRDUHZUUF - UUSYQUVMYRDUUSYQYFYPWAZXSUOZUVMYFYPXSVRUUSUVOXOUKZUVPUVMUIYOYLUURUVQA - YLYMVSZYFYPFGWBWCUUSXOIUUDDJUPBUVOAXOUPUKZYNUURAFUPUKZUUNUVSOPFGWDWEZ - VCAUUPYNUURSVCZAJXOWFZYNUURAUVEUVCUWCTUVFXOBJWGWHVCUVOJUOZUUDUIUUSUVQ - USUUDUWDYFYPJVRWIWJWKWLWMWNUUSUULUUPUUTUVAUVNUUFUIUVBUWBUVHUVLBCDIUUD - YRLRWOWPXBWQWRYOUUBUUIIDYOBHCDUDEYFYGFGVAJKNLRUUMAUVTYNOVBZUUOAHUPUKZ - YNQVBZAUVEYNTVBAUVKYNUAVBZUVRAYLYMWSZWTWRXAYOBHCDUDEYFYGFGVAXSKNLRUUM - UWEUUOUWGAXSUVDUKZYNAUULXQUVDUKZUVEUWJMAUWKXOBXQVJZAUUPUWLSXOIBXCVFAB - VTUKZUVSUWKUWLXDBCXELXLZUWABXOXQVTUPXFXGXHTBCDXOXQJLRXIVMZVBUWHUVRUWI - WTYOYIYFYGWAZYDUOZUUCYFYGYDVRYOUWPYAUKZUWQUUCUIYNUWRAYFYGFHWBXJYOYAIU - UBDYCUPBUWPAYAUPUKZYNAUVTUWFUWSOQFHWDWEZVBUUQAYCYAWFZYNAYCBYAVDUHZUKZ - YABYCVJUXAABHCEFGJKLMNOPQTUAXMZYCBYAVEYABYCWGWHVBUWPYCUOZUUBUIYOUWRUS - UUBUXEYFYGYCVRWIWJWKWLWMXAXKAXTYAWFZYDYAWFZYEYKXDAXTUXBUKYABXTVJUXFAB - HCEFGXSKLMNOPQUWOUAXMXTBYAVEYABXTWGWHAYDUXBUKZYABYDVJUXGAUULYBUXBUKZU - XCUXHMAUXIYABYBVJZAUUPUXJSYAIBXCVFAUWMUWSUXIUXJXDUWNUWTBYAYBVTUPXFXGX - HUXDBCDYAYBYCLRXIVMYDBYAVEYABYDWGWHUBUCFHXTYDXNWEXH $. + cv wa eqid crg adantr ad2antrr wf cmap elmapi simplrl fovcdmd simplrr + syl simpr ringcl syl3anc cvv ovexd fvexd fsuppmptdm gsummulc2 opelxpi + cop df-ov simprl sylan xpfi syl2anc wfn 3syl eqcomi ofc1 mpdan eqtrid + ffn a1i oveq1d ringass syl13anc eqtrd mpteq2dva oveq2d simprr 3eqtr4d + mamufv fconst6g cbs fvexi elmapg sylancr mpbird ringvcl adantl mamucl + wb ralrimivva eqfnov2 ) AFGUEZIUFZUEZJDUGZUHZKEUHZFHUEZXPUEZJKEUHZXRU + HZUIZUBURZUCURZXTUHZYFYGYDUHZUIZUCHUJUBFUJZAYJUBUCFHAYFFUKZYGHUKZUSZU + SZCUDGYFUDURZXSUHZYPYGKUHZDUHZULZUMUHZIYFYGYCUHZDUHZYHYIYOCUDGIYFYPJU + HZYRDUHZDUHZULZUMUHICUDGUUEULZUMUHZDUHUUAUUCYOGBCUNUOZCDUDUPUUEICUQUO + ZLUUKUTUUJUTRACVAUKZYNMVBZAGUPUKZYNPVBZAIBUKZYNSVBZYOYPGUKZUSZUULUUDB + UKZYRBUKZUUEBUKAUULYNUURMVCZUUSYFYPBFGJAXOBJVDZYNUURAJBXOVEUHZUKZUVCT + JBXOVFZVJVCAYLYMUURVGYOUURVKZVHZUUSYPYGBGHKAGHUEZBKVDZYNUURAKBUVIVEUH + UKZUVJUAKBUVIVFVJVCUVGAYLYMUURVIVHZBCDUUDYRLRVLVMYOUDGUUHVNVNUUEUUKUU + HUTUUOUUSUUDYRDVOYOCUQVPVQVRYOYTUUGCUMYOUDGYSUUFUUSYSIUUDDUHZYRDUHZUU + FUUSYQUVMYRDUUSYQYFYPVTZXSUOZUVMYFYPXSWAUUSUVOXOUKZUVPUVMUIYOYLUURUVQ + AYLYMWBZYFYPFGVSWCUUSXOIUUDDJUPBUVOAXOUPUKZYNUURAFUPUKZUUNUVSOPFGWDWE + ZVCAUUPYNUURSVCZAJXOWFZYNUURAUVEUVCUWCTUVFXOBJWLWGVCUVOJUOZUUDUIUUSUV + QUSUUDUWDYFYPJWAWHWMWIWJWKWNUUSUULUUPUUTUVAUVNUUFUIUVBUWBUVHUVLBCDIUU + DYRLRWOWPWQWRWSYOUUBUUIIDYOBHCDUDEYFYGFGVAJKNLRUUMAUVTYNOVBZUUOAHUPUK + ZYNQVBZAUVEYNTVBAUVKYNUAVBZUVRAYLYMWTZXBWSXAYOBHCDUDEYFYGFGVAXSKNLRUU + MUWEUUOUWGAXSUVDUKZYNAUULXQUVDUKZUVEUWJMAUWKXOBXQVDZAUUPUWLSXOIBXCVJA + BVNUKZUVSUWKUWLXLBCXDLXEZUWABXOXQVNUPXFXGXHTBCDXOXQJLRXIVMZVBUWHUVRUW + IXBYOYIYFYGVTZYDUOZUUCYFYGYDWAYOUWPYAUKZUWQUUCUIYNUWRAYFYGFHVSXJYOYAI + UUBDYCUPBUWPAYAUPUKZYNAUVTUWFUWSOQFHWDWEZVBUUQAYCYAWFZYNAYCBYAVEUHZUK + ZYABYCVDUXAABHCEFGJKLMNOPQTUAXKZYCBYAVFYABYCWLWGVBUWPYCUOZUUBUIYOUWRU + SUUBUXEYFYGYCWAWHWMWIWJWKXAXMAXTYAWFZYDYAWFZYEYKXLAXTUXBUKYABXTVDUXFA + BHCEFGXSKLMNOPQUWOUAXKXTBYAVFYABXTWLWGAYDUXBUKZYABYDVDUXGAUULYBUXBUKZ + UXCUXHMAUXIYABYBVDZAUUPUXJSYAIBXCVJAUWMUWSUXIUXJXLUWNUWTBYAYBVNUPXFXG + XHUXDBCDYAYBYCLRXIVMYDBYAVFYABYDWLWGUBUCFHXTYDXNWEXH $. $} $} $} @@ -280659,35 +280654,35 @@ represented as an element of (the base set of) ` ( ( 1 ... n ) Mat R ) ` . ( ( ( M X. O ) X. { Y } ) oF .x. ( X F Z ) ) ) $= ( vi vk vj cxp csn cof co wceq cv wral wcel cmpt cgsu cop cfv df-ov simpr simplrr opelxpi syl2anc cfn xpfi ad2antrr wfn cmap elmapi ffn 3syl eqcomi - wa wf a1i ofc1 mpdan eqtrid oveq2d cmgp ccmn ccrg eqid crngmgp syl fovrnd - mgpbas mgpplusg cmn12 syl13anc eqtrd mpteq2dva cplusg c0g crngring adantr - simplrl crg ringcl syl3anc cvv ovexd fsuppmptdm gsummulc2 fconst6g wb cbs - elmapg sylancr mpbird ringvcl simprl simprr mamufv adantl eqtr3id 3eqtr4d - fvexd fvexi mamucl ralrimivva eqfnov2 ) AIGHUEZJUFZUEZKDUGZUHZEUHZFHUEZYB - UEZIKEUHZYDUHZUIZUBUJZUCUJZYFUHZYLYMYJUHZUIZUCHUKUBFUKZAYPUBUCFHAYLFULZYM - HULZVKZVKZCUDGYLUDUJZIUHZUUBYMYEUHZDUHZUMZUNUHZJCUDGUUCUUBYMKUHZDUHZUMZUN - UHZDUHZYNYOUUAUUGCUDGJUUIDUHZUMZUNUHUULUUAUUFUUNCUNUUAUDGUUEUUMUUAUUBGULZ - VKZUUEUUCJUUHDUHZDUHZUUMUUPUUDUUQUUCDUUPUUDUUBYMUOZYEUPZUUQUUBYMYEUQUUPUU - SYAULZUUTUUQUIUUPUUOYSUVAUUAUUOURZAYRYSUUOUSZUUBYMGHUTVAUUPYAJUUHDKVBBUUS - AYAVBULZYTUUOAGVBULZHVBULZUVDQRGHVCVAZVDAJBULZYTUUOTVDZAKYAVEZYTUUOAKBYAV - 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[Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.) $) mamutpos $p |- ( ph -> tpos ( X F Y ) = ( tpos Y G tpos X ) ) $= ( co vj vi vk cv cmulr cfv cmpt cgsu cmpo ctpos eqid tposmpo wcel wa ccrg - w3a wceq simpl1 syl cxp cmap wf elmapi simpl3 simpr fovrnd simpl2 crngcom + w3a wceq simpl1 syl cxp cmap wf elmapi 3syl simpl3 fovcdmd simpl2 crngcom syl3anc ovtpos oveq12i eqtr4di mpteq2dva oveq2d mpoeq3dva mamuval tposeqd - 3syl eqtrid tposmap 3eqtr4d ) AUAUBGCDUCHUAUDZUCUDZITZWCUBUDZJTZDUEUFZTZU - GZUHTZUIZUJZUBUACGDUCHWEWCJUJZTZWCWBIUJZTZWGTZUGZUHTZUIZIJETZUJWMWOFTAWLU - BUACGWJUIWTUAUBGCWJWKWKUKULAUBUACGWJWSAWECUMZWBGUMZUPZWIWRDUHXDUCHWHWQXDW - CHUMZUNZWHWFWDWGTZWQXFDUOUMZWDBUMWFBUMWHXGUQXFAXHAXBXCXEURZNUSXFWBWCBGHIX - FAIBGHUTZVATUMZXJBIVBXIRIBXJVCVRAXBXCXEVDXDXEVEZVFXFWCWEBHCJXFAJBHCUTZVAT - UMZXMBJVBXISJBXMVCVRXLAXBXCXEVGVFBDWGWDWFMWGUKZVHVIWNWFWPWDWGWEWCJVJWCWBI - VJVKVLVMVNVOVSAXAWKABCDWGUAUCUBEGHUOIJKMXONOPQRSVPVQABGDWGUBUCUAFCHUOWMWO - LMXONQPOAXNWMBCHUTVATUMSJBHCVTUSAXKWOBHGUTVATUMRIBGHVTUSVPWA $. + simpr eqtrid tposmap 3eqtr4d ) AUAUBGCDUCHUAUDZUCUDZITZWCUBUDZJTZDUEUFZTZ + UGZUHTZUIZUJZUBUACGDUCHWEWCJUJZTZWCWBIUJZTZWGTZUGZUHTZUIZIJETZUJWMWOFTAWL + UBUACGWJUIWTUAUBGCWJWKWKUKULAUBUACGWJWSAWECUMZWBGUMZUPZWIWRDUHXDUCHWHWQXD + WCHUMZUNZWHWFWDWGTZWQXFDUOUMZWDBUMWFBUMWHXGUQXFAXHAXBXCXEURZNUSXFWBWCBGHI + XFAIBGHUTZVATUMZXJBIVBXIRIBXJVCVDAXBXCXEVEXDXEVRZVFXFWCWEBHCJXFAJBHCUTZVA + TUMZXMBJVBXISJBXMVCVDXLAXBXCXEVGVFBDWGWDWFMWGUKZVHVIWNWFWPWDWGWEWCJVJWCWB + IVJVKVLVMVNVOVSAXAWKABCDWGUAUCUBEGHUOIJKMXONOPQRSVPVQABGDWGUBUCUAFCHUOWMW + OLMXONQPOAXNWMBCHUTVATUMSJBHCVTUSAXKWOBHGUTVATUMRIBGHVTUSVPWA $. $} ${ @@ -281601,10 +281596,10 @@ represented as an element of (the base set of) ` ( ( 1 ... n ) Mat R ) ` . matgsumcl $p |- ( ( R e. CRing /\ M e. B ) -> ( U gsum ( r e. N |-> ( r M r ) ) ) e. ( Base ` R ) ) $= ( ccrg wcel wa cbs cfv cv co eqid adantr simpr mgpbas ccmn crngmgp matrcl - cfn cvv adantl simpld cxp cmap matbas2i elmapi fovrnd ralrimiva gsummptcl - wf 3syl ) CKLZEBLZMZCNOZGDFGPZVBEQZVACDJVARZUAURDUBLUSCDJUCSUTFUELZCUFLZU - SVEVFMURABCFEHIUDUGUHUTVCVALGFUTVBFLZMVBVBVAFFEUTFFUIZVAEUPZVGUTUSEVAVHUJ - QLVIURUSTABCVAEFHVDIUKEVAVHULUQSUTVGTZVJUMUNUO $. + cfn cvv adantl simpld wf cmap matbas2i elmapi fovcdmd ralrimiva gsummptcl + cxp 3syl ) CKLZEBLZMZCNOZGDFGPZVBEQZVACDJVARZUAURDUBLUSCDJUCSUTFUELZCUFLZ + USVEVFMURABCFEHIUDUGUHUTVCVALGFUTVBFLZMVBVBVAFFEUTFFUPZVAEUIZVGUTUSEVAVHU + JQLVIURUSTABCVAEFHVDIUKEVAVHULUQSUTVGTZVJUMUNUO $. $d P r $. madetsumid.y $e |- Y = ( ZRHom ` R ) $. @@ -283168,14 +283163,14 @@ seems more natural to use the definition of a free (left) module, avoiding to mavmulcl $p |- ( ph -> ( X .X. Y ) e. ( B ^m N ) ) $= ( vi co wcel vj cv cfv cmpt cgsu cmap crg mavmulval wral ccmn ringcmn syl wa adantr cfn ad2antrr cxp cbs wceq matbas2 syl2anc eleqtrrd elmapi simpr - wf fovrnd ffvelcdmd ringcl syl3anc ralrimiva gsummptcl eqid fmpt wb fvexi - cvv elmapg sylancr bitr4id mpbid eqeltrd ) AHIFSRGDUAGRUBZUAUBZHSZWCIUCZE - SZUDUESZUDZCGUFSZABCDEFRUAGUGHIJKLMNOPQUHAWGCTZRGUIZWHWITZAWJRGAWBGTZUMZC - UADGWFLADUJTZWMADUGTZWONDUKULUNAGUOTZWMOUNWNWFCTZUAGWNWCGTZUMZWPWDCTWECTW - RAWPWMWSNUPWTWBWCCGGHAGGUQZCHVEZWMWSAHCXAUFSZTXBAHBURUCZXCPAWQWPXCXDUSONB - DCGUGJLUTVAVBHCXAVCULUPWNWMWSAWMVDUNWNWSVDZVFWTGCWCIAGCIVEZWMWSAIWITXFQIC - GVCULUPXEVGCDEWDWELMVHVIVJVKVJAWKGCWHVEZWLRGCWGWHWHVLVMACVPTWQWLXGVNCDURL - VOOCGWHVPUOVQVRVSVTWA $. + wf fovcdmd ffvelcdmd ringcl syl3anc ralrimiva gsummptcl eqid cvv wb fvexi + fmpt elmapg sylancr bitr4id mpbid eqeltrd ) AHIFSRGDUAGRUBZUAUBZHSZWCIUCZ + ESZUDUESZUDZCGUFSZABCDEFRUAGUGHIJKLMNOPQUHAWGCTZRGUIZWHWITZAWJRGAWBGTZUMZ + CUADGWFLADUJTZWMADUGTZWONDUKULUNAGUOTZWMOUNWNWFCTZUAGWNWCGTZUMZWPWDCTWECT + WRAWPWMWSNUPWTWBWCCGGHAGGUQZCHVEZWMWSAHCXAUFSZTXBAHBURUCZXCPAWQWPXCXDUSON + BDCGUGJLUTVAVBHCXAVCULUPWNWMWSAWMVDUNWNWSVDZVFWTGCWCIAGCIVEZWMWSAIWITXFQI + CGVCULUPXEVGCDEWDWELMVHVIVJVKVJAWKGCWHVEZWLRGCWGWHWHVLVPACVMTWQWLXGVNCDUR + LVOOCGWHVMUOVQVRVSVTWA $. $} ${ @@ -283226,35 +283221,35 @@ seems more natural to use the definition of a free (left) module, avoiding to mavmulass $p |- ( ph -> ( ( X .X. Z ) .x. Y ) = ( X .x. ( Z .x. Y ) ) ) $= ( co vi vj vk cmap wcel wf wfn cmulr cfv eqid cxp cbs cfn matbas2 syl2anc crg wceq eleqtrrd mamucl eleqtrd mavmulcl elmapi ffn 3syl cv wa cmpt cgsu - ccmn ringcmn syl adantr ad2antrr simplr simprr fovrnd simprl ffvelcdm imp - wi ex ad2ant2r ringcl syl3anc gsumcom3fi mamufv oveq1d cplusg c0g adantlr - simpr ovexd fvexd fsuppmptdm gsummulc1 ringass syl13anc anassrs mpteq2dva - cvv oveq2d 3eqtr2d mavmulfv gsummulc2 eqtr4d 3eqtr4d eqfnfvd ) AUAGHJFTZI - ETZHJIETZETZAXICGUDTZUEGCXIUFXIGUGABCDDUHUIZEGXHIKMLXMUJZNOAXHCGGUKZUDTZB - ULUIZACGDFGGHJLNQOOOAHXQXPRAGUMUEZDUPUEZXPXQUQONBDCGUPKLUNUOZURZAJXQXPSXT - URZUSXTUTZPVAXICGVBGCXIVCVDAXKXLUEGCXKUFXKGUGABCDXMEGHXJKMLXNNORABCDXMEGJ - IKMLXNNOSPVAZVAXKCGVBGCXKVCVDAUAVEZGUEZVFZDUBGYEUBVEZXHTZYHIUIZXMTZVGZVHT - ZDUCGYEUCVEZHTZYNXJUIZXMTZVGZVHTZYEXIUIYEXKUIYGDUBGDUCGYOYNYHJTZYJXMTZXMT - ZVGZVHTZVGZVHTDUCGDUBGUUBVGVHTZVGZVHTYMYSYGGCGUBUCDUUBLADVIUEZYFAXSUUHNDV - JVKVLAXRYFOVLZUUIYGYHGUEZYNGUEZVFZVFZXSYOCUEZUUACUEZUUBCUEAXSYFUULNVMZUUM - YEYNCGGHAXOCHUFZYFUULAHXPUEZUUQYAHCXOVBVKZVMAYFUULVNYGUUJUUKVOZVPZUUMXSYT - CUEZYJCUEZUUOUUPUUMYNYHCGGJAXOCJUFZYFUULAJXPUEZUVDYBJCXOVBVKZVMUUTYGUUJUU - KVQVPZAUUJUVCYFUUKAUUJUVCAIXLUEZGCIUFZUUJUVCVTZPICGVBUVIUUJUVCGCYHIVRWAVD - ZVSZWBZCDXMYTYJLXNWCZWDCDXMYOUUALXNWCWDWEYGYLUUEDVHYGUBGYKUUDYGUUJVFZYKDU - CGYOYTXMTZVGZVHTZYJXMTDUCGUVPYJXMTZVGZVHTUUDUVOYIUVRYJXMUVOCGDXMUCFYEYHGG - UPHJQLXNAXSYFUUJNVMZAXRYFUUJOVMZUWBUWBAUURYFUUJYAVMAUVEYFUUJYBVMAYFUUJVNY - GUUJWKWFWGUVOGCDWHUIZDXMUCUMUVPYJDWIUIZLUWDUJZUWCUJZXNUWAUWBAUUJUVCYFUVLW - JUVOUUKVFZXSUUNUVBUVPCUEYGXSUUJUUKAXSYFNVLZVMYGUUKUUNUUJYGUUKVFZYEYNCGGHA - UUQYFUUKUUSVMAYFUUKVNYGUUKWKZVPZWJUWGYNYHCGGJYGUVDUUJUUKAUVDYFUVFVLZVMUVO - UUKWKYGUUJUUKVNVPCDXMYOYTLXNWCWDUVOUCGUVQWTWTUVPUWDUVQUJUWBUWGYOYTXMWLUVO - DWIWMWNWOUVOUVTUUCDVHUVOUCGUVSUUBYGUUJUUKUVSUUBUQZUUMXSUUNUVBUVCUWMUUPUVA - UVGUVMCDXMYOYTYJLXNWPWQWRWSXAXBWSXAYGYRUUGDVHYGUCGYQUUFUWIYQYODUBGUUAVGZV - HTZXMTUUFUWIYPUWOYOXMUWIBCDXMEUBYNGUPJIKMLXNAXSYFUUKNVMZAXRYFUUKOVMZAJXQU - EYFUUKSVMAUVHYFUUKPVMUWJXCXAUWIGCUWCDXMUBUMUUAYOUWDLUWEUWFXNUWPUWQUWKUWIU - UJVFZXSUVBUVCUUOYGXSUUKUUJUWHVMUWRYNYHCGGJYGUVDUUKUUJUWLVMYGUUKUUJVNUWIUU - JWKVPUWIUUJUVCAUVJYFUUKUVKVMVSUVNWDUWIUBGUWNWTWTUUAUWDUWNUJUWQUWRYTYJXMWL - UWIDWIWMWNXDXEWSXAXFYGBCDXMEUBYEGUPXHIKMLXNUWHUUIAXHXQUEYFYCVLAUVHYFPVLAY - FWKZXCYGBCDXMEUCYEGUPHXJKMLXNUWHUUIAHXQUEYFRVLAXJXLUEYFYDVLUWSXCXFXG $. + ccmn ringcmn syl adantr ad2antrr simplr simprr fovcdmd simprl wi ffvelcdm + imp ad2ant2r ringcl syl3anc gsumcom3fi simpr mamufv oveq1d cplusg adantlr + ex c0g ovexd fvexd fsuppmptdm gsummulc1 ringass syl13anc mpteq2dva oveq2d + cvv anassrs 3eqtr2d mavmulfv gsummulc2 eqtr4d 3eqtr4d eqfnfvd ) AUAGHJFTZ + IETZHJIETZETZAXICGUDTZUEGCXIUFXIGUGABCDDUHUIZEGXHIKMLXMUJZNOAXHCGGUKZUDTZ + BULUIZACGDFGGHJLNQOOOAHXQXPRAGUMUEZDUPUEZXPXQUQONBDCGUPKLUNUOZURZAJXQXPSX + TURZUSXTUTZPVAXICGVBGCXIVCVDAXKXLUEGCXKUFXKGUGABCDXMEGHXJKMLXNNORABCDXMEG + JIKMLXNNOSPVAZVAXKCGVBGCXKVCVDAUAVEZGUEZVFZDUBGYEUBVEZXHTZYHIUIZXMTZVGZVH + TZDUCGYEUCVEZHTZYNXJUIZXMTZVGZVHTZYEXIUIYEXKUIYGDUBGDUCGYOYNYHJTZYJXMTZXM + TZVGZVHTZVGZVHTDUCGDUBGUUBVGVHTZVGZVHTYMYSYGGCGUBUCDUUBLADVIUEZYFAXSUUHND + VJVKVLAXRYFOVLZUUIYGYHGUEZYNGUEZVFZVFZXSYOCUEZUUACUEZUUBCUEAXSYFUULNVMZUU + MYEYNCGGHAXOCHUFZYFUULAHXPUEZUUQYAHCXOVBVKZVMAYFUULVNYGUUJUUKVOZVPZUUMXSY + TCUEZYJCUEZUUOUUPUUMYNYHCGGJAXOCJUFZYFUULAJXPUEZUVDYBJCXOVBVKZVMUUTYGUUJU + UKVQVPZAUUJUVCYFUUKAUUJUVCAIXLUEZGCIUFZUUJUVCVRZPICGVBUVIUUJUVCGCYHIVSWJV + DZVTZWAZCDXMYTYJLXNWBZWCCDXMYOUUALXNWBWCWDYGYLUUEDVHYGUBGYKUUDYGUUJVFZYKD + UCGYOYTXMTZVGZVHTZYJXMTDUCGUVPYJXMTZVGZVHTUUDUVOYIUVRYJXMUVOCGDXMUCFYEYHG + GUPHJQLXNAXSYFUUJNVMZAXRYFUUJOVMZUWBUWBAUURYFUUJYAVMAUVEYFUUJYBVMAYFUUJVN + YGUUJWEWFWGUVOGCDWHUIZDXMUCUMUVPYJDWKUIZLUWDUJZUWCUJZXNUWAUWBAUUJUVCYFUVL + WIUVOUUKVFZXSUUNUVBUVPCUEYGXSUUJUUKAXSYFNVLZVMYGUUKUUNUUJYGUUKVFZYEYNCGGH + AUUQYFUUKUUSVMAYFUUKVNYGUUKWEZVPZWIUWGYNYHCGGJYGUVDUUJUUKAUVDYFUVFVLZVMUV + OUUKWEYGUUJUUKVNVPCDXMYOYTLXNWBWCUVOUCGUVQWTWTUVPUWDUVQUJUWBUWGYOYTXMWLUV + ODWKWMWNWOUVOUVTUUCDVHUVOUCGUVSUUBYGUUJUUKUVSUUBUQZUUMXSUUNUVBUVCUWMUUPUV + AUVGUVMCDXMYOYTYJLXNWPWQXAWRWSXBWRWSYGYRUUGDVHYGUCGYQUUFUWIYQYODUBGUUAVGZ + VHTZXMTUUFUWIYPUWOYOXMUWIBCDXMEUBYNGUPJIKMLXNAXSYFUUKNVMZAXRYFUUKOVMZAJXQ + UEYFUUKSVMAUVHYFUUKPVMUWJXCWSUWIGCUWCDXMUBUMUUAYOUWDLUWEUWFXNUWPUWQUWKUWI + UUJVFZXSUVBUVCUUOYGXSUUKUUJUWHVMUWRYNYHCGGJYGUVDUUKUUJUWLVMYGUUKUUJVNUWIU + UJWEVPUWIUUJUVCAUVJYFUUKUVKVMVTUVNWCUWIUBGUWNWTWTUUAUWDUWNUJUWQUWRYTYJXMW + LUWIDWKWMWNXDXEWRWSXFYGBCDXMEUBYEGUPXHIKMLXNUWHUUIAXHXQUEYFYCVLAUVHYFPVLA + YFWEZXCYGBCDXMEUCYEGUPHXJKMLXNUWHUUIAHXQUEYFRVLAXJXLUEYFYDVLUWSXCXFXG $. $} ${ @@ -283949,35 +283944,35 @@ is alternating (see ~ mdetralt ) and normalized (see ~ mdet1 ). adantl cbs eqid ccmn crg crngring ringcmn syl adantr cfn matrcl symgbasfi simpld ad2antrr cmgp cmhm czrh cpsgn coeq12i zrhpsgnmhm eqeltrid syl2an2r cvv wf mgpbas mhmf ffvelcdmda crngmgp cxp cmap simpr matbas2i elmapi 3syl - wf1o symgbasf1o f1of fovrnd ralrimiva gsummptcl syl3anc symggrp grpinvf1o - ringcl cgrp gsummptfif1o feqmptd eqidd fveq2 fveq1 oveq1d mpteq2dv oveq2d - oveq12d fmptco ccnv symginv fveq2d zrhpsgninv eqtrd fveq1d f1ocnv eqeltrd - ad2antlr eleqtrdi id f1ocnvfv1 sylan mpteq2dva 3eqtrd ) FUDUEZJCUEZUFZJDU - GZFUBEUBUHZLGUIZUGZIUCKUCUHZYKUGZYNJUJZUKZULUJZHUJZUKZULUJZFYTKUMUGZUNUGZ - UIZULUJFMEMUHZYLUGZIAKAUHZUUGUUEUGZJUJZUKZULUJZHUJZUKZULUJYHYJUUAUOYGUCBC - DEFGHIJKLUBNOPQRSTUAUPUQYIFURUGZEUBYTFUUCEYSUUNUSZYGFUTUEZYHYGFVAUEZUUPFV - BZFVCVDVEYIKVFUEZEVFUEYIUUSFVSUEZYHUUSUUTUFYGBCFKJOPVGUQVIZKEUUBUUBUSZQVH - VDYIYSUUNUEZUBEYIYKEUEZUFZUUQYMUUNUEYRUUNUEUVCYGUUQYHUVDUURVJYIEUUNYKYLYI - YLUUBFVKUGZVLUJZUEZEUUNYLVTYGUUQYHUUSUVHUURUVAUUQUUSUFYLFVMUGZKVNUGZUIUVG - LUVIGUVJRSVOKFVPVQVREUUNUUBUVFYLQUUNFUVFUVFUSUUOWAWBVDWCUVEUUNUCIKYPUUNFI - UAUUOWAZYGIUTUEZYHUVDFIUAWDZVJYIUUSUVDUVAVEUVEYPUUNUEUCKUVEYNKUEZUFYOYNUU - NKKJYIKKWEZUUNJVTZUVDUVNYIYHJUUNUVOWFUJUEUVPYGYHWGBCFUUNJKOUUOPWHJUUNUVOW - IWJZVJUVEKKYNYKUVEKKYKWKZKKYKVTUVDUVRYIKEYKUUBUVBQWLUQKKYKWMVDWCUVEUVNWGW - 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mgpbas cxp cmap matbas2i ad3antlr elmapi simpr fovrnd ralrimiva gsummptcl - symgbasf ringcl syl3anc mdetfval fmptd ) DUANZKBDLFUBOZUCOZLPZDUDOZFUEOZU - FZOZDUGOZMFMPZWHOZWNKPZQZUHUIQZDUJOZQZUHUIQECWEWPBNZRZELDWGWTJXBDUKNZDULN - WEXCXADUMZUNZDUOSXBFUPNZWGUPNXBXFDUTNZXAXFXGRWEABDFWPHIUQURUSZFWGWFWFTZWG - TZVASXBWTENZLWGXBWHWGNZRZXCWLENWRENXKWEXCXAXLXDVBXBWGEWHWKXBWKWFWMVCQNZWG - EWKVDXBXCXFXNXEXHFDVEVFWGEWFWMWKXJEDWMWMTZJVJZVGSVHXMEMWMFWQXPWEWMULNXAXL - DWMXOVIVBXBXFXLXHUNXMWQENMFXMWNFNZRZWOWNEFFWPXRWPEFFVKZVLQNZXSEWPVDXAXTWE - XLXQABDEWPFHJIVMVNWPEXSVOSXMFFWNWHXLFFWHVDXBFWGWHWFXIXJVTURVHXMXQVPVQVRVS - EDWSWLWRJWSTZWAWBVRVSMABCWGDWJWSWMKFWILGHIXJWITWJTYAXOWCWD $. + mgpbas cxp cmap matbas2i ad3antlr elmapi symgbasf simpr fovcdmd ralrimiva + gsummptcl ringcl syl3anc mdetfval fmptd ) DUANZKBDLFUBOZUCOZLPZDUDOZFUEOZ + UFZOZDUGOZMFMPZWHOZWNKPZQZUHUIQZDUJOZQZUHUIQECWEWPBNZRZELDWGWTJXBDUKNZDUL + NWEXCXADUMZUNZDUOSXBFUPNZWGUPNXBXFDUTNZXAXFXGRWEABDFWPHIUQURUSZFWGWFWFTZW + GTZVASXBWTENZLWGXBWHWGNZRZXCWLENWRENXKWEXCXAXLXDVBXBWGEWHWKXBWKWFWMVCQNZW + GEWKVDXBXCXFXNXEXHFDVEVFWGEWFWMWKXJEDWMWMTZJVJZVGSVHXMEMWMFWQXPWEWMULNXAX + LDWMXOVIVBXBXFXLXHUNXMWQENMFXMWNFNZRZWOWNEFFWPXRWPEFFVKZVLQNZXSEWPVDXAXTW + EXLXQABDEWPFHJIVMVNWPEXSVOSXMFFWNWHXLFFWHVDXBFWGWHWFXIXJVPURVHXMXQVQVRVSV + TEDWSWLWRJWSTZWAWBVSVTMABCWGDWJWSWMKFWILGHIXJWITWJTYAXOWCWD $. $( The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.) $) @@ -284180,45 +284175,45 @@ bijection from the singleton containing the empty set (empty matrix) wfn eqid f1ofn syl fnnfpeq0 adantl bicomd necon3bid wex n0 csn mgpplusg cmulr cbs ccmn crngmgp 3ad2ant1 ad2antrr simpll2 cmap matbas2i 3ad2ant3 cxp elmapi mgpbas eqcomi feq3d mpbird ad3antrrr symgbasf ad2antrl simpr - ffvelcdmda fovrnd cin disjdif cun wss difss dmss ax-mp fssdm sseld impr - snssd undif sylib eqcomd gsummptfidmsplit cmnd cvv crg crngring ringmgp - adantr 3adant3 vex ffvelcdmd weq oveq12d syl2anc eqeq1d imbi12d eqtrd - wf fveq2 id gsumsn syl3anc simprr fnelnfp mpbid neeq1 oveq1 neeq2 oveq2 - rspc2v impancom imp mpd oveq1d simpl2 sylancl eldifi ralrimiva ad2ant2r - ssfi gsummptcl ringlz 3eqtrd exlimdv syl5bi sylbid expimpd 3impia simpl - expr 3simpa cmgp cmhm zrhpsgnmhm sylan mhmf ringrz syl2an 3adant2 ) EUG - UHZNUIUHZMBUHZUJZHUKZIUKZULZUWFUWGMUMZOUNZUOZINUPHNUPZDLUHZDUQNURZULZUS - 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eqtrd oveq1d crg ccrg fovrnd - crngring mgpbas ccmn difssd ssfid eldifi ad2antrr simpr ffvelcdmda sylan2 - crngmgp ralrimiva gsummptcl ringass syl13anc mgpplusg cin disjdif a1i cun + wss df-ov oveq2i 3eqtr4g 3eqtr3d oveq2d eqtrd oveq1d crg crngring fovcdmd + ofc1 ccrg mgpbas ccmn crngmgp difssd ssfid eldifi simpr ffvelcdmda sylan2 + ad2antrr ralrimiva gsummptcl ringass syl13anc mgpplusg cin c0 disjdif a1i undif gsummptfidmsplit oveq12d gsumsn syl3anc mpteq2dva 3eqtr4d mdetleib2 - c0 c0g sylib eqcomd cmnd cmnmnd fveq2 cmhm zrhpsgnmhm mhmf crngcom cplusg - id symgbasfi ringcl ovexd fvexd fsuppmptdm gsummulc2 ) AEUEIUGUHZUIUHZUEU - JZEUKUHZIULUHZUMZUHZEUNUHZUFIUFUJZUVFUUTUHZJVAZUOUPVAZFVAZUOZUPVAZKEUEUUS - UVDUVEUFIUVFUVGLVAZUOUPVAZFVAZUOZUPVAZFVAZJDUHZKLDUHZFVAAUVLEUEUUSKUVOFVA - ZUOZUPVAUVRAUVKUWBEUPAUEUUSUVJUWAAUUTUUSUQZVBZUVJUVDKUVNFVAZFVAZUWAUWDUVI - UWEUVDFUWDGGUUTUHZJVAZUVEUFIGURZUSZUVMUOZUPVAZFVAZKGUWGLVAZUWLFVAZFVAZUVI - 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X |-> ( k e. I |-> A ) ) e. ( J Cn K ) ) $= ( wcel cfv wa adantr ctop vz cmpt ccn co cuni wf cv ccnp wral cixp ctopon - ffvelcdmda toptopon2 sylib syl3anc fvmptelrn an32s ralrimiva wb mptelixpg - cnf2 syl mpbird wceq ptuni syl2anc eleqtrd fmpttd adantlr simplr toponuni - simpr ad2antrr eqid cncnpi ptcnp cpt pttop eqeltrid cncnp mpbir2and ) ABJ - DFCUBZUBZGHUCUDPZJHUEZWCUFZWCUAUGZGHUHUDQPZUAJUIZABJWBWEABUGJPZRZWBDFDUGZ - EQZUEZUJZWEWKWBWOPZCWNPZDFUIZWKWQDFAWLFPZWJWQAWSRZBJCWNWTGJUKQPZWMWNUKQPZ - BJCUBZGWMUCUDPZJWNXCUFAXAWSLSWTWMTPXBAFTWLENULWMUMUNOXCGWMJWNVAUOUPUQURWK - FIPZWPWRUSAXEWJMSDFCWNIUTVBVCAWOWEVDZWJAXEFTEUFZXFMNDFEHIKVEVFSVGVHAWHUAJ - AWGJPZRZBCWGDEFGHIJKAXAXHLSAXEXHMSAXGXHNSAXHVLXIWSRZXDWGGUEZPXCWGGWMUHUDQ - PAWSXDXHOVIXJWGJXKAXHWSVJAJXKVDZXHWSAXAXLLJGVKVBVMVGWGXCGWMXKXKVNVOVFVPUR - AXAHWEUKQPZWDWFWIRUSLAHTPXMAHEVQQZTKAXEXGXNTPMNFEIVRVFVSHUMUNUAWCGHJWEVTV - FWA $. + ffvelcdmda toptopon2 sylib cnf2 syl3anc fvmptelcdm ralrimiva wb mptelixpg + an32s syl mpbird wceq ptuni syl2anc eleqtrd fmpttd simpr adantlr toponuni + simplr ad2antrr eqid cncnpi ptcnp cpt pttop eqeltrid cncnp mpbir2and ) AB + JDFCUBZUBZGHUCUDPZJHUEZWCUFZWCUAUGZGHUHUDQPZUAJUIZABJWBWEABUGJPZRZWBDFDUG + ZEQZUEZUJZWEWKWBWOPZCWNPZDFUIZWKWQDFAWLFPZWJWQAWSRZBJCWNWTGJUKQPZWMWNUKQP + ZBJCUBZGWMUCUDPZJWNXCUFAXAWSLSWTWMTPXBAFTWLENULWMUMUNOXCGWMJWNUOUPUQVAURW + KFIPZWPWRUSAXEWJMSDFCWNIUTVBVCAWOWEVDZWJAXEFTEUFZXFMNDFEHIKVEVFSVGVHAWHUA + JAWGJPZRZBCWGDEFGHIJKAXAXHLSAXEXHMSAXGXHNSAXHVIXIWSRZXDWGGUEZPXCWGGWMUHUD + QPAWSXDXHOVJXJWGJXKAXHWSVLAJXKVDZXHWSAXAXLLJGVKVBVMVGWGXCGWMXKXKVNVOVFVPU + RAXAHWEUKQPZWDWFWIRUSLAHTPXMAHEVQQZTKAXEXGXNTPMNFEIVRVFVSHUMUNUAWCGHJWEVT + VFWA $. $} ${ @@ -305890,19 +305885,19 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= 22-Aug-2015.) $) cnmpt11 $p |- ( ph -> ( x e. X |-> C ) e. ( J Cn L ) ) $= ( vz wceq cfv wcel cmpt ccom co cv wral wa simpr ctopon wf cnf2 syl3anc - ccn fvmptelrn eqid fvmpt2 syl2anc fveq2d cuni eleq1d ctop syl toptopon2 - cntop2 sylib fmpt sylibr adantr rspcdva fvmptd3 eqtrd 3eqtr4d ralrimiva - fvco3 sylan nfv nfcv nfmpt1 nfco nffv nfeq fveq2 eqeq12d cbvralw wfn wb - fco ffnd fmpttd eqfnfv mpbird cnco eqeltrrd ) ACKEUAZBJDUAZUBZBJFUAZGIU - LUCZAWOWPRZQUDZWOSZWSWPSZRZQJUEZABUDZWOSZXDWPSZRZBJUEXCAXGBJAXDJTZUFZXD - WNSZWMSZFXEXFXIXKDWMSFXIXJDWMXIXHDKTXJDRAXHUGZABJDKAGJUHSTHKUHSTZWNGHUL - UCTZJKWNUIZLNMWNGHJKUJUKZUMZBJDKWNWNUNUOUPUQXICDEFKWMIURZWMUNZPXQXIEXRT - ZFXRTZCKDCUDDREFXRPUSAXTCKUEZXHAKXRWMUIZYBAXMIXRUHSTZWMHIULUCTZYCNAIUTT - ZYDAYEYFOWMHIVCVAIVBVDOWMHIKXRUJUKZCKXREWMXSVEVFVGXQVHZVIVJAXOXHXEXKRXP - JKXDWMWNVMVNXIXHYAXFFRXLYHBJFXRWPWPUNUOUPVKVLXGXBBQJXGQVOBWTXABWSWOBWMW - NBWMVPBJDVQVRBWSVPZVSBWSWPBJFVQYIVSVTXDWSRXEWTXFXAXDWSWOWAXDWSWPWAWBWCV - DAWOJWDWPJWDWRXCWEAJXRWOAYCXOJXRWOUIYGXPJKXRWMWNWFUPWGAJXRWPABJFXRYHWHW - GQJWOWPWIUPWJAXNYEWOWQTMOWNWMGHIWKUPWL $. + fvmptelcdm eqid fvmpt2 syl2anc fveq2d cuni eleq1d ctop cntop2 toptopon2 + ccn syl sylib sylibr adantr rspcdva fvmptd3 eqtrd fvco3 sylan ralrimiva + fmpt 3eqtr4d nfv nfcv nfmpt1 nfco nffv fveq2 eqeq12d cbvralw wfn wb fco + nfeq ffnd fmpttd eqfnfv mpbird cnco eqeltrrd ) ACKEUAZBJDUAZUBZBJFUAZGI + VBUCZAWOWPRZQUDZWOSZWSWPSZRZQJUEZABUDZWOSZXDWPSZRZBJUEXCAXGBJAXDJTZUFZX + DWNSZWMSZFXEXFXIXKDWMSFXIXJDWMXIXHDKTXJDRAXHUGZABJDKAGJUHSTHKUHSTZWNGHV + BUCTZJKWNUIZLNMWNGHJKUJUKZULZBJDKWNWNUMUNUOUPXICDEFKWMIUQZWMUMZPXQXIEXR + TZFXRTZCKDCUDDREFXRPURAXTCKUEZXHAKXRWMUIZYBAXMIXRUHSTZWMHIVBUCTZYCNAIUS + TZYDAYEYFOWMHIUTVCIVAVDOWMHIKXRUJUKZCKXREWMXSVMVEVFXQVGZVHVIAXOXHXEXKRX + PJKXDWMWNVJVKXIXHYAXFFRXLYHBJFXRWPWPUMUNUOVNVLXGXBBQJXGQVOBWTXABWSWOBWM + WNBWMVPBJDVQVRBWSVPZVSBWSWPBJFVQYIVSWFXDWSRXEWTXFXAXDWSWOVTXDWSWPVTWAWB + VDAWOJWCWPJWCWRXCWDAJXRWOAYCXOJXRWOUIYGXPJKXRWMWNWEUOWGAJXRWPABJFXRYHWH + WGQJWOWPWIUOWJAXNYEWOWQTMOWNWMGHIWKUOWL $. $} ${ @@ -305925,15 +305920,15 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cnmpt1t $p |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( J Cn ( K tX L ) ) ) $= ( vy cuni cmpt cfv cop co ccn wcel wceq cv ctopon toponuni mpteq1 3syl wa - ctx simpr wf ctop cntop2 syl toptopon2 cnf2 syl3anc fvmptelrn eqid fvmpt2 - sylib syl2anc opeq12d mpteq2dva eqtr3d nfcv nffvmpt1 fveq2 cbvmpt txcnmpt - nfop eqeltrrd ) ABEMZBUAZBHCNZOZVLBHDNZOZPZNZBHCDPZNZEFGUGQRQZABHVQNZVRVT - AEHUBOSZHVKTWBVRTIHEUCBHVKVQUDUEABHVQVSAVLHSZUFZVNCVPDWEWDCFMZSVNCTAWDUHZ - ABHCWFAWCFWFUBOSZVMEFRQSZHWFVMUIIAFUJSZWHAWIWJJVMEFUKULFUMUSJVMEFHWFUNUOU - PBHCWFVMVMUQURUTWEWDDGMZSVPDTWGABHDWKAWCGWKUBOSZVOEGRQSZHWKVOUIIAGUJSZWLA - WMWNKVOEGUKULGUMUSKVOEGHWKUNUOUPBHDWKVOVOUQURUTVAVBVCAWIWMVRWASJKLFGEVMVO - VRVKVKUQBLVKVQLUAZVMOZWOVOOZPLVQVDBWPWQBHCWOVEBHDWOVEVIVLWOTVNWPVPWQVLWOV - MVFVLWOVOVFVAVGVHUTVJ $. + ctx simpr wf ctop cntop2 syl toptopon2 sylib cnf2 syl3anc fvmptelcdm eqid + fvmpt2 syl2anc opeq12d mpteq2dva eqtr3d nfcv nffvmpt1 nfop cbvmpt txcnmpt + fveq2 eqeltrrd ) ABEMZBUAZBHCNZOZVLBHDNZOZPZNZBHCDPZNZEFGUGQRQZABHVQNZVRV + TAEHUBOSZHVKTWBVRTIHEUCBHVKVQUDUEABHVQVSAVLHSZUFZVNCVPDWEWDCFMZSVNCTAWDUH + ZABHCWFAWCFWFUBOSZVMEFRQSZHWFVMUIIAFUJSZWHAWIWJJVMEFUKULFUMUNJVMEFHWFUOUP + UQBHCWFVMVMURUSUTWEWDDGMZSVPDTWGABHDWKAWCGWKUBOSZVOEGRQSZHWKVOUIIAGUJSZWL + AWMWNKVOEGUKULGUMUNKVOEGHWKUOUPUQBHDWKVOVOURUSUTVAVBVCAWIWMVRWASJKLFGEVMV + OVRVKVKURBLVKVQLUAZVMOZWOVOOZPLVQVDBWPWQBHCWOVEBHDWOVEVFVLWOTVNWPVPWQVLWO + VMVIVLWOVOVIVAVGVHUTVJ $. ${ cnmpt12f.f $e |- ( ph -> F e. ( ( K tX L ) Cn M ) ) $. @@ -305954,17 +305949,17 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) $) cnmpt12 $p |- ( ph -> ( x e. X |-> D ) e. ( J Cn M ) ) $= - ( cmpo co cmpt ccn cv wcel wa cuni wceq ctopon cfv cnf2 syl3anc fvmptelrn - wf wi wal jca wral cxp ctx txtopon syl2anc ctop cntop2 syl toptopon2 eqid - sylib fmpo sylibr r2al adantr eleq1 bi2anan9 eleq1d imbi12d syl3c ovmpoga - spc2gv mpteq2dva cnmpt12f eqeltrrd ) ABMEFCDNOGUCZUDZUEBMHUEILUFUDABMWGHA - BUGMUHZUIZENUHZFOUHZHLUJZUHZWGHUKABMENAIMULUMUHZJNULUMUHZBMEUEZIJUFUDUHMN - WPUQPSQWPIJMNUNUOUPZABMFOAWNKOULUMUHZBMFUEZIKUFUDUHMOWSUQPTRWSIKMOUNUOUPZ - WIWJWKUIZCUGZNUHZDUGZOUHZUIZGWLUHZURZDUSCUSZXAWMWIWJWKWQWTUTZAXIWHAXGDOVA - CNVAZXIANOVBZWLWFUQZXKAJKVCUDZXLULUMUHZLWLULUMUHZWFXNLUFUDUHZXMAWOWRXOSTJ - KNOVDVEALVFUHZXPAXQXRUAWFXNLVGVHLVIVKUAWFXNLXLWLUNUOCDNOGWLWFWFVJZVLVMXGC - DNOVNVKVOXJXHXAWMURCDEFNOXBEUKZXDFUKZUIZXFXAXGWMXTXCWJYAXEWKXBENVPXDFOVPV - QYBGHWLUBVRVSWBVTCDEFNOGHWFWLUBXSWAUOWCABEFWFIJKLMPQRUAWDWE $. + ( cmpo co cmpt ccn cv wcel wa cuni wceq ctopon wf cnf2 syl3anc fvmptelcdm + cfv wal jca wral cxp ctx txtopon syl2anc ctop cntop2 toptopon2 sylib eqid + wi syl fmpo sylibr r2al adantr eleq1 bi2anan9 eleq1d imbi12d spc2gv syl3c + ovmpoga mpteq2dva cnmpt12f eqeltrrd ) ABMEFCDNOGUCZUDZUEBMHUEILUFUDABMWGH + ABUGMUHZUIZENUHZFOUHZHLUJZUHZWGHUKABMENAIMULUQUHZJNULUQUHZBMEUEZIJUFUDUHM + NWPUMPSQWPIJMNUNUOUPZABMFOAWNKOULUQUHZBMFUEZIKUFUDUHMOWSUMPTRWSIKMOUNUOUP + ZWIWJWKUIZCUGZNUHZDUGZOUHZUIZGWLUHZVJZDURCURZXAWMWIWJWKWQWTUSZAXIWHAXGDOU + TCNUTZXIANOVAZWLWFUMZXKAJKVBUDZXLULUQUHZLWLULUQUHZWFXNLUFUDUHZXMAWOWRXOST + JKNOVCVDALVEUHZXPAXQXRUAWFXNLVFVKLVGVHUAWFXNLXLWLUNUOCDNOGWLWFWFVIZVLVMXG + CDNOVNVHVOXJXHXAWMVJCDEFNOXBEUKZXDFUKZUIZXFXAXGWMXTXCWJYAXEWKXBENVPXDFOVP + VQYBGHWLUBVRVSVTWACDEFNOGHWFWLUBXSWBUOWCABEFWFIJKLMPQRUAWDWE $. $} ${ @@ -306224,15 +306219,15 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= by Mario Carneiro, 22-Aug-2015.) $) cnmptkp $p |- ( ph -> ( x e. X |-> C ) e. ( J Cn L ) ) $= ( cfv wcel vw cmpt co cv wa cuni eqid adantr wceq eleq1d wf wral ctopon - ccn ctop topontop syl toptopon2 cxko xkotopon syl2anc syl3anc fvmptelrn - sylib cnf2 fmpt sylibr rspcdva fvmptd3 mpteq2dva toponuni eleqtrd fveq1 - xkopjcn cnmpt11 eqeltrrd ) ABJECKDUBZSZUBBJFUBGIUNUCABJVRFABUDJTZUEZCED - FKVQIUFZVQUGZRAEKTVSQUHZVTDWATZFWATCKECUDEUIDFWARUJVTKWAVQUKZWDCKULVTHK - UMSTZIWAUMSTZVQHIUNUCZTWEAWFVSNUHVTIUOTZWGAWIVSAILUMSTWIOLIUPUQZUHIURVD - ABJVQWHAGJUMSTIHUSUCZWHUMSTZBJVQUBZGWKUNUCTJWHWMUKMAHUOTZWIWLAWFWNNKHUP - UQZWJHIWKWKUGUTVAZPWMGWKJWHVEVBVCVQHIKWAVEVBCKWADVQWBVFVGWCVHVIVJABUAVQ - EUAUDZSZVRGWKIJWHMPWPAWNWIEHUFZTUAWHWRUBWKIUNUCTWOWJAEKWSQAWFKWSUINKHVK - UQVLEHIUAWSWSUGVNVBEWQVQVMVOVP $. + ccn ctop topontop toptopon2 sylib cxko xkotopon syl2anc cnf2 fvmptelcdm + syl syl3anc fmpt sylibr rspcdva fvmptd3 mpteq2dva eleqtrd xkopjcn fveq1 + toponuni cnmpt11 eqeltrrd ) ABJECKDUBZSZUBBJFUBGIUNUCABJVRFABUDJTZUEZCE + DFKVQIUFZVQUGZRAEKTVSQUHZVTDWATZFWATCKECUDEUIDFWARUJVTKWAVQUKZWDCKULVTH + KUMSTZIWAUMSTZVQHIUNUCZTWEAWFVSNUHVTIUOTZWGAWIVSAILUMSTWIOLIUPVDZUHIUQU + RABJVQWHAGJUMSTIHUSUCZWHUMSTZBJVQUBZGWKUNUCTJWHWMUKMAHUOTZWIWLAWFWNNKHU + PVDZWJHIWKWKUGUTVAZPWMGWKJWHVBVEVCVQHIKWAVBVECKWADVQWBVFVGWCVHVIVJABUAV + QEUAUDZSZVRGWKIJWHMPWPAWNWIEHUFZTUAWHWRUBWKIUNUCTWOWJAEKWSQAWFKWSUINKHV + NVDVKEHIUAWSWSUGVLVEEWQVQVMVOVP $. $} ${ @@ -306246,13 +306241,13 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cnmptk1 $p |- ( ph -> ( x e. X |-> ( y e. Y |-> C ) ) e. ( J Cn ( M ^ko K ) ) ) $= ( vw cmpt ccom cxko co ccn cv wcel wral ctopon cfv adantr ctop topontop - wa syl eqid xkotopon syl2anc cnf2 syl3anc fvmptelrn fmpt sylibr fmptcof - wf eqidd mpteq2dva xkoco2cn coeq2 cnmpt11 eqeltrrd ) ABLDNFUBZCMEUBZUCZ - UBBLCMGUBZUBHKIUDUEZUFUEABLVOVPABUGLUHZUOZCDMNEFGVNVMVSMNVNVFZENUHCMUIV - SIMUJUKUHZJNUJUKUHZVNIJUFUEZUHVTAWAVRPULAWBVRQULABLVNWCAHLUJUKUHJIUDUEZ - WCUJUKUHZBLVNUBZHWDUFUEUHLWCWFVFOAIUMUHZJUMUHZWEAWAWGPMIUNUPZAWBWHQNJUN - UPIJWDWDUQURUSZRWFHWDLWCUTVAVBVNIJMNUTVACMNEVNVNUQVCVDVSVNVGVSVMVGTVEVH - ABUAVNVMUAUGZUCVOHWDVQLWCORWJAIJKUAVMWISVIWKVNVMVJVKVL $. + wa wf eqid xkotopon syl2anc cnf2 syl3anc fvmptelcdm fmpt sylibr fmptcof + syl eqidd mpteq2dva xkoco2cn coeq2 cnmpt11 eqeltrrd ) ABLDNFUBZCMEUBZUC + ZUBBLCMGUBZUBHKIUDUEZUFUEABLVOVPABUGLUHZUOZCDMNEFGVNVMVSMNVNUPZENUHCMUI + VSIMUJUKUHZJNUJUKUHZVNIJUFUEZUHVTAWAVRPULAWBVRQULABLVNWCAHLUJUKUHJIUDUE + ZWCUJUKUHZBLVNUBZHWDUFUEUHLWCWFUPOAIUMUHZJUMUHZWEAWAWGPMIUNVFZAWBWHQNJU + NVFIJWDWDUQURUSZRWFHWDLWCUTVAVBVNIJMNUTVACMNEVNVNUQVCVDVSVNVGVSVMVGTVEV + HABUAVNVMUAUGZUCVOHWDVQLWCORWJAIJKUAVMWISVIWKVNVMVJVKVL $. $} ${ @@ -306297,16 +306292,16 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cnmptkk $p |- ( ph -> ( x e. X |-> ( y e. Y |-> C ) ) e. ( J Cn ( M ^ko K ) ) ) $= ( vf vg cmpt ccom cxko co ccn cv wcel wa wf wral ctopon cfv ctop topontop - adantr syl ccmp nllytop eqid xkotopon syl2anc cnf2 syl3anc fvmptelrn fmpt - cnlly sylibr eqidd fmptcof mpteq2dva cmpo ctx xkococn wceq coeq1 sylan9eq - coeq2 cnmpt12 eqeltrrd ) ABMDOFUFZCNEUFZUGZUFBMCNGUFZUFHKIUHUIZUJUIABMWGW - HABUKMULZUMZCDNOEFGWFWEWKNOWFUNZEOULCNUOWKINUPUQULZJOUPUQULZWFIJUJUIZULWL - AWMWJQUTAWNWJRUTABMWFWOAHMUPUQULJIUHUIZWOUPUQULZBMWFUFZHWPUJUIULMWOWRUNPA - IURULZJURULZWQAWMWSQNIUSVAZAJVBVKULZWTTVBJVCVAZIJWPWPVDVEVFZUAWRHWPMWOVGV - HVIWFIJNOVGVHCNOEWFWFVDVJVLWKWFVMWKWEVMUCVNVOABUDUEWEWFUDUKZUEUKZUGZWGHKJ - UHUIZWPWIMJKUJUIZWOPUBUAAWTKURULZXHXIUPUQULXCAKLUPUQULXJSLKUSVAZJKXHXHVDV - EVFXDAWSXBXJUDUEXIWOXGVPZXHWPVQUIWIUJUIULXATXKIJKUDUEXLXLVDVRVHXEWEVSXFWF - VSXGWEXFUGWGXEWEXFVTXFWFWEWBWAWCWD $. + syl ccmp cnlly nllytop eqid xkotopon syl2anc cnf2 syl3anc fvmptelcdm fmpt + adantr eqidd fmptcof mpteq2dva cmpo ctx xkococn wceq coeq1 coeq2 sylan9eq + sylibr cnmpt12 eqeltrrd ) ABMDOFUFZCNEUFZUGZUFBMCNGUFZUFHKIUHUIZUJUIABMWG + WHABUKMULZUMZCDNOEFGWFWEWKNOWFUNZEOULCNUOWKINUPUQULZJOUPUQULZWFIJUJUIZULW + LAWMWJQVKAWNWJRVKABMWFWOAHMUPUQULJIUHUIZWOUPUQULZBMWFUFZHWPUJUIULMWOWRUNP + AIURULZJURULZWQAWMWSQNIUSUTZAJVAVBULZWTTVAJVCUTZIJWPWPVDVEVFZUAWRHWPMWOVG + VHVIWFIJNOVGVHCNOEWFWFVDVJWBWKWFVLWKWEVLUCVMVNABUDUEWEWFUDUKZUEUKZUGZWGHK + JUHUIZWPWIMJKUJUIZWOPUBUAAWTKURULZXHXIUPUQULXCAKLUPUQULXJSLKUSUTZJKXHXHVD + VEVFXDAWSXBXJUDUEXIWOXGVOZXHWPVPUIWIUJUIULXATXKIJKUDUEXLXLVDVQVHXEWEVRXFW + FVRXGWEXFUGWGXEWEXFVSXFWFWEVTWAWCWD $. $} ${ @@ -306354,17 +306349,17 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) $) cnmptk1p $p |- ( ph -> ( x e. X |-> C ) e. ( J Cn L ) ) $= - ( wcel vf vz cmpt cfv ccn co cv eqid ctopon cnf2 syl3anc fvmptelrn wceq - wa wf eleq1d wral adantr cxko ctop ccmp cnlly nllytop topontop xkotopon - syl syl2anc fmpt sylibr rspcdva fvmptd3 mpteq2dva cmpo cuni ctx mpoeq12 - toponuni sylancr xkofvcn eqeltrd fveq1 fveq2 sylan9eq cnmpt12 eqeltrrd - ) ABJECKDUCZUDZUCBJFUCGIUEUFABJWGFABUGJTZUNZCEDFKWFLWFUHZSABJEKAGJUIUDT - ZHKUIUDTZBJEUCZGHUEUFTJKWMUOMNRWMGHJKUJUKULZWIDLTZFLTCKECUGEUMDFLSUPWIK - LWFUOZWOCKUQWIWLILUIUDTZWFHIUEUFZTWPAWLWHNURAWQWHOURABJWFWRAWKIHUSUFZWR - UIUDTZBJWFUCZGWSUEUFTJWRXAUOMAHUTTZIUTTZWTAHVAVBTZXBPVAHVCVFAWQXCOLIVDV - FZHIWSWSUHVEVGZQXAGWSJWRUJUKULWFHIKLUJUKCKLDWFWJVHVIWNVJVKVLABUAUBWFEUB - UGZUAUGZUDZWGGWSHIJWRKMQRXFNAUAUBWRKXIVMZUAUBWRHVNZXIVMZWSHVOUFIUEUFZAW - RWRUMKXKUMZXJXLUMWRUHAWLXNNKHVQVFUAUBWRKWRXKXIVPVRAXDXCXLXMTPXEUBHIUAXL + ( wcel vf vz cmpt cfv co cv wa eqid ctopon cnf2 syl3anc fvmptelcdm wceq + ccn wf eleq1d wral adantr cxko ctop cnlly nllytop syl topontop xkotopon + ccmp syl2anc fmpt sylibr rspcdva fvmptd3 mpteq2dva ctx toponuni mpoeq12 + cmpo cuni sylancr xkofvcn eqeltrd fveq1 fveq2 sylan9eq cnmpt12 eqeltrrd + ) ABJECKDUCZUDZUCBJFUCGIUNUEABJWGFABUFJTZUGZCEDFKWFLWFUHZSABJEKAGJUIUDT + ZHKUIUDTZBJEUCZGHUNUETJKWMUOMNRWMGHJKUJUKULZWIDLTZFLTCKECUFEUMDFLSUPWIK + LWFUOZWOCKUQWIWLILUIUDTZWFHIUNUEZTWPAWLWHNURAWQWHOURABJWFWRAWKIHUSUEZWR + UIUDTZBJWFUCZGWSUNUETJWRXAUOMAHUTTZIUTTZWTAHVFVATZXBPVFHVBVCAWQXCOLIVDV + CZHIWSWSUHVEVGZQXAGWSJWRUJUKULWFHIKLUJUKCKLDWFWJVHVIWNVJVKVLABUAUBWFEUB + UFZUAUFZUDZWGGWSHIJWRKMQRXFNAUAUBWRKXIVPZUAUBWRHVQZXIVPZWSHVMUEIUNUEZAW + RWRUMKXKUMZXJXLUMWRUHAWLXNNKHVNVCUAUBWRKWRXKXIVOVRAXDXCXLXMTPXEUBHIUAXL XKXKUHXLUHVSVGVTXHWFUMXGEUMXIXGWFUDWGXGXHWFWAXGEWFWBWCWDWE $. $} @@ -306376,20 +306371,20 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cnmptk2 $p |- ( ph -> ( x e. X , y e. Y |-> A ) e. ( ( J tX K ) Cn L ) ) $= ( vw vk cfv wceq wcel vf vz cv cmpt cmpo ctx co nffvmpt1 nfcv nffv nfmpt1 ccn nfmpt fveq2 fveq1d sylan9eq cbvmpo wa simplr ctopon cxko wf ctop ccmp - cnlly nllytop syl topontop eqid xkotopon syl2anc syl3anc fvmptelrn adantr - cnf2 fvmpt2 simpr eqtrd 3impa mpoeq3dva eqtrid cnmpt1st cnmpt21f cnmpt2nd - cuni toponuni mpoeq12 sylancr xkofvcn eqeltrd fveq1 cnmpt22 eqeltrrd ) AP - QHIQUCZPUCZBHCIDUDZUDZRZRZUEZBCHIDUEZEFUFUGGULUGAWTBCHICUCZBUCZWQRZRZUEXA - PQBCHIWSXEBWNWRBHWPWOUHBWNUIUJCWNWRCWOWQCBHWPCHUICIDUKUMCWOUIUJCWNUIUJPXE - UIQXEUIWOXCSZWNXBSWSWNXDRXEXFWNWRXDWOXCWQUNUOWNXBXDUNUPUQABCHIXEDAXCHTZXB - ITZXEDSAXGURZXHURZXEXBWPRZDXJXBXDWPXJXGWPFGULUGZTZXDWPSAXGXHUSXIXMXHABHWP - XLAEHUTRTGFVAUGZXLUTRTZWQEXNULUGTHXLWQVBKAFVCTZGVCTZXOAFVDVETZXPNVDFVFVGA - GJUTRTZXQMJGVHVGZFGXNXNVIVJVKZOWQEXNHXLVOVLVMZVNBHWPXLWQWQVIVPVKUOXJXHDJT - XKDSXIXHVQXICIDJXIFIUTRTZXSXMIJWPVBAYCXGLVNAXSXGMVNYBWPFGIJVOVLVMCIDJWPWP - VIVPVKVRVSVTWAAPQUAUBWRWNUBUCZUAUCZRZWSEFXNFGIHIXLKLAPQWOWQEFEXNHIKLAPQEF - HIKLWBOWCAPQEFHIKLWDYALAUAUBXLIYFUEZUAUBXLFWEZYFUEZXNFUFUGGULUGZAXLXLSIYH - SZYGYISXLVIAYCYKLIFWFVGUAUBXLIXLYHYFWGWHAXRXQYIYJTNXTUBFGUAYIYHYHVIYIVIWI - VKWJYEWRSYDWNSYFYDWRRWSYDYEWRWKYDWNWRUNUPWLWM $. + nllytop syl topontop eqid xkotopon syl2anc cnf2 syl3anc fvmptelcdm adantr + cnlly fvmpt2 simpr eqtrd mpoeq3dva eqtrid cnmpt1st cnmpt21f cnmpt2nd cuni + 3impa toponuni mpoeq12 sylancr xkofvcn eqeltrd fveq1 cnmpt22 eqeltrrd ) A + PQHIQUCZPUCZBHCIDUDZUDZRZRZUEZBCHIDUEZEFUFUGGULUGAWTBCHICUCZBUCZWQRZRZUEX + APQBCHIWSXEBWNWRBHWPWOUHBWNUIUJCWNWRCWOWQCBHWPCHUICIDUKUMCWOUIUJCWNUIUJPX + EUIQXEUIWOXCSZWNXBSWSWNXDRXEXFWNWRXDWOXCWQUNUOWNXBXDUNUPUQABCHIXEDAXCHTZX + BITZXEDSAXGURZXHURZXEXBWPRZDXJXBXDWPXJXGWPFGULUGZTZXDWPSAXGXHUSXIXMXHABHW + PXLAEHUTRTGFVAUGZXLUTRTZWQEXNULUGTHXLWQVBKAFVCTZGVCTZXOAFVDVOTZXPNVDFVEVF + AGJUTRTZXQMJGVGVFZFGXNXNVHVIVJZOWQEXNHXLVKVLVMZVNBHWPXLWQWQVHVPVJUOXJXHDJ + TXKDSXIXHVQXICIDJXIFIUTRTZXSXMIJWPVBAYCXGLVNAXSXGMVNYBWPFGIJVKVLVMCIDJWPW + PVHVPVJVRWEVSVTAPQUAUBWRWNUBUCZUAUCZRZWSEFXNFGIHIXLKLAPQWOWQEFEXNHIKLAPQE + FHIKLWAOWBAPQEFHIKLWCYALAUAUBXLIYFUEZUAUBXLFWDZYFUEZXNFUFUGGULUGZAXLXLSIY + HSZYGYISXLVHAYCYKLIFWFVFUAUBXLIXLYHYFWGWHAXRXQYIYJTNXTUBFGUAYIYHYHVHYIVHW + IVJWJYEWRSYDWNSYFYDWRRWSYDYEWRWKYDWNWRUNUPWLWM $. $} ${ @@ -306635,11 +306630,11 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= elqtop $p |- ( ( J e. V /\ F : Z -onto-> Y /\ Z C_ X ) -> ( A e. ( J qTop F ) <-> ( A C_ Y /\ ( `' F " A ) e. J ) ) ) $= ( vs wcel wfo wss w3a cqtop co ccnv cv cima wa cvv cpw crab qtopval2 wceq - eleq2d imaeq2 eleq1d elrab wb uniexg eqeltrid 3ad2ant1 simp3 ssexd fornex - cuni simp2 sylc elpw2g syl anbi1d bitrid bitrd ) CDJZGFBKZGELZMZACBNOZJAB - PZIQZRZCJZIFUAZUBZJZAFLZVIARZCJZSZVGVHVNABCDEFGIHUCUEVOAVMJZVRSVGVSVLVRIA - VMVJAUDVKVQCVJAVIUFUGUHVGVTVPVRVGFTJZVTVPUIVGGTJVEWAVGGETVDVEETJVFVDECUPT - HCDUJUKULVDVEVFUMUNVDVEVFUQGFTBUOURAFTUSUTVAVBVC $. + eleq2d imaeq2 eleq1d elrab wb cuni uniexg eqeltrid 3ad2ant1 simp3 focdmex + ssexd simp2 sylc elpw2g syl anbi1d bitrid bitrd ) CDJZGFBKZGELZMZACBNOZJA + BPZIQZRZCJZIFUAZUBZJZAFLZVIARZCJZSZVGVHVNABCDEFGIHUCUEVOAVMJZVRSVGVSVLVRI + AVMVJAUDVKVQCVJAVIUFUGUHVGVTVPVRVGFTJZVTVPUIVGGTJVEWAVGGETVDVEETJVFVDECUJ + THCDUKULUMVDVEVFUNUPVDVEVFUQGFTBUOURAFTUSUTVAVBVC $. $( The quotient topology is unaffected by restriction to the base set. This property makes it slightly more convenient to use, since we don't @@ -306977,35 +306972,35 @@ K C_ ( J qTop F ) ) $= cima toponuni sseqtrd eqid cnrest qtoptopon df-ima imaeq2d foimacnv eqtrd wb eqtr3id eqimss cnrest2 syl3anc mpbid resttopon qtopss cv wfun sseqtrrd fnfun fores foeq3 elqtop3 cin cnvresima imass2 adantl adantr df-ss eqtrid - sylib eleq1d ccld simplrl ctop cvv topontop toponmax fornex sylc restopn2 - ssexd sstrd sylan simprbda adantrl an32s mpbir2and eqeltrrd cdif ssdifssd - elrestr difeq1d funcnvcnv imadif 3syl eqtr4d simpr simplrr opncld eqeltrd - restcldr qtopcld difssd restcldi isopn2 mpbird wo mpjaodan sylbid expimpd - expr ssrdv eqssd ) AEDUBUCZCUDUCZEBUDUCZDBUEZUBUCZAYSYRYQUGUCMZYQCUFNMZYS - UHZCOZYQYTPAYSYRYPUGUCMZUUAADEYPUGUCMZBEUIZPUUEAEFUFNMZDFUJZUUFHAFGDUKZUU - IIFGDULQZDEFUMRABFUUGABDUNZCUTZFKADUOZUUMFDCUPAFDUUKUQZURUSZAUUHFUUGOHFEV - AQVBBDEYPUUGUUGVCVDRAYPGUFNMZUUCCPZCGPZUUEUUAVJAUUHUUJUUQHIDEFGVERZAUUDUU - RAUUCDBUTZCDBVFAUVADUUMUTZCABUUMDKVGAUUJUUSUVBCOIJFGCDVHRVIZVKZUUCCVLQJCY - SYRYPGVMVNVOAUUQUUSUUBUUTJCYPGVPRZUVDYSYRYQCVQVNAUAYTYQAUAVRZYTMZUVFCPZYS - UNUVFUTZYRMZSZUVFYQMZAYRBUFNMZBCYSUKZUVGUVKVJAUUHBFPUVMHUUPBEFVPRZABUVAYS - UKZUVNADVSZBUUNPUVPAUUIUVQUUKFDWAQZABFUUNUUPUUOVTBDWBRAUVACOUVPUVNVJUVCUV - ACBYSWCQVOUVFYSYRBCWDRAUVHUVJUVLAUVHSZUVJUULUVFUTZYRMZUVLUVSUVIUVTYRUVSUV - IUVTBWEZUVTBUVFDWFUVSUVTBPZUWBUVTOUVSUVTUUMBUVHUVTUUMPAUVFCUULWGWHABUUMOZ - UVHKWIVTUVTBWJWLWKWMAUVHUWAUVLAUVHUWASZSZBEMZUVLBEWNNZMZUWFUWGSZUVFCWEZUV - FYQUWJUVHUWKUVFOAUVHUWAUWGWOZUVFCWJWLUWJYPWPMZCWQMZUVFYPMZUWKYQMAUWMUWEUW - GAUUQUWMUUTGYPWRQTAUWNUWEUWGACGWQAFEMZUUJGWQMAUUHUWPHFEWSQIFGEDWTXAJXCTUW - JUWOUVFGPZUVTEMZUWJUVFCGUWLAUUSUWEUWGJTXDAUWGUWEUWRAUWGSZUWAUWRUVHUWSUWAU - WRUWCAEWPMZUWGUWAUWRUWCSVJAUUHUWTHFEWRQBUVTEXBXEXFXGXHAUWOUWQUWRSVJZUWEUW - GAUUHUUJUXAHIUVFDEFGWDRTXIUVFCYPWPWQXMVNXJUWFUWISZUVLYQUIZUVFXKZYQWNNZMZU - XBCUVFXKZUXDUXEUXBCUXCUVFUXBUUBCUXCOAUUBUWEUWIUVETZCYQVAQZXNUXBCYPUIZPUXG - YPWNNMZUXGCPUXGUXEMUXBCGUXJAUUSUWEUWIJTZUXBUUQGUXJOAUUQUWEUWIUUTTGYPVAQVB - UXBUXKUXGGPZUULUXGUTZUWHMZUXBCGUVFUXLXLUXBUXNBUVTXKZUWHUXBUXNUUMUVTXKZUXP - UXBUVQUULUNVSUXNUXQOAUVQUWEUWIUVRTDXOCUVFUULXPXQUXBBUUMUVTAUWDUWEUWIKTXNX - RUXBUWIUXPYRWNNZMUXPUWHMUWFUWIXSUXBUXPYRUIZUVTXKZUXRUXBBUXSUVTUXBUVMBUXSO - AUVMUWEUWIUVOTZBYRVAQXNUXBYRWPMZUWAUXTUXRMUXBUVMUYBUYABYRWRQAUVHUWAUWIXTU - VTYRUXSUXSVCYARYBBUXPEYCRYBAUXKUXMUXOSVJZUWEUWIAUUHUUJUYCHIUXGDEFGYDRTXIU - XBCUVFYECUXGYPUXJUXJVCYFVNXJUXBYQWPMZUVFUXCPUVLUXFVJUXBUUBUYDUXHCYQWRQUXB - UVFCUXCAUVHUWAUWIWOUXIVBUVFYQUXCUXCVCYGRYHAUWGUWIYIUWELWIYJYMYKYLYKYNYO + sylib eleq1d ccld simplrl ctop topontop toponmax focdmex sylc ssexd sstrd + cvv restopn2 sylan simprbda adantrl an32s mpbir2and elrestr eqeltrrd cdif + difeq1d ssdifssd imadif 3syl eqtr4d simpr simplrr opncld eqeltrd restcldr + funcnvcnv qtopcld difssd restcldi isopn2 mpbird wo mpjaodan expimpd ssrdv + expr sylbid eqssd ) AEDUBUCZCUDUCZEBUDUCZDBUEZUBUCZAYSYRYQUGUCMZYQCUFNMZY + SUHZCOZYQYTPAYSYRYPUGUCMZUUAADEYPUGUCMZBEUIZPUUEAEFUFNMZDFUJZUUFHAFGDUKZU + UIIFGDULQZDEFUMRABFUUGABDUNZCUTZFKADUOZUUMFDCUPAFDUUKUQZURUSZAUUHFUUGOHFE + VAQVBBDEYPUUGUUGVCVDRAYPGUFNMZUUCCPZCGPZUUEUUAVJAUUHUUJUUQHIDEFGVERZAUUDU + URAUUCDBUTZCDBVFAUVADUUMUTZCABUUMDKVGAUUJUUSUVBCOIJFGCDVHRVIZVKZUUCCVLQJC + YSYRYPGVMVNVOAUUQUUSUUBUUTJCYPGVPRZUVDYSYRYQCVQVNAUAYTYQAUAVRZYTMZUVFCPZY + SUNUVFUTZYRMZSZUVFYQMZAYRBUFNMZBCYSUKZUVGUVKVJAUUHBFPUVMHUUPBEFVPRZABUVAY + SUKZUVNADVSZBUUNPUVPAUUIUVQUUKFDWAQZABFUUNUUPUUOVTBDWBRAUVACOUVPUVNVJUVCU + VACBYSWCQVOUVFYSYRBCWDRAUVHUVJUVLAUVHSZUVJUULUVFUTZYRMZUVLUVSUVIUVTYRUVSU + VIUVTBWEZUVTBUVFDWFUVSUVTBPZUWBUVTOUVSUVTUUMBUVHUVTUUMPAUVFCUULWGWHABUUMO + ZUVHKWIVTUVTBWJWLWKWMAUVHUWAUVLAUVHUWASZSZBEMZUVLBEWNNZMZUWFUWGSZUVFCWEZU + VFYQUWJUVHUWKUVFOAUVHUWAUWGWOZUVFCWJWLUWJYPWPMZCXCMZUVFYPMZUWKYQMAUWMUWEU + WGAUUQUWMUUTGYPWQQTAUWNUWEUWGACGXCAFEMZUUJGXCMAUUHUWPHFEWRQIFGEDWSWTJXATU + WJUWOUVFGPZUVTEMZUWJUVFCGUWLAUUSUWEUWGJTXBAUWGUWEUWRAUWGSZUWAUWRUVHUWSUWA + UWRUWCAEWPMZUWGUWAUWRUWCSVJAUUHUWTHFEWQQBUVTEXDXEXFXGXHAUWOUWQUWRSVJZUWEU + WGAUUHUUJUXAHIUVFDEFGWDRTXIUVFCYPWPXCXJVNXKUWFUWISZUVLYQUIZUVFXLZYQWNNZMZ + UXBCUVFXLZUXDUXEUXBCUXCUVFUXBUUBCUXCOAUUBUWEUWIUVETZCYQVAQZXMUXBCYPUIZPUX + GYPWNNMZUXGCPUXGUXEMUXBCGUXJAUUSUWEUWIJTZUXBUUQGUXJOAUUQUWEUWIUUTTGYPVAQV + BUXBUXKUXGGPZUULUXGUTZUWHMZUXBCGUVFUXLXNUXBUXNBUVTXLZUWHUXBUXNUUMUVTXLZUX + PUXBUVQUULUNVSUXNUXQOAUVQUWEUWIUVRTDYCCUVFUULXOXPUXBBUUMUVTAUWDUWEUWIKTXM + XQUXBUWIUXPYRWNNZMUXPUWHMUWFUWIXRUXBUXPYRUIZUVTXLZUXRUXBBUXSUVTUXBUVMBUXS + OAUVMUWEUWIUVOTZBYRVAQXMUXBYRWPMZUWAUXTUXRMUXBUVMUYBUYABYRWQQAUVHUWAUWIXS + UVTYRUXSUXSVCXTRYABUXPEYBRYAAUXKUXMUXOSVJZUWEUWIAUUHUUJUYCHIUXGDEFGYDRTXI + UXBCUVFYECUXGYPUXJUXJVCYFVNXKUXBYQWPMZUVFUXCPUVLUXFVJUXBUUBUYDUXHCYQWQQUX + BUVFCUXCAUVHUWAUWIWOUXIVBUVFYQUXCUXCVCYGRYHAUWGUWIYIUWELWIYJYMYNYKYNYLYO $. $} @@ -308009,13 +308004,13 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario indishmph $p |- ( A ~~ B -> { (/) , A } ~= { (/) , B } ) $= ( vf wbr wf1o c0 cpr wcel ccn cmap f1of cvv elmapd mpbird ctopon cfv wceq co wf syl cen wex chmph bren chmeo ccnv wfo cdm f1odm vex eqeltrrdi f1ofo - cv dmex fornex indistopon cnindis syl2anc eleqtrrd f1ocnv ishmeo sylanbrc - sylc hmphi exlimiv sylbi ) ABUADABCUMZEZCUBFAGZFBGZUCDZABCUDVHVKCVHVGVIVJ - UERHZVKVHVGVIVJIRZHVGUFZVJVIIRZHVLVHVGBAJRZVMVHVGVPHABVGSABVGKVHBAVGLLVHA - LHZABVGUGBLHZVHAVGUHLABVGUIVGCUJUNUKZABVGULABLVGUOVCZVSMNVHVIAOPHZVRVMVPQ - VHVQWAVSALUPTVTBVILAUQURUSVHVNABJRZVOVHVNWBHBAVNSZVHBAVNEWCABVGUTBAVNKTVH - ABVNLLVSVTMNVHVJBOPHZVQVOWBQVHVRWDVTBLUPTVSAVJLBUQURUSVGVIVJVAVBVGVIVJVDT - VEVF $. + dmex focdmex sylc indistopon cnindis syl2anc eleqtrrd f1ocnv ishmeo hmphi + cv sylanbrc exlimiv sylbi ) ABUADABCVCZEZCUBFAGZFBGZUCDZABCUDVHVKCVHVGVIV + JUERHZVKVHVGVIVJIRZHVGUFZVJVIIRZHVLVHVGBAJRZVMVHVGVPHABVGSABVGKVHBAVGLLVH + ALHZABVGUGBLHZVHAVGUHLABVGUIVGCUJUMUKZABVGULABLVGUNUOZVSMNVHVIAOPHZVRVMVP + QVHVQWAVSALUPTVTBVILAUQURUSVHVNABJRZVOVHVNWBHBAVNSZVHBAVNEWCABVGUTBAVNKTV + HABVNLLVSVTMNVHVJBOPHZVQVOWBQVHVRWDVTBLUPTVSAVJLBUQURUSVGVIVJVAVDVGVIVJVB + TVEVF $. $} ${ @@ -313558,11 +313553,11 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the Carneiro, 23-Aug-2015.) $) cnmpt1plusg $p |- ( ph -> ( x e. X |-> ( A .+ B ) ) e. ( K Cn J ) ) $= ( cfv co cmpt ccn wcel cplusf cv cbs wceq ctopon ctmd eqid tmdtopon syl - wa cnf2 syl3anc fvmptelrn plusfval syl2anc mpteq2dva ctx tmdcn cnmpt12f - wf eqeltrrd ) ABICDFUAPZQZRBICDEQZRHGSQZABIVCVDABUBITUJCFUCPZTDVFTVCVDU - DABICVFAHIUEPTZGVFUEPTZBICRZVETIVFVIUTMAFUFTZVHLFGVFJVFUGZUHUIZNVIHGIVF - UKULUMABIDVFAVGVHBIDRZVETIVFVMUTMVLOVMHGIVFUKULUMVFEVBFCDVKKVBUGZUNUOUP - ABCDVBHGGGIMNOAVJVBGGUQQGSQTLVBFGJVNURUIUSVA $. + wa wf cnf2 syl3anc fvmptelcdm plusfval syl2anc mpteq2dva tmdcn cnmpt12f + ctx eqeltrrd ) ABICDFUAPZQZRBICDEQZRHGSQZABIVCVDABUBITUJCFUCPZTDVFTVCVD + UDABICVFAHIUEPTZGVFUEPTZBICRZVETIVFVIUKMAFUFTZVHLFGVFJVFUGZUHUIZNVIHGIV + FULUMUNABIDVFAVGVHBIDRZVETIVFVMUKMVLOVMHGIVFULUMUNVFEVBFCDVKKVBUGZUOUPU + QABCDVBHGGGIMNOAVJVBGGUTQGSQTLVBFGJVNURUIUSVA $. $} cnmpt2plusg.l $e |- ( ph -> L e. ( TopOn ` Y ) ) $. @@ -315447,71 +315442,71 @@ discrete topology (which is Hausdorff), this theorem can be used to turn A. x e. K ( ( H ` x ) .- ( G gsum ( F |` ( { x } X. n ) ) ) ) e. L ) ) $= ( vf vz vg vy cpw cfn cin cv wf cfv wss co cmpt cres cgsu crn wcel wi wral wa csn cxp wrex wex elin2d elfpw simplbi sselda eqid ccmn adantr - syl ctps ctgp tgptps fovrn syl3an1 fmpttd cima df-ima ctopon tgptopon - 3expa toponss syl2anc resmptd rneqd eqtrid chmeo ccom wceq ffvelcdmda - cminusg grpsubval sylan mpteq2dva cgrp grpinvcl grpinvf feqmptd eqidd - tgpgrp oveq2 fmptco eqtr4d eqeltrd eqeltrrd cvv syldan ralbidv adantl - ovex sstrdi sylib sylanbrc a1i adantlr oveq2d eleq1d ad2antrr reseq2d - 3syl weq tgplacthmeo hmeoco hmeoima grpsubid1 elrnmpt1s sylancl tsmsi - grpinvhmeo ralrimiva sseq1 imbi1d ac6sfi cun frn inss1 sspwuni rnxpss - cuni rnss unssd wfo wfn ffn dffn4 fofi inss2 unifi elinel2 rnfi ssun2 - unfi adantrr wb fvssunirn ssun1 sstri id sseqtrrid pm5.5 reseq2 bitrd - rspcv cmnmnd simplr jca ovexd c0g fvexi fsuppmptdm gsumcl velsn ovres - cmnd sylanbr oveq1 eqtrd gsumsn syl3anc snfi snssd fssresd fdmfifsupp - xpss12 xpfi sylancr gsumxp 3eqtr4rd elrnmpti isabl ad3antrrr ablnncan - cabl simpr syl5ibrcom rexlimdva syl5bi syld an32s ralimdva impr fveq2 - sylbid xpeq1d oveq12d cbvralvw sseq2 anbi12d rspcev syl12anc exlimddv - sneq xpeq2 ) AOEVBZVCVDZURVEZVFZHVEZUYOVGZUSVEZVHZLIEUYQIVEZKVIZVJZUY - SVKZVLVIZUTPUYQMVGZUTVEZQVIZVJZVMZVNZVOZUSUYNVPZHOVPZVQZFVMZJVEZVHZBV - EZMVGZLKVUSVRZVUQVSZVKZVLVIZQVIZPVNZBOVPZVQZJUYNVTZURAOVCVNZVAVEZUYSV - HZVUKVOZUSUYNVPZVAUYNVTZHOVPVUOURWAACVBZVCOUOWBZAVVOHOAUYQOVNZUYQCVNZ - VVOAOCUYQAOVVPVCVDVNZOCVHZUOVVTVWAVVJOCWCWDWIWEZAVVSVQZUSVAEDVUFUYNVU - JVUCLNSUAUIUYNWFALWGVNZVVSUBWHALWJVNZVVSALWKVNZVWEUCLWLWIWHAESVNVVSUE - WHVWCIEVUBDAVVSVUAEVNZVUBDVNZACEVSZDKVFZVVSVWGVWHUFUYQVUADCEKWMZWNWTW - OUHVWCUTDVUHVJZPWPZVUJNVWCVWMVWLPVKZVMVUJVWLPWQVWCVWNVUIVWCUTDPVUHAPD - VHZVVSANDWRVGVNZPNVNZVWOAVWFVWPUCLNDUIUAWSWIUMPNDXAXBZWHXCXDXEVWCVWLN - NXFVIZVNVWQVWMNVNVWCVWLVADVUFVVKGVIZVJZLXJVGZXGZVWSVWCVWLUTDVUFVUGVXB - VGZGVIZVJVXCVWCUTDVUHVXEVWCVUFDVNZVUGDVNZVUHVXEXHACDUYQMUGXIZDGLVXBQV - 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cima df-ima tgptopon + ctopon toponss syl2anc resmptd rneqd eqtrid chmeo ccom wceq grpsubval + cminusg ffvelcdmda sylan mpteq2dva cgrp tgpgrp grpinvcl grpinvf eqidd + feqmptd fmptco eqtr4d eqeltrd eqeltrrd cvv ovex syldan ralbidv adantl + oveq2 sstrdi 3syl sylanbrc a1i adantlr oveq2d eleq1d ad2antrr reseq2d + sylib weq grpinvhmeo hmeoco hmeoima grpsubid1 elrnmpt1s sylancl tsmsi + tgplacthmeo ralrimiva sseq1 imbi1d ac6sfi cuni cun inss1 sspwuni rnss + frn rnxpss unssd wfo wfn ffn dffn4 fofi inss2 unifi elinel2 rnfi unfi + adantrr ssun2 wb fvssunirn ssun1 sstri id sseqtrrid pm5.5 bitrd rspcv + reseq2 cmnd cmnmnd simplr jca ovexd c0g fvexi fsuppmptdm gsumcl velsn + ovres sylanbr oveq1 eqtrd gsumsn syl3anc snssd xpss12 fssresd sylancr + snfi xpfi fdmfifsupp gsumxp 3eqtr4rd elrnmpti cabl ad3antrrr ablnncan + isabl simpr syl5ibrcom rexlimdva syl5bi syld an32s ralimdva impr sneq + sylbid fveq2 xpeq1d oveq12d cbvralvw anbi12d rspcev syl12anc exlimddv + sseq2 xpeq2 ) 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S A. d e. T ( c .+ d ) e. U ) $. @@ -315528,9 +315523,9 @@ discrete topology (which is Hausdorff), this theorem can be used to turn xpfi fdmfifsupp gsumcl ablpncan3 syl12anc cv wral cfv csn cof wa adantr cmpt snfi sylancr sselda snssd cvv c0g fvexi a1i fmpttd eqid fsuppmptdm wf ovexd gsumsub fvexd feqresmpt eqidd offval2 oveq2d cmnd cmnmnd simpr - fovrnd velsn ovres sylanbr oveq1 mpteq2dva gsumxp oveq2 eleq1d eqeltrrd - weq eqtrd gsumsn syl3anc adantll 3eqtr4d eqtr4d 3eqtr3d cmap fveq2 sneq - xpeq1d reseq2d oveq12d rspccva elmapd mpbird rspcdva rspc2va syl21anc + fovcdmd weq velsn ovres sylanbr oveq1 mpteq2dva gsumxp eleq1d eqeltrrd + oveq2 eqtrd gsumsn syl3anc adantll 3eqtr4d eqtr4d 3eqtr3d fveq2 reseq2d + cmap sneq xpeq1d oveq12d rspccva elmapd mpbird rspcdva rspc2va syl21anc sylan ) AONRUAVKZVLZVMWCZOPRVLZVMWCZUVCTWCZGWCZUVEJAOVNVOZUVCDVOUVEDVOU VGUVEVPAOVQVOZOVRVOZUVHAOVSVOUVIUIOVTWAUHOWBWDZAUVADUVBOWEUDUGUPUHARWEV OZUAWEVOZUVAWEVOACWFZWERVAWGZAEWFZWEUAVCWGZRUAWOWHZACEVKZDUVANULARCWIZU @@ -315549,16 +315544,16 @@ discrete topology (which is Hausdorff), this theorem can be used to turn UAUWPUYCNWCZXGZVMWCZUXAUYFUXIOYGVOZUXHUYODVOUYLUYOVPUXIUVJUYPUXJOYHWAAU XHYIUXIUADUYNOWEUDUGUPUXJUXMUXIVIUAUYMDUXIUYCUAVOZXEZUWPUYCDCENUXIUXOUY QUXPXFUXIUWPCVOUYQUXQXFUXIUAEUYCUXRXJYJXPUXIVIUAUYNXLXLUYMUDUYNXQUXMUYR - UWPUYCNXTUXTXRWQUYKDUYOVJOUWPRUGVJVHYTZUYJUYNOVMUYSVIUAUYIUYMUYSUYQXEUY - IUYHUYCNWCZUYMUYSUYHUWRVOUYQUYIUYTVPVJUWPYKUYHUYCUWRUANYLYMUYSUYTUYMVPU - YQUYHUWPUYCNYNXFUUAYOYFUUBUUCUXIUWRDUAVJVIUWTOWEWEUDUGUPUXJUXKUXIUXLXOU - XMUXSUYAYPUXIUYEUYNOVMUXIVIUAUYDUYMUXHUYQUYDUYMVPAUWPUYCRUANYLUUDYOYFUU - EYOYFARDUAVHVIUVBOWEWEUDUGUPUHUVOUVQUWGUWHYPUUFYFUUGAOKWTZVMWCZIVOUXDIV - OKSRUUHWCZUXCVUAUXCVPVUBUXDIVUAUXCOVMYQYRVGAUXCVUCVORSUXCXSAVHRUXBSABWT - ZPXBZONVUDXCZUAVKZVLZVMWCZTWCZSVOZBRXAUXHUXBSVOZVEVUKVULBUWPRBVHYTZVUJU - XBSVUMVUEUWQVUIUXATVUDUWPPUUIVUMVUHUWTOVMVUMVUGUWSNVUMVUFUWRUAVUDUWPUUJ - UUKUULYFUUMYRUUNUUTXPASRUXCQUWBUSVAUUOUUPUUQYSVBUWNUWOUVCUWLGWCZJVOUEUF - UVCUVFHIUWKUVCVPUWMVUNJUWKUVCUWLGYNYRUWLUVFVPVUNUVGJUWLUVFUVCGYQYRUURUU + UWPUYCNXTUXTXRWQUYKDUYOVJOUWPRUGVJVHYKZUYJUYNOVMUYSVIUAUYIUYMUYSUYQXEUY + IUYHUYCNWCZUYMUYSUYHUWRVOUYQUYIUYTVPVJUWPYLUYHUYCUWRUANYMYNUYSUYTUYMVPU + YQUYHUWPUYCNYOXFUUAYPYFUUBUUCUXIUWRDUAVJVIUWTOWEWEUDUGUPUXJUXKUXIUXLXOU + XMUXSUYAYQUXIUYEUYNOVMUXIVIUAUYDUYMUXHUYQUYDUYMVPAUWPUYCRUANYMUUDYPYFUU + EYPYFARDUAVHVIUVBOWEWEUDUGUPUHUVOUVQUWGUWHYQUUFYFUUGAOKWTZVMWCZIVOUXDIV + OKSRUUJWCZUXCVUAUXCVPVUBUXDIVUAUXCOVMYTYRVGAUXCVUCVORSUXCXSAVHRUXBSABWT + ZPXBZONVUDXCZUAVKZVLZVMWCZTWCZSVOZBRXAUXHUXBSVOZVEVUKVULBUWPRBVHYKZVUJU + XBSVUMVUEUWQVUIUXATVUDUWPPUUHVUMVUHUWTOVMVUMVUGUWSNVUMVUFUWRUAVUDUWPUUK + UULUUIYFUUMYRUUNUUTXPASRUXCQUWBUSVAUUOUUPUUQYSVBUWNUWOUVCUWLGWCZJVOUEUF + UVCUVFHIUWKUVCVPUWMVUNJUWKUVCUWLGYOYRUWLUVFVPVUNUVGJUWLUVFUVCGYTYRUURUU SYS $. $} @@ -315947,13 +315942,13 @@ unit group (that is, the nonzero numbers) to the field. (Contributed (Contributed by Mario Carneiro, 5-Oct-2015.) $) cnmpt1vsca $p |- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) ) $= ( wcel cscaf cfv co cmpt ccn cv cbs wceq ctopon ctps ctlm tlmscatps syl - wa wf eqid istps sylib cnf2 syl3anc fvmptelrn scafval syl2anc mpteq2dva - tlmtps ctx vscacn cnmpt12f eqeltrrd ) ABKCDJUAUBZUCZUDBKCDEUCZUDIGUEUCZ - ABKVKVLABUFKTUNCFUGUBZTDJUGUBZTVKVLUHABKCVNAIKUIUBTZHVNUIUBTZBKCUDZIHUE - UCTKVNVRUOQAFUJTZVQAJUKTZVSPFJLULUMVNHFVNUPZOUQURRVRIHKVNUSUTVAABKDVOAV - PGVOUIUBTZBKDUDZVMTKVOWCUOQAJUJTZWBAVTWDPJVEUMVOGJVOUPZNUQURSWCIGKVOUSU - TVAVOVJEFVNJCDWELWAVJUPZMVBVCVDABCDVJIHGGKQRSAVTVJHGVFUCGUEUCTPVJFGHJWF - NLOVGUMVHVI $. + wa wf eqid istps sylib cnf2 syl3anc fvmptelcdm tlmtps scafval mpteq2dva + syl2anc ctx vscacn cnmpt12f eqeltrrd ) ABKCDJUAUBZUCZUDBKCDEUCZUDIGUEUC + ZABKVKVLABUFKTUNCFUGUBZTDJUGUBZTVKVLUHABKCVNAIKUIUBTZHVNUIUBTZBKCUDZIHU + EUCTKVNVRUOQAFUJTZVQAJUKTZVSPFJLULUMVNHFVNUPZOUQURRVRIHKVNUSUTVAABKDVOA + VPGVOUIUBTZBKDUDZVMTKVOWCUOQAJUJTZWBAVTWDPJVBUMVOGJVOUPZNUQURSWCIGKVOUS + UTVAVOVJEFVNJCDWELWAVJUPZMVCVEVDABCDVJIHGGKQRSAVTVJHGVFUCGUEUCTPVJFGHJW + FNLOVGUMVHVI $. $} cnmpt2vsca.m $e |- ( ph -> M e. ( TopOn ` Y ) ) $. @@ -317944,8 +317939,8 @@ unit group (that is, the nonzero numbers) to the field. (Contributed by Thierry Arnoux, 7-Feb-2018.) $) psmetcl $p |- ( ( D e. ( PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* ) $= - ( cpsmet cfv wcel cxp cxr wf co psmetf fovrn syl3an1 ) CDEFGDDHICJADGBDGA - BCKIGCDLABIDDCMN $. + ( cpsmet cfv wcel cxp cxr wf co psmetf fovcdm syl3an1 ) CDEFGDDHICJADGBDG + ABCKIGCDLABIDDCMN $. $d a A $. $( The distance function of a pseudometric space is zero if its arguments @@ -318185,25 +318180,25 @@ the base set to the (extended) reals and which is nonnegative, symmetric except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.) $) isxmet2d $p |- ( ph -> D e. ( *Met ` X ) ) $= - ( wcel wa co cc0 wceq cle wbr cpnf cv cxr fovrnda 0xr xrletri3 biantrud - wb sylancl 3bitr2d w3a cr cxad cmnf caddc 3expa rexadd breqtrrd anassrs - adantl 3adantr3 pnfge syl ad2antrr oveq2 wne cicc cxp wral ffnd elxrge0 - wf wfn sylanbrc ralrimivva adantr simpr3 simpr1 fovrnd eliccxr xaddpnf1 - ffnov renemnf syl2an sylan9eqr simpr2 simprbi ge0nemnf syl2anc necon4bd - wi wn a1d imp w3o elxr sylib mpjao3dan oveq1 xaddpnf2 isxmetd ) ABCDEFG - HIABUAZGMZCUAZGMZNNZXAXCEOZPQZXFPRSZPXFRSZNZXHXAXCQXEXFUBMZPUBMXGXJUGAX - AXCUBGGEIUCZUDXFPUEUHXEXIXHJUFKUIAXBXDDUAZGMZUJZNZXMXAEOZUKMZXFXQXMXCEO - ZULOZRSZXQTQZXQUMQZXPXRNZXSUKMZYAXSTQZXSUMQZXPXRYEYAXPXRYENZNXFXQXSUNOZ - XTRAXOYHXFYIRSLUOYHXTYIQXPXQXSUPUSUQURYDYFNXFTXTRXPXFTRSZXRYFXPXKYJAXBX - DXKXNXLUTXFVAVBZVCYFYDXTXQTULOZTXSTXQULVDXPXQUBMZXQUMVEZYLTQXRXPXQPTVFO - ZMZYMXPXMXAYOGGEAGGVGZYOEVKZXOAEYQVLXFYOMZCGVHBGVHYRAYQUBEIVIAYSBCGGXEX - KXIYSXLJXFVJVMVNBCGGYOEWAVMVOZAXBXDXNVPZAXBXDXNVQVRZXQPTVSVBZXQWBXQVTWC - WDUQYDYGYAXPYGYAWJXRXPYAXSUMXPXSUMVEZYAWKZXPXSUBMZPXSRSZUUDXPXSYOMZUUFX - PXMXCYOGGEYTUUAAXBXDXNWEVRZXSPTVSVBZXPUUHUUGUUIUUHUUFUUGXSVJWFVBXSWGWHZ - WLWIVOWMYDUUFYEYFYGWNXPUUFXRUUJVOXSWOWPWQXPYBNXFTXTRXPYJYBYKVOYBXPXTTXS - ULOZTXQTXSULWRXPUUFUUDUULTQUUJUUKXSWSWHWDUQXPYCYAXPYAXQUMXPYNUUEXPYMPXQ - RSZYNUUCXPYPUUMUUBYPYMUUMXQVJWFVBXQWGWHWLWIWMXPYMXRYBYCWNUUCXQWOWPWQWT - $. + ( wcel wa co cc0 wceq cle wbr cpnf cv cxr fovcdmda 0xr xrletri3 sylancl + wb biantrud 3bitr2d w3a cr cxad cmnf caddc 3expa rexadd adantl breqtrrd + anassrs 3adantr3 pnfge syl ad2antrr oveq2 wne cicc cxp wf wfn wral ffnd + elxrge0 sylanbrc ralrimivva ffnov adantr simpr3 fovcdmd eliccxr renemnf + simpr1 xaddpnf1 syl2an sylan9eqr wi simpr2 simprbi ge0nemnf syl2anc a1d + wn necon4bd imp w3o elxr sylib mpjao3dan oveq1 xaddpnf2 isxmetd ) ABCDE + FGHIABUAZGMZCUAZGMZNNZXAXCEOZPQZXFPRSZPXFRSZNZXHXAXCQXEXFUBMZPUBMXGXJUG + AXAXCUBGGEIUCZUDXFPUEUFXEXIXHJUHKUIAXBXDDUAZGMZUJZNZXMXAEOZUKMZXFXQXMXC + EOZULOZRSZXQTQZXQUMQZXPXRNZXSUKMZYAXSTQZXSUMQZXPXRYEYAXPXRYENZNXFXQXSUN + OZXTRAXOYHXFYIRSLUOYHXTYIQXPXQXSUPUQURUSYDYFNXFTXTRXPXFTRSZXRYFXPXKYJAX + BXDXKXNXLUTXFVAVBZVCYFYDXTXQTULOZTXSTXQULVDXPXQUBMZXQUMVEZYLTQXRXPXQPTV + FOZMZYMXPXMXAYOGGEAGGVGZYOEVHZXOAEYQVIXFYOMZCGVJBGVJYRAYQUBEIVKAYSBCGGX + EXKXIYSXLJXFVLVMVNBCGGYOEVOVMVPZAXBXDXNVQZAXBXDXNWAVRZXQPTVSVBZXQVTXQWB + WCWDURYDYGYAXPYGYAWEXRXPYAXSUMXPXSUMVEZYAWKZXPXSUBMZPXSRSZUUDXPXSYOMZUU + FXPXMXCYOGGEYTUUAAXBXDXNWFVRZXSPTVSVBZXPUUHUUGUUIUUHUUFUUGXSVLWGVBXSWHW + IZWJWLVPWMYDUUFYEYFYGWNXPUUFXRUUJVPXSWOWPWQXPYBNXFTXTRXPYJYBYKVPYBXPXTT + XSULOZTXQTXSULWRXPUUFUUDUULTQUUJUUKXSWSWIWDURXPYCYAXPYAXQUMXPYNUUEXPYMP + XQRSZYNUUCXPYPUUMUUBYPYMUUMXQVLWGVBXQWHWIWJWLWMXPYMXRYBYCWNUUCXQWOWPWQW + T $. $} $} @@ -318237,15 +318232,15 @@ the base set to the (extended) reals and which is nonnegative, symmetric of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) $) xmetcl $p |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* ) $= - ( cxmet cfv wcel cxp cxr wf co xmetf fovrn syl3an1 ) CDEFGDDHICJADGBDGABC - KIGCDLABIDDCMN $. + ( cxmet cfv wcel cxp cxr wf co xmetf fovcdm syl3an1 ) CDEFGDDHICJADGBDGAB + CKIGCDLABIDDCMN $. $( Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.) $) metcl $p |- ( ( D e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR ) $= - ( cmet cfv wcel cxp cr wf co metf fovrn syl3an1 ) CDEFGDDHICJADGBDGABCKIG - CDLABIDDCMN $. + ( cmet cfv wcel cxp cr wf co metf fovcdm syl3an1 ) CDEFGDDHICJADGBDGABCKI + GCDLABIDDCMN $. $( An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, @@ -318253,15 +318248,15 @@ the base set to the (extended) reals and which is nonnegative, symmetric ismet2 $p |- ( D e. ( Met ` X ) <-> ( D e. ( *Met ` X ) /\ D : ( X X. X ) --> RR ) ) $= ( vx vy vz cmet cfv wcel cvv cxmet cr wf wa elfvex cv wceq cle wral cxr - co cxp adantr cc0 wb caddc wbr cxad simpllr simpr simplrl simplrr rexaddd - fovrnd breq2d ralbidva anbi2d 2ralbidva wss ressxr fss biantrurd pm5.32da - sylancl bitr3d biancomd ismet isxmet anbi1d 3bitr4d pm5.21nii ) ABFGHZBIH - ZABJGHZBBUAZKALZMZABFNVMVLVOABJNUBVLVOCOZDOZATZUCPVQVRPUDZVSEOZVQATZWAVRA - TZUETZQUFZEBRZMZDBRCBRZMZVNSALZVTVSWBWCUGTZQUFZEBRZMZDBRCBRZMZVOMVKVPVLWI - WPVOVLVOWHWPVLVOMZWOWHWPWQWNWGCDBBWQVQBHZVRBHZMZMZWMWFVTXAWLWEEBXAWABHZMZ - WKWDVSQXCWBWCXCWAVQKBBAVLVOWTXBUHZXAXBUIZWQWRWSXBUJUMXCWAVRKBBAXDXEWQWRWS - XBUKUMULUNUOUPUQWQWJWOWQVOKSURWJVLVOUIUSVNKSAUTVCVAVDVBVECDEIABVFVLVMWPVO - CDEIABVGVHVIVJ $. + co cxp adantr cc0 wb caddc wbr cxad simpllr simpr simplrl fovcdmd simplrr + rexaddd breq2d ralbidva anbi2d 2ralbidva wss ressxr fss sylancl biantrurd + bitr3d pm5.32da biancomd ismet isxmet anbi1d 3bitr4d pm5.21nii ) ABFGHZBI + HZABJGHZBBUAZKALZMZABFNVMVLVOABJNUBVLVOCOZDOZATZUCPVQVRPUDZVSEOZVQATZWAVR + ATZUETZQUFZEBRZMZDBRCBRZMZVNSALZVTVSWBWCUGTZQUFZEBRZMZDBRCBRZMZVOMVKVPVLW + IWPVOVLVOWHWPVLVOMZWOWHWPWQWNWGCDBBWQVQBHZVRBHZMZMZWMWFVTXAWLWEEBXAWABHZM + ZWKWDVSQXCWBWCXCWAVQKBBAVLVOWTXBUHZXAXBUIZWQWRWSXBUJUKXCWAVRKBBAXDXEWQWRW + SXBULUKUMUNUOUPUQWQWJWOWQVOKSURWJVLVOUIUSVNKSAUTVAVBVCVDVECDEIABVFVLVMWPV + OCDEIABVGVHVIVJ $. $( A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) $) @@ -318622,63 +318617,63 @@ the base set to the (extended) reals and which is nonnegative, symmetric Carneiro, 20-Aug-2015.) $) prdsxmetlem $p |- ( ph -> D e. ( *Met ` B ) ) $= ( vf vg vh vz cvv wcel cbs fvexi a1i cxp cc0 cpnf co wf cxr wss prdsdsf - cicc iccssxr fss sylancl cv wa cle wbr fovrnda elxrge0 simprbi syl cmpt - cfv crn csn cun clt csup wral adantr ralrimiva simprl simprr prdsdsval3 - wceq breq1d cxmet adantlr prdsbascl r19.21bi xmetcl syl3anc fmpttd frnd - wb 0xr snssd unssd supxrleub 0le0 c0ex breq1 ralsn ralunb mpbiran2 ovex - mpbir rgenw ralrnmptw ax-mp bitri xmetge0 biantrud xrletri3 xmeteq0 wfn - eqid prdsbasfn syl2anc w3a cr caddc 3adantr3 3adant3 3ad2ant1 ssun1 cab - elabrex adantl rnmpt eleqtrrdi sselid supxrub syl2an2r breqtrrd xrrege0 - wrex syl22anc mpbird fovrnd 3bitr2d ralbidva fnmpt eqfnfv bitr4d bitrid - 3bitrd 3ad2antl1 simp23 simp3l simp21 simp3r readdcld xmettri2 syl13anc - simp22 rexaddd breqtrd readdcl 3ad2ant3 le2addd addge0d sylibr sylanbrc - cxad letrd rexrd eqbrtrd isxmet2d ) AUCUDUEDUGCCUGUHACLUIOUJUKACCULZUMU - NUTUOZDUPZUVKUQURUVJUQDUPABCDEFGHIJKLMNOPQRSTUAUBUSZUMUNVAUVJUVKUQDVBVC - AUCVDZCUHZUDVDZCUHZVEZVEZUVNUVPDUOZUVKUHZUMUVTVFVGZAUVNUVPUVKCCDUVMVHUW - AUVTUQUHUWBUVTVIVJVKUVSUVTUMVFVGBHBVDZUVNVMZUWCUVPVMZGUOZVLZVNZUMVOZVPZ - UQVQVRZUMVFVGZUFVDZUMVFVGZUFUWJVSZUVNUVPWEZUVSUVTUWKUMVFUVSBCDEFGUVNUVP - HIJKMLNOAFJUHZUVRSVTZAHKUHZUVRTVTZAEMUHZBHVSZUVRAUXABHUAWAZVTZAUVOUVQWB - ZAUVOUVQWCZPQRWDZWFUVSUWJUQURZUMUQUHZUWLUWOWOUVSUWHUWIUQUVSHUQUWGUVSBHU - WFUQUVSUWCHUHZVEZGIWGVMUHZUWDIUHZUWEIUHZUWFUQUHZAUXJUXLUVRUBWHZUVSUXMBH - UVSBCEFUVNHIJKMLNOUWRUWTUXDPUXEWIZWJZUVSUXNBHUVSBCEFUVPHIJKMLNOUWRUWTUX - DPUXFWIZWJZUWDUWEGIWKZWLZWMWNUVSUMUQUXIUVSWPUKWQWRZWPUFUWJUMWSVCUWOUWFU - MVFVGZBHVSZUVSUWPUWOUWNUFUWHVSZUYEUWOUYFUWNUFUWIVSZUYGUMUMVFVGZWTUWNUYH - UFUMXAUWMUMUMVFXBXCXGUWNUFUWHUWIXDXEUWFUGUHZBHVSUYFUYEWOUYIBHUWDUWEGXFX - HUWNUYDBUFHUWFUWGUGUWGXQZUWMUWFUMVFXBXIXJXKUVSUYEUWDUWEWEZBHVSZUWPUVSUY - DUYKBHUXKUYDUYDUMUWFVFVGZVEZUWFUMWEZUYKUXKUYMUYDUXKUXLUXMUXNUYMUXPUXRUX - TUWDUWEGIXLZWLXMUXKUXOUXIUYOUYNWOUYBWPUWFUMXNVCUXKUXLUXMUXNUYOUYKWOUXPU - XRUXTUWDUWEGIXOWLUUAUUBUVSUVNHXPUVPHXPUWPUYLWOUVSCBHEVLZFUVNHJKLNOUWRUW - TAUYQHXPZUVRAUXBUYRUXCBHEUYQMUYQXQUUCVKVTZUXEXRUVSCUYQFUVPHJKLNOUWRUWTU - 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Eliminate disjoint variable @@ -318983,34 +318978,34 @@ the base set to the (extended) reals and which is nonnegative, symmetric f1ofn oveq2 ralrn forn raleqdv bitr3d ralbidv mpbid oveq1 ffnov xmeteq0 sylanbrc syl3anc eqeq1d wf1 f1of1 f1fveq 3bitr4d simpr xmettri2 oveq12d 3brtr4d ralrimiva breq2d jca eqeq2 bibi12d oveq2d breq12d anbi12d eqeq1 - syl13anc oveq1d cvv elfvexd fornex sylc isxmet mpbir2and ) ACBUFUGUHZBB - UIZUJCUKZUAUOZQUOZCRZULSZYBYCSZUPZYDUBUOZYBCRZYHYCCRZUMRZUNUQZUBBTZURZQ - BTZUABTZACXTUSYDUJUHZQBTZUABTZYAABCDEDUTUGZGHIJKAHBGVAZHBGVBZLHBGVCVDZM - YTVEOVFAUCUOZGUGZYCCRZUJUHZQBTZUCHTZYSAUUEUDUOZGUGZCRZUJUHZUDHTZUCHTUUI - AUUMUCUDHHAUUDHUHZUUJHUHZURZURZUULUUDUUJFRZUJUURBCDEFGHUUDUUJIAEGDVJRSZ - UUQJVGZAHDVHUGSZUUQKVGZAUUAUUQLVGZADIUHZUUQMVGZNOAFHUFUGUHZUUQPVGZAUUOU - UPVIZAUUOUUPVKZVLZAUVGUUQUUSUJUHZPUVGUUOUUPUVLUUDUUJFHVMVNVOVPVQAUUNUUH - UCHAUUGQGVRZTZUUNUUHAGHUSZUVNUUNUPAUUAUVOLHBGVTVDZUUGUUMQUDHGYCUUKSZUUF - UULUJYCUUKUUECWAZVSWBVDAUUGQUVMBAUUBUVMBSUUCHBGWCVDZWDWEWFWGAYRUAUVMTZU - UIYSAUVOUVTUUIUPUVPYRUUHUAUCHGYBUUESZYQUUGQBUWAYDUUFUJYBUUEYCCWHZVSWFWB - VDAYRUAUVMBUVSWDWEWGUAQBBUJCWIWKAUUFULSZUUEYCSZUPZUUFYHUUECRZYJUMRZUNUQ - ZUBBTZURZQBTZUCHTZYPAUULULSZUUEUUKSZUPZUULUWFYHUUKCRZUMRZUNUQZUBBTZURZU - DHTZUCHTUWLAUWTUCUDHHUURUWOUWSUURUUSULSZUUDUUJSZUWMUWNUURUVGUUOUUPUXBUX - CUPUVHUVIUVJUUDUUJFHWJWLUURUULUUSULUVKWMAHBGWNZUUQUWNUXCUPAUUAUXDLHBGWO - VDHBUUDUUJGWPVOWQUURUULUEUOZGUGZUUECRZUXFUUKCRZUMRZUNUQZUEHTZUWSUURUXJU - EHUURUXEHUHZURZUUSUXEUUDFRZUXEUUJFRZUMRZUULUXIUNUXMUVGUXLUUOUUPUUSUXPUN - UQUURUVGUXLUVHVGZUURUXLWRZUURUUOUXLUVIVGZUURUUPUXLUVJVGZUUDUUJUXEFHWSXK - UURUULUUSSUXLUVKVGUXMUXGUXNUXHUXOUMUXMBCDEFGHUXEUUDIUURUUTUXLUVAVGZUURU - VBUXLUVCVGZUURUUAUXLUVDVGZUURUVEUXLUVFVGZNOUXQUXRUXSVLUXMBCDEFGHUXEUUJI - UYAUYBUYCUYDNOUXQUXRUXTVLWTXAXBAUXKUWSUPUUQAUWRUBUVMTZUXKUWSAUVOUYEUXKU - PUVPUWRUXJUBUEHGYHUXFSZUWQUXIUULUNUYFUWFUXGUWPUXHUMYHUXFUUECWHYHUXFUUKC - WHWTXCWBVDAUWRUBUVMBUVSWDWEVGWGXDVQAUXAUWKUCHAUWJQUVMTZUXAUWKAUVOUYGUXA - UPUVPUWJUWTQUDHGUVQUWEUWOUWIUWSUVQUWCUWMUWDUWNUVQUUFUULULUVRWMYCUUKUUEX - EXFUVQUWHUWRUBBUVQUUFUULUWGUWQUNUVRUVQYJUWPUWFUMYCUUKYHCWAXGXHWFXIWBVDA - UWJQUVMBUVSWDWEWFWGAYOUAUVMTZUWLYPAUVOUYHUWLUPUVPYOUWKUAUCHGUWAYNUWJQBU - WAYGUWEYMUWIUWAYEUWCYFUWDUWAYDUUFULUWBWMYBUUEYCXJXFUWAYLUWHUBBUWAYDUUFY - KUWGUNUWBUWAYIUWFYJUMYBUUEYHCWAXLXHWFXIWFWBVDAYOUAUVMBUVSWDWEWGABXMUHZX - SYAYPURUPAHXMUHUUBUYIAFUFHPXNUUCHBXMGXOXPUAQUBXMCBXQVDXR $. + syl13anc oveq1d cvv elfvexd focdmex sylc isxmet mpbir2and ) ACBUFUGUHZB + BUIZUJCUKZUAUOZQUOZCRZULSZYBYCSZUPZYDUBUOZYBCRZYHYCCRZUMRZUNUQZUBBTZURZ + QBTZUABTZACXTUSYDUJUHZQBTZUABTZYAABCDEDUTUGZGHIJKAHBGVAZHBGVBZLHBGVCVDZ + MYTVEOVFAUCUOZGUGZYCCRZUJUHZQBTZUCHTZYSAUUEUDUOZGUGZCRZUJUHZUDHTZUCHTUU + IAUUMUCUDHHAUUDHUHZUUJHUHZURZURZUULUUDUUJFRZUJUURBCDEFGHUUDUUJIAEGDVJRS + ZUUQJVGZAHDVHUGSZUUQKVGZAUUAUUQLVGZADIUHZUUQMVGZNOAFHUFUGUHZUUQPVGZAUUO + UUPVIZAUUOUUPVKZVLZAUVGUUQUUSUJUHZPUVGUUOUUPUVLUUDUUJFHVMVNVOVPVQAUUNUU + HUCHAUUGQGVRZTZUUNUUHAGHUSZUVNUUNUPAUUAUVOLHBGVTVDZUUGUUMQUDHGYCUUKSZUU + FUULUJYCUUKUUECWAZVSWBVDAUUGQUVMBAUUBUVMBSUUCHBGWCVDZWDWEWFWGAYRUAUVMTZ + UUIYSAUVOUVTUUIUPUVPYRUUHUAUCHGYBUUESZYQUUGQBUWAYDUUFUJYBUUEYCCWHZVSWFW + BVDAYRUAUVMBUVSWDWEWGUAQBBUJCWIWKAUUFULSZUUEYCSZUPZUUFYHUUECRZYJUMRZUNU + QZUBBTZURZQBTZUCHTZYPAUULULSZUUEUUKSZUPZUULUWFYHUUKCRZUMRZUNUQZUBBTZURZ + UDHTZUCHTUWLAUWTUCUDHHUURUWOUWSUURUUSULSZUUDUUJSZUWMUWNUURUVGUUOUUPUXBU + XCUPUVHUVIUVJUUDUUJFHWJWLUURUULUUSULUVKWMAHBGWNZUUQUWNUXCUPAUUAUXDLHBGW + OVDHBUUDUUJGWPVOWQUURUULUEUOZGUGZUUECRZUXFUUKCRZUMRZUNUQZUEHTZUWSUURUXJ + UEHUURUXEHUHZURZUUSUXEUUDFRZUXEUUJFRZUMRZUULUXIUNUXMUVGUXLUUOUUPUUSUXPU + NUQUURUVGUXLUVHVGZUURUXLWRZUURUUOUXLUVIVGZUURUUPUXLUVJVGZUUDUUJUXEFHWSX + KUURUULUUSSUXLUVKVGUXMUXGUXNUXHUXOUMUXMBCDEFGHUXEUUDIUURUUTUXLUVAVGZUUR + UVBUXLUVCVGZUURUUAUXLUVDVGZUURUVEUXLUVFVGZNOUXQUXRUXSVLUXMBCDEFGHUXEUUJ + IUYAUYBUYCUYDNOUXQUXRUXTVLWTXAXBAUXKUWSUPUUQAUWRUBUVMTZUXKUWSAUVOUYEUXK + UPUVPUWRUXJUBUEHGYHUXFSZUWQUXIUULUNUYFUWFUXGUWPUXHUMYHUXFUUECWHYHUXFUUK + CWHWTXCWBVDAUWRUBUVMBUVSWDWEVGWGXDVQAUXAUWKUCHAUWJQUVMTZUXAUWKAUVOUYGUX + AUPUVPUWJUWTQUDHGUVQUWEUWOUWIUWSUVQUWCUWMUWDUWNUVQUUFUULULUVRWMYCUUKUUE + XEXFUVQUWHUWRUBBUVQUUFUULUWGUWQUNUVRUVQYJUWPUWFUMYCUUKYHCWAXGXHWFXIWBVD + AUWJQUVMBUVSWDWEWFWGAYOUAUVMTZUWLYPAUVOUYHUWLUPUVPYOUWKUAUCHGUWAYNUWJQB + UWAYGUWEYMUWIUWAYEUWCYFUWDUWAYDUUFULUWBWMYBUUEYCXJXFUWAYLUWHUBBUWAYDUUF + YKUWGUNUWBUWAYIUWFYJUMYBUUEYHCWAXLXHWFXIWFWBVDAYOUAUVMBUVSWDWEWGABXMUHZ + XSYAYPURUPAHXMUHUUBUYIAFUFHPXNUUCHBXMGXOXPUAQUBXMCBXQVDXR $. $} imasf1omet.m $e |- ( ph -> E e. ( Met ` V ) ) $. @@ -319618,8 +319613,8 @@ the base set to the (extended) reals and which is nonnegative, symmetric NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) $) blssm $p |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( P ( ball ` D ) R ) C_ X ) $= - ( cxmet cfv wcel cxr w3a cbl co cxp cpw wf blf fovrn syl3an1 elpwid ) ADE - FGZBDGZCHGZIBCAJFZKZDSDHLDMZUBNTUAUCUDGADOBCUDDHUBPQR $. + ( cxmet cfv wcel cxr w3a cbl co cxp cpw wf blf fovcdm syl3an1 elpwid ) AD + EFGZBDGZCHGZIBCAJFZKZDSDHLDMZUBNTUAUCUDGADOBCUDDHUBPQR $. $( The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, @@ -320881,32 +320876,32 @@ S C_ ( P ( ball ` D ) T ) ) $= vc cxp cicc wfn cv wral wf xmetf syl ffnd wa w3a xmetge0 elxrge0 sylanbrc xmetcl 3expb sylan ralrimivva ffnov cop opelxpi fvco3 syl2an df-ov fveq2i fcod 3eqtr4g eqeq1d fveq2 bibi12d ralrimiva adantr rspcdva xmeteq0 3bitrd - wb eqeq1 3adantr3 ffvelcdmd simpr3 simpr1 fovrnd simpr2 ge0xaddcl syl2anc - xaddcld 3anrot xmettri2 sylan2br wi breq1 breq1d imbi12d rspc2va syl21anc - breq2 breq2d fvoveq1 oveq1d breq12d fveq2d oveq2d xrletrd opelxpd oveq12d - mpd oveq2 fvco3d 3brtr4d isxmetd ) AUAUBUHEDUCZUDFADUEFGUFAFFUIZLUGUJMZNE - DHADXTUKUAULZUBULZDMZYAOZUBFUMUAFUMXTYADUNZAXTNDADFUEPOZXTNDUNZGDFUOUPZUQ - AYEUAUBFFAYGYBFOZYCFOZURZYEGYGYJYKYEYGYJYKUSYDNOLYDQRYEYBYCDFVCYBYCDFUTYD - VAVBVDVEZVFUAUBFFYADVGVBZVNAYLURZYBYCXSMZLSYDEPZLSZYDLSZYBYCSZYOYPYQLYOYB - YCVHZXSPZUUADPZEPZYPYQAYHUUAXTOUUBUUDSYLYIYBYCFFVIXTNUUAEDVJVKYBYCXSVLYDU - UCEYBYCDVLVMVOZVPYOBULZEPZLSZUUFLSZWDZYRYSWDBYAYDUUFYDSZUUHYRUUIYSUUKUUGY - QLUUFYDEVQZVPUUFYDLWEVRAUUJBYAUMYLAUUJBYAIVSVTYMWAAYGYLYSYTWDZGYGYJYKUUMY - BYCDFWBVDVEWCAYJYKUHULZFOZUSZURZYQUUNYBDMZEPZUUNYCDMZEPZTMZYPUUNYBXSMZUUN - YCXSMZTMQUUQYQUURUUTTMZEPZUVBUUQYANYDEAYANEUNUUPHVTZAYJYKYEUUOYMWFZWGUUQY - ANUVEEUVGUUQUURYAOZUUTYAOZUVEYAOZUUQUUNYBYAFFDAYFUUPYNVTZAYJYKUUOWHZAYJYK - UUOWIZWJZUUQUUNYCYAFFDUVLUVMAYJYKUUOWKZWJZUURUUTWLWMZWGUUQUUSUVAUUQYANUUR - EUVGUVOWGUUQYANUUTEUVGUVQWGWNUUQYDUVEQRZYQUVFQRZAYGUUPUVSGUUPYGUUOYJYKUSU - VSUUOYJYKWOYBYCUUNDFWPWQVEUUQYEUVKUUFCULZQRZUUGUWAEPZQRZWRZCYAUMBYAUMZUVS - UVTWRZUVHUVRAUWFUUPAUWEBCYAYAJVFVTUWEUWGYDUWAQRZYQUWCQRZWRBCYDUVEYAYAUUKU - WBUWHUWDUWIUUFYDUWAQWSUUKUUGYQUWCQUULWTXAUWAUVESZUWHUVSUWIUVTUWAUVEYDQXDU - WJUWCUVFYQQUWAUVEEVQXEXAXBXCXNUUQUVIUVJUUFUWATMEPZUUGUWCTMZQRZCYAUMBYAUMZ - UVFUVBQRZUVOUVQAUWNUUPAUWMBCYAYAKVFVTUWMUWOUURUWATMZEPZUUSUWCTMZQRBCUURUU - TYAYAUUFUURSZUWKUWQUWLUWRQUUFUURUWAETXFUWSUUGUUSUWCTUUFUUREVQXGXHUWAUUTSZ - UWQUVFUWRUVBQUWTUWPUVEEUWAUUTUURTXOXIUWTUWCUVAUUSTUWAUUTEVQXJXHXBXCXKAYJY - KYPYQSUUOUUEWFUUQUVCUUSUVDUVATUUQUUNYBVHZXSPUXADPZEPUVCUUSUUQXTNUXAEDAYHU - UPYIVTZUUQUUNYBFFUVMUVNXLXPUUNYBXSVLUURUXBEUUNYBDVLVMVOUUQUUNYCVHZXSPUXDD - PZEPUVDUVAUUQXTNUXDEDUXCUUQUUNYCFFUVMUVPXLXPUUNYCXSVLUUTUXEEUUNYCDVLVMVOX - MXQXR $. + wb eqeq1 ffvelcdmd simpr3 simpr1 fovcdmd simpr2 ge0xaddcl syl2anc xaddcld + 3adantr3 3anrot xmettri2 sylan2br wi breq1 breq1d imbi12d breq2d syl21anc + breq2 rspc2va fvoveq1 oveq1d breq12d fveq2d oveq2d xrletrd opelxpd fvco3d + mpd oveq2 oveq12d 3brtr4d isxmetd ) AUAUBUHEDUCZUDFADUEFGUFAFFUIZLUGUJMZN + EDHADXTUKUAULZUBULZDMZYAOZUBFUMUAFUMXTYADUNZAXTNDADFUEPOZXTNDUNZGDFUOUPZU + QAYEUAUBFFAYGYBFOZYCFOZURZYEGYGYJYKYEYGYJYKUSYDNOLYDQRYEYBYCDFVCYBYCDFUTY + DVAVBVDVEZVFUAUBFFYADVGVBZVNAYLURZYBYCXSMZLSYDEPZLSZYDLSZYBYCSZYOYPYQLYOY + BYCVHZXSPZUUADPZEPZYPYQAYHUUAXTOUUBUUDSYLYIYBYCFFVIXTNUUAEDVJVKYBYCXSVLYD + UUCEYBYCDVLVMVOZVPYOBULZEPZLSZUUFLSZWDZYRYSWDBYAYDUUFYDSZUUHYRUUIYSUUKUUG + YQLUUFYDEVQZVPUUFYDLWEVRAUUJBYAUMYLAUUJBYAIVSVTYMWAAYGYLYSYTWDZGYGYJYKUUM + YBYCDFWBVDVEWCAYJYKUHULZFOZUSZURZYQUUNYBDMZEPZUUNYCDMZEPZTMZYPUUNYBXSMZUU + NYCXSMZTMQUUQYQUURUUTTMZEPZUVBUUQYANYDEAYANEUNUUPHVTZAYJYKYEUUOYMWNZWFUUQ + YANUVEEUVGUUQUURYAOZUUTYAOZUVEYAOZUUQUUNYBYAFFDAYFUUPYNVTZAYJYKUUOWGZAYJY + KUUOWHZWIZUUQUUNYCYAFFDUVLUVMAYJYKUUOWJZWIZUURUUTWKWLZWFUUQUUSUVAUUQYANUU + REUVGUVOWFUUQYANUUTEUVGUVQWFWMUUQYDUVEQRZYQUVFQRZAYGUUPUVSGUUPYGUUOYJYKUS + UVSUUOYJYKWOYBYCUUNDFWPWQVEUUQYEUVKUUFCULZQRZUUGUWAEPZQRZWRZCYAUMBYAUMZUV + SUVTWRZUVHUVRAUWFUUPAUWEBCYAYAJVFVTUWEUWGYDUWAQRZYQUWCQRZWRBCYDUVEYAYAUUK + UWBUWHUWDUWIUUFYDUWAQWSUUKUUGYQUWCQUULWTXAUWAUVESZUWHUVSUWIUVTUWAUVEYDQXD + UWJUWCUVFYQQUWAUVEEVQXBXAXEXCXNUUQUVIUVJUUFUWATMEPZUUGUWCTMZQRZCYAUMBYAUM + ZUVFUVBQRZUVOUVQAUWNUUPAUWMBCYAYAKVFVTUWMUWOUURUWATMZEPZUUSUWCTMZQRBCUURU + UTYAYAUUFUURSZUWKUWQUWLUWRQUUFUURUWAETXFUWSUUGUUSUWCTUUFUUREVQXGXHUWAUUTS + ZUWQUVFUWRUVBQUWTUWPUVEEUWAUUTUURTXOXIUWTUWCUVAUUSTUWAUUTEVQXJXHXEXCXKAYJ + YKYPYQSUUOUUEWNUUQUVCUUSUVDUVATUUQUUNYBVHZXSPUXADPZEPUVCUUSUUQXTNUXAEDAYH + UUPYIVTZUUQUUNYBFFUVMUVNXLXMUUNYBXSVLUURUXBEUUNYBDVLVMVOUUQUUNYCVHZXSPUXD + DPZEPUVDUVAUUQXTNUXDEDUXCUUQUUNYCFFUVMUVPXLXMUUNYCXSVLUUTUXEEUUNYCDVLVMVO + XPXQXR $. $} ${ @@ -326629,11 +326624,11 @@ Normed space homomorphisms (bounded linear operators) (Contributed by Mario Carneiro, 5-Oct-2015.) $) cnmpt1ds $p |- ( ph -> ( x e. X |-> ( A D B ) ) e. ( K Cn R ) ) $= ( co cmpt wcel cbs cfv cxp cres ccn cv wa ctopon wf ctps cms mstps eqid - syl istps cnf2 syl3anc fvmptelrn ovresd mpteq2dva ctx cnmpt12f eqeltrrd - sylib msdcn ) ABJCDEGUAUBZVFUCUDZRZSBJCDERZSIFUERABJVHVIABUFJTUGCDEVFAB - JCVFAIJUHUBTZHVFUHUBTZBJCSZIHUERZTJVFVLUIOAGUJTZVKAGUKTZVNNGULUNVFHGVFU - MZLUOVDZPVLIHJVFUPUQURABJDVFAVJVKBJDSZVMTJVFVRUIOVQQVRIHJVFUPUQURUSUTAB - CDVGIHHFJOPQAVOVGHHVARFUERTNEHFGVFVPKLMVEUNVBVC $. + syl istps sylib cnf2 syl3anc fvmptelcdm ovresd mpteq2dva msdcn cnmpt12f + ctx eqeltrrd ) ABJCDEGUAUBZVFUCUDZRZSBJCDERZSIFUERABJVHVIABUFJTUGCDEVFA + BJCVFAIJUHUBTZHVFUHUBTZBJCSZIHUERZTJVFVLUIOAGUJTZVKAGUKTZVNNGULUNVFHGVF + UMZLUOUPZPVLIHJVFUQURUSABJDVFAVJVKBJDSZVMTJVFVRUIOVQQVRIHJVFUQURUSUTVAA + BCDVGIHHFJOPQAVOVGHHVDRFUERTNEHFGVFVPKLMVBUNVCVE $. $} cnmpt2ds.l $e |- ( ph -> L e. ( TopOn ` Y ) ) $. @@ -327493,7 +327488,7 @@ Normed space homomorphisms (bounded linear operators) $( Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.) $) - cncffvrn $p |- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> + cncfcdm $p |- ( ( C C_ CC /\ F e. ( A -cn-> B ) ) -> ( F e. ( A -cn-> C ) <-> F : A --> C ) ) $= ( vw vx vz vy cc wss ccncf co wcel wa cv cmin cabs cfv wral crp wf clt wi wbr wrex cncfi 3expb ralrimivva adantl wb cncfrss simpl elcncf2 mpbiran2d @@ -327504,9 +327499,9 @@ Normed space homomorphisms (bounded linear operators) $( The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.) $) cncfss $p |- ( ( B C_ C /\ C C_ CC ) -> ( A -cn-> B ) C_ ( A -cn-> C ) ) $= - ( vf wss cc wa ccncf co cv wcel cncff adantl simpll fssd cncffvrn adantll - wf wb mpbird ex ssrdv ) BCEZCFEZGZDABHIZACHIZUEDJZUFKZUHUGKZUEUIGZUJACUHR - ZUKABCUHUIABUHRUEABUHLMUCUDUINOUDUIUJULSUCABCUHPQTUAUB $. + ( vf wss cc wa ccncf co cv wcel wf cncff adantl simpll wb cncfcdm adantll + fssd mpbird ex ssrdv ) BCEZCFEZGZDABHIZACHIZUEDJZUFKZUHUGKZUEUIGZUJACUHLZ + UKABCUHUIABUHLUEABUHMNUCUDUIOSUDUIUJULPUCABCUHQRTUAUB $. $} ${ @@ -327639,11 +327634,11 @@ Normed space homomorphisms (bounded linear operators) $( Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) $) cncfcompt2 $p |- ( ph -> ( x e. A |-> T ) e. ( A -cn-> E ) ) $= - ( cmpt ccncf co wcel wss ccom cv wa adantr wf cncff syl fvmptelrn ralrimi - sseldd ex eqidd fmptcof eqcomd cc cncfrss cncfss syl2anc cncfco eqeltrd ) - ABDIPZCFHPZBDGPZUAZDJQRAVDVAABCDFGHIVCVBAGFSZBDKABUBDSZVEAVFUCEFGAEFTZVFN - UDABDGEAVCDEQRZSDEVCUELDEVCUFUGUHUJUKUIAVCULAVBULOUMUNADFJVCVBAVHDFQRZVCA - VGFUOTZVHVITNAVBFJQRSVJMFJVBUPUGDEFUQURLUJMUSUT $. + ( cmpt ccncf co wcel wss ccom cv wa adantr wf cncff syl fvmptelcdm sseldd + ex ralrimi eqidd fmptcof eqcomd cc cncfrss cncfss syl2anc cncfco eqeltrd + ) ABDIPZCFHPZBDGPZUAZDJQRAVDVAABCDFGHIVCVBAGFSZBDKABUBDSZVEAVFUCEFGAEFTZV + FNUDABDGEAVCDEQRZSDEVCUELDEVCUFUGUHUIUJUKAVCULAVBULOUMUNADFJVCVBAVHDFQRZV + CAVGFUOTZVHVITNAVBFJQRSVJMFJVBUPUGDEFUQURLUIMUSUT $. $} ${ @@ -327765,11 +327760,11 @@ Normed space homomorphisms (bounded linear operators) $( Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.) $) cncfmpt2ss $p |- ( ph -> ( x e. X |-> ( A F B ) ) e. ( X -cn-> S ) ) $= - ( co cmpt wcel wf cc wss ccncf cv wa syl fvmptelrn syl2anc fmpttd ctx ccn - cncff wb a1i ssid cncfss mp2an sselid cncfmpt2f cncffvrn sylancr mpbird ) - ABHCDFOZPZHEUAOZQZHEVBRZABHVAEABUBHQUCCEQDEQVAEQABHCEABHCPZVCQHEVFRKHEVFU - JUDUEABHDEABHDPZVCQHEVGRLHEVGUJUDUENUFUGAESTZVBHSUAOZQVDVEUKMABCDFGHIFGGU - HOGUIOQAJULAVCVIVFVHSSTVCVITMSUMHESUNUOZKUPAVCVIVGVJLUPUQHSEVBURUSUT $. + ( co cmpt wcel wf cc wss ccncf cv cncff syl fvmptelcdm syl2anc fmpttd ctx + wa wb ccn a1i ssid cncfss mp2an sselid cncfmpt2f cncfcdm sylancr mpbird ) + ABHCDFOZPZHEUAOZQZHEVBRZABHVAEABUBHQUICEQDEQVAEQABHCEABHCPZVCQHEVFRKHEVFU + CUDUEABHDEABHDPZVCQHEVGRLHEVGUCUDUENUFUGAESTZVBHSUAOZQVDVEUJMABCDFGHIFGGU + HOGUKOQAJULAVCVIVFVHSSTVCVITMSUMHESUNUOZKUPAVCVIVGVJLUPUQHSEVBURUSUT $. $} ${ @@ -333932,26 +333927,26 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner cxp cmopn ctx ccn wf cv wbr cabs cmin ccom wi wrex csca cphphl phlipf syl cphl cclm wss cphclm clmsscn fssd cip c2 cdiv cnm c1 caddc simpll simplrl simplrr simpr ipcnlem1 ralrimiva simprl ovresd breq1d simprr anbi12d wceq - ad2antrr fovrnd cnmetdval syl2anc ipfval adantl oveq12d imbi12d 2ralbidva - fveq2d eqtrd rexbidv ralbidv mpbird ralrimivva cxmet cxms cms cngp cphngp - ngpms msxms xmsxmet cnxmet a1i cnfldtopn txmetcn syl3anc mpbir2and mstopn - wb oveq1d eleqtrrd ) DUCLZADUDMZDUEMZYBUGZUFZUHMZYEUINZCUJNZBBUINZCUJNXTA - YGLZYCOAUKZUAULZHULZYDNZIULZPUMZUBULZJULZYDNZYNPUMZQZYKYPANZYLYQANZUNUOUP - ZNZKULZPUMZUQZJYBRHYBRZISURZKSRZUBYBRUAYBRZXTYCDUSMZUEMZOAXTDVCLYCUUMAUKD - UTUULAUUMYBDYBTZEUULTZUUMTZVAVBXTDVDLUUMOVEDVFUULUUMDUUOUUPVGVBVHZXTUUJUA - UBYBYBXTYKYBLZYPYBLZQZQZUUJYKYLYANZYNPUMZYPYQYANZYNPUMZQZYKYPDVIMZNZYLYQU - VGNZUONZUNMZUUEPUMZUQZJYBRHYBRZISURZKSRUVAUVOKSUVAUUESLZQHJYKYPYAUUEUUEVJ - VKNZYKDVLMZMVMVNNVKNZUVQYPUVRMUVSVNNVKNZUVGUVRYBDIUUNUVGTZYATUVRTUVSTUVTT - XTUUTUVPVOXTUURUUSUVPVPXTUURUUSUVPVQUVAUVPVRVSVTUVAUUIUVOKSUVAUUHUVNISUVA - UUGUVMHJYBYBUVAYLYBLZYQYBLZQZQZYTUVFUUFUVLUWEYOUVCYSUVEUWEYMUVBYNPUWEYKYL - YAYBXTUURUUSUWDVPZUVAUWBUWCWAZWBWCUWEYRUVDYNPUWEYPYQYAYBXTUURUUSUWDVQZUVA - UWBUWCWDZWBWCWEUWEUUDUVKUUEPUWEUUDUUAUUBUONZUNMZUVKUWEUUAOLUUBOLUUDUWKWFU - WEYKYPOYBYBAXTYJUUTUWDUUQWGZUWFUWHWHUWEYLYQOYBYBAUWLUWGUWIWHUUAUUBUUCUUCT - WIWJUWEUWJUVJUNUWEUUAUVHUUBUVIUOUWEUURUUSUUAUVHWFUWFUWHAUVGYBDYKYPUUNUWAE - WKWJUWDUUBUVIWFUVAAUVGYBDYLYQUUNUWAEWKWLWMWPWQWCWNWOWRWSWTXAXTYDYBXBMLZUW - MUUCOXBMLZYIYJUUKQXQXTDXCLZUWMXTDXDLZUWOXTDXELUWPDXFDXGVBZDXHVBYDDYBUUNYD - TZXIVBZUWSUWNXTXJXKUAUBKIJHYDYDUUCAYEYECYBYBOYETZUWTCGXLXMXNXOXTYHYFCUJXT - BYEBYEUIXTUWPBYEWFUWQYDBDYBFUUNUWRXPVBZUXAWMXRXS $. + ad2antrr fovcdmd cnmetdval syl2anc ipfval adantl oveq12d fveq2d 2ralbidva + eqtrd imbi12d rexbidv ralbidv mpbird ralrimivva cxmet wb cxms cngp cphngp + cms ngpms msxms xmsxmet cnxmet cnfldtopn txmetcn syl3anc mpbir2and mstopn + a1i oveq1d eleqtrrd ) DUCLZADUDMZDUEMZYBUGZUFZUHMZYEUINZCUJNZBBUINZCUJNXT + AYGLZYCOAUKZUAULZHULZYDNZIULZPUMZUBULZJULZYDNZYNPUMZQZYKYPANZYLYQANZUNUOU + PZNZKULZPUMZUQZJYBRHYBRZISURZKSRZUBYBRUAYBRZXTYCDUSMZUEMZOAXTDVCLYCUUMAUK + DUTUULAUUMYBDYBTZEUULTZUUMTZVAVBXTDVDLUUMOVEDVFUULUUMDUUOUUPVGVBVHZXTUUJU + AUBYBYBXTYKYBLZYPYBLZQZQZUUJYKYLYANZYNPUMZYPYQYANZYNPUMZQZYKYPDVIMZNZYLYQ + UVGNZUONZUNMZUUEPUMZUQZJYBRHYBRZISURZKSRUVAUVOKSUVAUUESLZQHJYKYPYAUUEUUEV + JVKNZYKDVLMZMVMVNNVKNZUVQYPUVRMUVSVNNVKNZUVGUVRYBDIUUNUVGTZYATUVRTUVSTUVT + TXTUUTUVPVOXTUURUUSUVPVPXTUURUUSUVPVQUVAUVPVRVSVTUVAUUIUVOKSUVAUUHUVNISUV + AUUGUVMHJYBYBUVAYLYBLZYQYBLZQZQZYTUVFUUFUVLUWEYOUVCYSUVEUWEYMUVBYNPUWEYKY + LYAYBXTUURUUSUWDVPZUVAUWBUWCWAZWBWCUWEYRUVDYNPUWEYPYQYAYBXTUURUUSUWDVQZUV + AUWBUWCWDZWBWCWEUWEUUDUVKUUEPUWEUUDUUAUUBUONZUNMZUVKUWEUUAOLUUBOLUUDUWKWF + UWEYKYPOYBYBAXTYJUUTUWDUUQWGZUWFUWHWHUWEYLYQOYBYBAUWLUWGUWIWHUUAUUBUUCUUC + TWIWJUWEUWJUVJUNUWEUUAUVHUUBUVIUOUWEUURUUSUUAUVHWFUWFUWHAUVGYBDYKYPUUNUWA + EWKWJUWDUUBUVIWFUVAAUVGYBDYLYQUUNUWAEWKWLWMWNWPWCWQWOWRWSWTXAXTYDYBXBMLZU + WMUUCOXBMLZYIYJUUKQXCXTDXDLZUWMXTDXGLZUWOXTDXELUWPDXFDXHVBZDXIVBYDDYBUUNY + DTZXJVBZUWSUWNXTXKXQUAUBKIJHYDYDUUCAYEYECYBYBOYETZUWTCGXLXMXNXOXTYHYFCUJX + TBYEBYEUIXTUWPBYEWFUWQYDBDYBFUUNUWRXPVBZUXAWMXRXS $. $} ${ @@ -333970,12 +333965,12 @@ of the norm of the sum of two orthogonal vectors (i.e., whose inner Carneiro, 13-Oct-2015.) $) cnmpt1ip $p |- ( ph -> ( x e. X |-> ( A ., B ) ) e. ( K Cn C ) ) $= ( cfv co wcel cipf cmpt ccn cv wa cbs wceq ctopon ctps ccph cngp cphngp - wf ngptps 3syl eqid istps sylib cnf2 syl3anc fvmptelrn ipfval mpteq2dva - syl2anc ctx ipcn syl cnmpt12f eqeltrrd ) ABJCDIUARZSZUBBJCDFSZUBHEUCSAB - JVKVLABUDJTUECIUFRZTDVMTVKVLUGABJCVMAHJUHRTZGVMUHRTZBJCUBZHGUCSZTJVMVPU - MOAIUITZVOAIUJTZIUKTVRNIULIUNUOVMGIVMUPZKUQURZPVPHGJVMUSUTVAABJDVMAVNVO - BJDUBZVQTJVMWBUMOWAQWBHGJVMUSUTVAVJFVMICDVTMVJUPZVBVDVCABCDVJHGGEJOPQAV - SVJGGVESEUCSTNVJGEIWCKLVFVGVHVI $. + ngptps 3syl eqid istps cnf2 syl3anc fvmptelcdm ipfval syl2anc mpteq2dva + wf sylib ctx ipcn syl cnmpt12f eqeltrrd ) ABJCDIUARZSZUBBJCDFSZUBHEUCSA + BJVKVLABUDJTUECIUFRZTDVMTVKVLUGABJCVMAHJUHRTZGVMUHRTZBJCUBZHGUCSZTJVMVP + VCOAIUITZVOAIUJTZIUKTVRNIULIUMUNVMGIVMUOZKUPVDZPVPHGJVMUQURUSABJDVMAVNV + OBJDUBZVQTJVMWBVCOWAQWBHGJVMUQURUSVJFVMICDVTMVJUOZUTVAVBABCDVJHGGEJOPQA + VSVJGGVESEUCSTNVJGEIWCKLVFVGVHVI $. $} cnmpt2ip.l $e |- ( ph -> L e. ( TopOn ` Y ) ) $. @@ -336854,28 +336849,28 @@ need not be complete (for instance if the given set is infinite ( vh wcel cfv crefld co c2 cexp cmpt wceq eqid cr cds cfrlm ctcph cnm csg ccom cv cmin cgsu csqrt cmpo rrxval fveq2d clmod cgrp crg recrng srngring csr ax-mp frlmlmod mpan lmodgrp tcphds 3syl cxp wf cbs grpsubf cc0 cfsupp - cmap crab rrxbase rebase re0g frlmbas eqtrd sqxpeqd feq23d mpbird fovrnda - wbr wfn ffnd fnov sylib rrxnm eqtr2d oveq1d mpteq2dv oveq2d fmpoco w3a wa - fveq1 cof simp1 simprl adantr eleqtrd 3impb frlmbasmap syl2anc elmapi syl - simprr inidm eqidd a1i frlmsubgval ffvelcdmda resubgval mpteq2dva 3eqtr4d - offval simpl resubcld fvmpt2d mpoeq3dva 3eqtr2rd ) FGKZEUALMFUBNZUCLZUALZ - YDUDLZYCUELZUFZCDBBMAFAUGZCUGZLZYIDUGZLZUHNZOPNZQZUINZUJLZUKZYBEYDUAEFGHU - LZUMYBYCUNKZYCUOKZYHYERMUPKZYBUUAMUSKUUCUQMURUTZMYCFGYCSZVAVBZYCVCZYDYGYF - YCYDSYFSYGSZVDVEYBYHCDBBMAFYIYJYLYGNZLZOPNZQZUINZUJLZUKYSYBCDJBBBUUIMAFYI - JUGZLZOPNZQZUINZUJLZUUNYGYFYBYJYLBBBYGYBBBVFZBYGVGYCVHLZUVBVFZUVBYGVGZYBU - UAUUBUVDUUFUUGUVBYCYGUVBSZUUHVIVEYBUVABUVCUVBYGYBBUVBYBBUUOVJVKWCJTFVLNZV - MZUVBBJEFGHIVNUUCYBUVGUVBRUUDUVGMJYCFTUPGVJUUEVOVPUVGSVQVBVRZVSUVHVTWAZWB - YBYGUVAWDYGCDBBUUIUKRYBUVABYGUVIWECDBBYGWFWGYBJBUUTQEUDLYFABJEFGHIWHYBEYD - UDYTUMWIUUOUUIRZUUSUUMUJUVJUURUULMUIUVJAFUUQUUKUVJUUPUUJOPYIUUOUUIWPWJWKW - LUMWMYBCDBBUUNYRYBYJBKZYLBKZWNZUUMYQUJUVMUULYPMUIUVMAFUUKYOUVMYIFKWOZUUJY - NOPUVMAFYNUUITUVMYJYLMUELZWQNZAFYKYMUVONZQUUIAFYNQUVMAFFYKYMUVOFYJYLGGUVM - FTYJUVMYJUVFKZFTYJVGUVMYBYJUVBKZUVRYBUVKUVLWRZYBUVKUVLUVSYBUVKUVLWOZWOZYJ - BUVBYBUVKUVLWSYBBUVBRUWAUVHWTZXAZXBUVBMYCFTGYJUUEVOUVEXCXDYJTFXEXFZWEUVMF - TYLUVMYLUVFKZFTYLVGUVMYBYLUVBKZUWFUVTYBUVKUVLUWGUWBYLBUVBYBUVKUVLXGUWCXAZ - XBUVBMYCFTGYLUUEVOUVEXCXDYLTFXEXFZWEUVTUVTFXHUVNYKXIUVNYMXIXPYBUVKUVLUUIU - VPRUWBUVBMYJYLFYGUVOGYCUUEUVEUUCUWBUUDXJYBUWAXQUWDUWHUVOSZUUHXKXBUVMAFYNU - VQUVNYKTKYMTKYNUVQRUVMFTYIYJUWEXLZUVMFTYIYLUWIXLZUVOYKYMUWJXMXDXNXOUVNYKY - MUWKUWLXRXSWJXNWLUMXTVRYA $. + wbr cmap crab rrxbase rebase frlmbas eqtrd sqxpeqd feq23d mpbird fovcdmda + re0g wfn ffnd fnov sylib rrxnm eqtr2d fveq1 oveq1d mpteq2dv oveq2d fmpoco + w3a wa simp1 simprl adantr eleqtrd 3impb frlmbasmap syl2anc elmapi simprr + cof syl inidm eqidd offval a1i frlmsubgval ffvelcdmda resubgval mpteq2dva + simpl 3eqtr4d resubcld fvmpt2d mpoeq3dva 3eqtr2rd ) FGKZEUALMFUBNZUCLZUAL + ZYDUDLZYCUELZUFZCDBBMAFAUGZCUGZLZYIDUGZLZUHNZOPNZQZUINZUJLZUKZYBEYDUAEFGH + ULZUMYBYCUNKZYCUOKZYHYERMUPKZYBUUAMUSKUUCUQMURUTZMYCFGYCSZVAVBZYCVCZYDYGY + FYCYDSYFSYGSZVDVEYBYHCDBBMAFYIYJYLYGNZLZOPNZQZUINZUJLZUKYSYBCDJBBBUUIMAFY + IJUGZLZOPNZQZUINZUJLZUUNYGYFYBYJYLBBBYGYBBBVFZBYGVGYCVHLZUVBVFZUVBYGVGZYB + UUAUUBUVDUUFUUGUVBYCYGUVBSZUUHVIVEYBUVABUVCUVBYGYBBUVBYBBUUOVJVKVLJTFVMNZ + VNZUVBBJEFGHIVOUUCYBUVGUVBRUUDUVGMJYCFTUPGVJUUEVPWCUVGSVQVBVRZVSUVHVTWAZW + BYBYGUVAWDYGCDBBUUIUKRYBUVABYGUVIWECDBBYGWFWGYBJBUUTQEUDLYFABJEFGHIWHYBEY + DUDYTUMWIUUOUUIRZUUSUUMUJUVJUURUULMUIUVJAFUUQUUKUVJUUPUUJOPYIUUOUUIWJWKWL + WMUMWNYBCDBBUUNYRYBYJBKZYLBKZWOZUUMYQUJUVMUULYPMUIUVMAFUUKYOUVMYIFKWPZUUJ + YNOPUVMAFYNUUITUVMYJYLMUELZXFNZAFYKYMUVONZQUUIAFYNQUVMAFFYKYMUVOFYJYLGGUV + MFTYJUVMYJUVFKZFTYJVGUVMYBYJUVBKZUVRYBUVKUVLWQZYBUVKUVLUVSYBUVKUVLWPZWPZY + JBUVBYBUVKUVLWRYBBUVBRUWAUVHWSZWTZXAUVBMYCFTGYJUUEVPUVEXBXCYJTFXDXGZWEUVM + FTYLUVMYLUVFKZFTYLVGUVMYBYLUVBKZUWFUVTYBUVKUVLUWGUWBYLBUVBYBUVKUVLXEUWCWT + ZXAUVBMYCFTGYLUUEVPUVEXBXCYLTFXDXGZWEUVTUVTFXHUVNYKXIUVNYMXIXJYBUVKUVLUUI + UVPRUWBUVBMYJYLFYGUVOGYCUUEUVEUUCUWBUUDXKYBUWAXPUWDUWHUVOSZUUHXLXAUVMAFYN + UVQUVNYKTKYMTKYNUVQRUVMFTYIYJUWEXMZUVMFTYIYLUWIXMZUVOYKYMUWJXNXCXOXQUVNYK + YMUWKUWLXRXSWKXOWMUMXTVRYA $. ${ rrxvsca.r $e |- .xb = ( .s ` H ) $. @@ -338147,15 +338142,15 @@ Hilbert space (in the algebraic sense, meaning that all algebraically $( The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) divcncf $p |- ( ph -> ( x e. X |-> ( A / B ) ) e. ( X -cn-> CC ) ) $= - ( vy cdiv co cmpt c1 cc ccncf cv wcel wf cncff syl cc0 cmul fvmptelrn csn - wa cdif eldifad wne eldifsni divrecd mpteq2dva ccom csb ralrimiva fmptcos - eqidd wceq csbov2g csbvarg oveq2d eqtr2d ax-1cn eqid cdivcncf mp1i cncfco - eqtrd eqeltrd mulcncf ) ABECDIJZKBECLDIJZUAJZKEMNJZABEVIVKABOEPUDZCDABECM - ABECKZVLPEMVNQFEMVNRSUBVMDMTUCZABEDMVOUEZABEDKZEVPNJPEVPVQQGEVPVQRSUBZUFZ - VMDVPPZDTUGVRDMTUHSUIUJABCVJEFABEVJKZHVPLHOZIJZKZVQUKZVLAWEBEHDWCULZKWAAB - HEVPDWCVQWDAVTBEVRUMAVQUOAWDUOUNABEWFVJVMWFLHDWBULZIJZVJVMDMPZWFWHUPVSHDL - WBIMUQSVMWGDLIVMWIWGDUPVSHDMURSUSVFUJUTAEVPMVQWDGLMPWDVPMNJPAVAHLWDWDVBVC - VDVEVGVHVG $. + ( vy cdiv co cmpt c1 cc ccncf cv wcel wf cncff syl cc0 cmul wa fvmptelcdm + csn cdif eldifad wne eldifsni divrecd mpteq2dva ccom ralrimiva eqidd wceq + csb fmptcos csbov2g csbvarg oveq2d eqtrd eqtr2d ax-1cn eqid cdivcncf mp1i + cncfco eqeltrd mulcncf ) ABECDIJZKBECLDIJZUAJZKEMNJZABEVIVKABOEPUBZCDABEC + MABECKZVLPEMVNQFEMVNRSUCVMDMTUDZABEDMVOUEZABEDKZEVPNJPEVPVQQGEVPVQRSUCZUF + ZVMDVPPZDTUGVRDMTUHSUIUJABCVJEFABEVJKZHVPLHOZIJZKZVQUKZVLAWEBEHDWCUOZKWAA + BHEVPDWCVQWDAVTBEVRULAVQUMAWDUMUPABEWFVJVMWFLHDWBUOZIJZVJVMDMPZWFWHUNVSHD + LWBIMUQSVMWGDLIVMWIWGDUNVSHDMURSUSUTUJVAAEVPMVQWDGLMPWDVPMNJPAVBHLWDWDVCV + DVEVFVGVHVG $. $} @@ -339560,51 +339555,51 @@ Hilbert space (in the algebraic sense, meaning that all algebraically fmptd supxrcl rpred readdcld rexrd cioo cuni wbr eliun w3a 3adant3 wb wrex ovolfioo syl2anc mpbid simp3 sylc ccnv simpl1 f1ocnv 4syl 2fveq3 rsp wceq fveq2d fveq12d eqtrdi breq1d breq2d anbi12d rexlimdva mpbird - vex mpd c1 wi ad2antlr adantlr simplr sylan simpl2 simpr fovrnd fvmpt - fvex df-ov fveq2i opelxpd f1ocnvfv2 eqtrid op1st op2nd biimprd rspcev - cop syl6an rexlimdv3a syl5bi ralrimiv ovollb cfz fzfi elfznn ffvelcdm - eqtrd cfn nnre syl2an fimaxre3 sylancr fllep1 flcl peano2zd zred letr - cfl syl3anc mpan2d ralimdva simpll crp cexp cdiv ad2antrl ovoliunlem1 - c2 cz simprr expr syld wfn ffn breq1 ralrn supxrleub xrletrd ) AHUKBU - LZUMUNZFUOZUPUQURZEUOUPUQURZCUSUTZAUWQVCVAZUWRUPVBABVCVAZHUKVDUXCAUXD - HUKOVEHUKBVCVFVGZUWQVHVIAUWSUPVAZUWTUPVBAUWSVJVKVLUTZUPAUKVMVCVCVNVQZ - KVOZUKUXGFVOZUWSUXGVAAGUKGVPZLUNZVRUNZUXLVSUNZIUNZUNZUXHKAUXKUKVBZVTZ - UKUXHUXMUXOUXRUXOUXHUKWAUTZVBUKUXHUXOVOUXRUKUXSUXNIAUKUXSIVOZUXQUCWHU - XRUXLUKUKVNZVBZUXNUKVBAUKUYAUXKLAUKUYALWBZUKUYALVOZUBUKUYALWCVIZWDZUX - LUKUKWEVIWFVMUXOWGWIUXRUYBUXMUKVBUYFUXLUKUKWJVIWFUAWTZFKWKWLWMKWMZUYH - WNTWOZUKUXGFWPWQVJVKWRWSZUWSXAVIAUXBAUXACQACRXBXCXDZAUXIUWQXEKWMUOXFV - AZUWRUWTVMXGUYGAUYLUFVPZKUNZVSUNZUGVPZUQXGZUYPUYNVRUNZUQXGZVTZUFUKXLZ - 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(Contributed by Mario Carneiro, @@ -343215,8 +343210,8 @@ or are almost disjoint (the interiors are disjoint). (Contributed by (Contributed by Mario Carneiro, 11-Aug-2014.) $) mbfmptcl $p |- ( ( ph /\ x e. A ) -> B e. CC ) $= ( cc cmpt cdm wf cmbf wcel mbff syl wral wceq ralrimiva dmmptg feq2d - mpbid fvmptelrn ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEFUCNOAUDCHUCADEMZBCPUDCQA - UFBCGRBCDESOTUAUB $. + mpbid fvmptelcdm ) ABCDHABCDIZJZHUCKZCHUCKAUCLMUEFUCNOAUDCHUCADEMZBCPUDCQ + AUFBCGRBCDESOTUAUB $. $( The domain of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Aug-2014.) $) @@ -344717,78 +344712,78 @@ or are almost disjoint (the interiors are disjoint). 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iftrue ovmpoa mp2an 0cn mul01i pm2.61d2 f1ofo - c0ex wfo fofi an32s dfin4 eqsstrri sseli eqeltrdi mul02d sylan2 3eqtr2d - elsni ) AGHNUDOZUEUFZHUGZDUGZPUHZUIZUAUKZVUECUKZUJOZVUFIOZULOZUAUMZCUMZ - VUAGUGZBUKZVUFNOZVUMVUFIOZULOZBUMZCUMVULVUAVUPCUMBUMAUYTVUDVUEUYSUNVUEU - HUOZUPUFZULOZUAUMZVUDVUAVUICUMZUAUMVUKAUYSUEUQZQVUDURQZUYSUGZVUCUIVUDUS - VUDRVUCUIUSUYTVVASAGHJKUTAVUBURQZVUDVUBUSZVVDADURQVVFADNVULVUAVAZVGZURM - ANVBVBVAZVCZVVHURQZVVIURQVVJVBNVDZVVKVEVVJVBNVFVHAVULURQZVUAURQZVVLAGVV - CQZVVNJGVIVQZAHVVCQZVVOKHVIVQZVULVUAVJVKVVJVVHNVLVMVNDVOVQZVUBVUCVRZVUB - VUDVSVPZAVVEVUBVUCARVUBUYSAUBBRRRNVULVUAVUBGHVTVTAUBUKZVULQVUMVUAQTZTVW - CVUMWAZNUFZNVVHUOZVWCVUMNOVUBAVWEVVHQZVWFVWGQZVWDANWBZVVHNUQZUSZVWHVWIW - JVVMVWJVEVVJVBNWCVHZAVVHVVJVWKAVULVBUSVUAVBUSVVHVVJUSAVULRVBARRGAVVPRRG - VDJGWDVQZWEZWFWGAVUARVBARRHAVVRRRHVDZKHWDVQZWEZWFWGVULVBVUAVBWHVKVVJVBN - VEWIWKZVVHVWENWLVMVWCVUMVULVUAUUAUUBVWCVUMNWMVUBVVIUGVWGDVVIMUUDNVVHUUC - 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(Contributed by Mario Carneiro, 28-Jun-2014.) (Proof modification is discouraged.) @@ -344801,137 +344796,137 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by 6-Aug-2014.) $) i1fposd $p |- ( ph -> ( x e. RR |-> if ( 0 <_ A , A , 0 ) ) e. dom S.1 ) $= ( vy cr cc0 cmpt cfv cle wbr cif nfcv wceq breq2d ifbieq1d wcel syl eqid - cv citg1 cdm nffvmpt1 nfbr nfif fveq2 cbvmpt wa simpr wf fvmptelrn fvmpt2 - i1ff syl2anc mpteq2dva eqtrid i1fpos eqeltrrd ) AEFGETZBFCHZIZJKZVAGLZHZB - FGCJKZCGLZHZUAUBZAVDBFGBTZUTIZJKZVJGLZHVGEBFVCVLVBBVAGBGVAJBGMZBJMBFCUSUC - ZUDVNVMUEEVLMUSVINZVBVKVAVJGVOVAVJGJUSVIUTUFZOVPPUGABFVLVFAVIFQZUHZVKVEVJ - CGVRVJCGJVRVQCFQVJCNAVQUIABFCFAUTVHQZFFUTUJDUTUMRUKBFCFUTUTSULUNZOVTPUOUP - AVSVDVHQDEUTVDVDSUQRUR $. + cv citg1 cdm nffvmpt1 nfbr nfif fveq2 cbvmpt simpr i1ff fvmptelcdm fvmpt2 + wa wf syl2anc mpteq2dva eqtrid i1fpos eqeltrrd ) AEFGETZBFCHZIZJKZVAGLZHZ + BFGCJKZCGLZHZUAUBZAVDBFGBTZUTIZJKZVJGLZHVGEBFVCVLVBBVAGBGVAJBGMZBJMBFCUSU + CZUDVNVMUEEVLMUSVINZVBVKVAVJGVOVAVJGJUSVIUTUFZOVPPUGABFVLVFAVIFQZULZVKVEV + JCGVRVJCGJVRVQCFQVJCNAVQUHABFCFAUTVHQZFFUTUMDUTUIRUJBFCFUTUTSUKUNZOVTPUOU + PAVSVDVHQDEUTVDVDSUQRUR $. $} $( The difference of two simple functions is a simple function. (Contributed @@ -345335,41 +345330,41 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cn wcel wa cr co cc0 cdiv cfv cv cneg cicc cle wbr cif cexp cmul cmpt c2 cfz crn wceq mbfi1fseqlem2 adantl clt wb cxp wral cpnf cico rge0ssre cfl simpr ffvelcdm syl2an sselid cn0 2nn nnnn0 nnexpcl sylancr ad2antrl - wf nnred remulcld reflcl syl nndivred ralrimivva fmpo sylib fovrn 3expa - syl3an1 nnre ad2antlr nngt0 jca lemul1 syl3anc biimpa cuz fveq2d oveq2d - cz oveq12d ovex ovmpoa ad4ant23 oveq1d adantr ffvelcdmda elrege0 simpld - simpl nnnn0d nn0ge0d mulge0 syl12anc flge0nn0 nn0cnd nncnd nnne0d eqtrd - syl2anc divcan1d eqeltrd nn0uz eleqtrdi nnmulcl mpdan nnzd elfz5 mpbird - oveq1 eqid fvmpt recnd divcan4d wfn elfznn0 nn0red nndivre syl2anr ffnd - fmpttd fnfvelrn eqeltrrd wn eluzfz2 ifclda eluzfz1 nnne0 div0d fmpt3d + nnred remulcld reflcl syl nndivred ralrimivva fmpo sylib fovcdm syl3an1 + wf 3expa nnre ad2antlr nngt0 jca lemul1 syl3anc biimpa cuz fveq2d simpl + cz oveq2d oveq12d ovex ovmpoa ad4ant23 oveq1d adantr ffvelcdmda elrege0 + simpld nnnn0d nn0ge0d mulge0 syl12anc syl2anc nn0cnd nncnd nnne0d eqtrd + flge0nn0 divcan1d eqeltrd nn0uz eleqtrdi nnmulcl mpdan nnzd elfz5 oveq1 + mpbird eqid fvmpt recnd divcan4d wfn elfznn0 nn0red nndivre fmpttd ffnd + syl2anr fnfvelrn eqeltrrd wn eluzfz2 ifclda eluzfz1 nnne0 div0d fmpt3d nncn ifcld ) ADMNZOZBPBUAZDUBDUCQNZDUUKHQZDUDUEZUUMDUFZRUFZERDUJDUGQZUH QZUKQZEUAZUUQSQZUIZULZDGTZUUIUVDBPUUPUIUMAABCDEFGHIJKLUNUOUUJUUKPNZOZUU LUUORUVCUVFUUNUUMDUVCUVFUUNOZUUMUUQUHQZUVBTZUUMUVCUVGUVIUVHUUQSQZUUMUVG UVHUUSNZUVIUVJUMUVGUVKUVHUURUDUEZUVFUUNUVLUVFUUMPNZDPNZUUQPNZRUUQUPUEZO - ZUUNUVLUQAUUIUVEUVMAMPURPHVNZUUIUVEUVMACUAZFTZUJUUTUGQZUHQZVCTZUWASQZPN + ZUUNUVLUQAUUIUVEUVMAMPURPHWDZUUIUVEUVMACUAZFTZUJUUTUGQZUHQZVCTZUWASQZPN ZCPUSEMUSUVRAUWEECMPAUUTMNZUVSPNZOZOZUWCUWAUWIUWBPNUWCPNUWIUVTUWAUWIRUT - VAQZPUVTVBAPUWJFVNZUWGUVTUWJNUWHJUWFUWGVDPUWJUVSFVEVFVGUWIUWAUWFUWAMNZA - UWGUWFUJMNZUUTVHNUWLVIUUTVJUJUUTVKVLVMZVOVPUWBVQVRUWNVSVTECMPUWDPHKWAWB - DUUKPMPHWCWEWDZUUIUVNAUVEDWFWGZUVFUUQMNZUVQUUIUWQAUVEUUIUWMDVHNUWQVIDVJ - UJDVKVLZWGZUWQUVOUVPUUQWFUUQWHWIVRUUMDUUQWJWKWLUVGUVHRWMTZNUURWPNUVKUVL + VAQZPUVTVBAPUWJFWDZUWGUVTUWJNUWHJUWFUWGVDPUWJUVSFVEVFVGUWIUWAUWFUWAMNZA + UWGUWFUJMNZUUTVHNUWLVIUUTVJUJUUTVKVLVMZVNVOUWBVPVQUWNVRVSECMPUWDPHKVTWA + DUUKPMPHWBWCWEZUUIUVNAUVEDWFWGZUVFUUQMNZUVQUUIUWQAUVEUUIUWMDVHNUWQVIDVJ + UJDVKVLZWGZUWQUVOUVPUUQWFUUQWHWIVQUUMDUUQWJWKWLUVGUVHRWMTZNUURWPNUVKUVL UQUVGUVHVHUWTUVGUVHUUKFTZUUQUHQZVCTZVHUVGUVHUXCUUQSQZUUQUHQUXCUVGUUMUXD UUQUHUUIUVEUUMUXDUMAUUNECDUUKMPUWDUXDHUUTDUMZUVSUUKUMZOZUWCUXCUWAUUQSUX - GUWBUXBVCUXGUVTUXAUWAUUQUHUXGUVSUUKFUXEUXFVDWNUXGUUTDUJUGUXEUXFXFWOZWQW - NUXHWQKUXCUUQSWRWSWTXAUVGUXCUUQUVGUXCUVFUXCVHNZUUNUVFUXBPNRUXBUDUEZUXIU - VFUXAUUQUVFUXAPNZRUXAUDUEZUVFUXAUWJNUXKUXLOZUUJPUWJUUKFAUWKUUIJXBXCUXAX - DWBZXEUVFUUQUWSVOZVPUVFUXMUVORUUQUDUEUXJUXNUXOUVFUUQUVFUUQUWSXGXHUXAUUQ - XIXJUXBXKXPXBZXLUVGUUQUVFUWQUUNUWSXBZXMZUVGUUQUXQXNZXQXOUXPXRXSXTUVGUUR - UVFUURMNZUUNUUIUXTAUVEUUIUWQUXTUWRDUUQYAYBWGZXBYCUVHRUURYDXPYEZEUVHUVAU - VJUUSUVBUUTUVHUUQSYFUVBYGZUVHUUQSWRYHVRUVGUUMUUQUVGUUMUVFUVMUUNUWOXBYIU + GUWBUXBVCUXGUVTUXAUWAUUQUHUXGUVSUUKFUXEUXFVDWNUXGUUTDUJUGUXEUXFWOWQZWRW + NUXHWRKUXCUUQSWSWTXAXBUVGUXCUUQUVGUXCUVFUXCVHNZUUNUVFUXBPNRUXBUDUEZUXIU + VFUXAUUQUVFUXAPNZRUXAUDUEZUVFUXAUWJNUXKUXLOZUUJPUWJUUKFAUWKUUIJXCXDUXAX + EWAZXFUVFUUQUWSVNZVOUVFUXMUVORUUQUDUEUXJUXNUXOUVFUUQUVFUUQUWSXGXHUXAUUQ + XIXJUXBXPXKXCZXLUVGUUQUVFUWQUUNUWSXCZXMZUVGUUQUXQXNZXQXOUXPXRXSXTUVGUUR + UVFUURMNZUUNUUIUXTAUVEUUIUWQUXTUWRDUUQYAYBWGZXCYCUVHRUURYDXKYFZEUVHUVAU + VJUUSUVBUUTUVHUUQSYEUVBYGZUVHUUQSWSYHVQUVGUUMUUQUVGUUMUVFUVMUUNUWOXCYIU XRUXSYJXOUVGUVBUUSYKZUVKUVIUVCNUVFUYDUUNUUJUYDUVEUUJUUSPUVBUUJEUUSUVAPU - UTUUSNZUUTPNUWQUVAPNUUJUYEUUTUUTUURYLYMUUIUWQAUWRUOUUTUUQYNYOYQYPXBZXBU - YBUUSUVHUVBYRXPYSUVFDUVCNUUNYTUVFUURUVBTZDUVCUVFUYGUURUUQSQZDUVFUURUUSN - ZUYGUYHUMUVFUURUWTNZUYIUVFUURVHUWTUVFUURUYAXGXSXTZRUURUUAVRZEUURUVAUYHU - USUVBUUTUURUUQSYFUYCUURUUQSWRYHVRUVFDUUQUVFDUWPYIUVFUUQUWSXMUVFUUQUWSXN - YJXOUVFUYDUYIUYGUVCNUYFUYLUUSUURUVBYRXPYSXBUUBUVFRUVBTZRUVCUVFUYMRUUQSQ - ZRUVFRUUSNZUYMUYNUMUVFUYJUYOUYKRUURUUCVRZERUVAUYNUUSUVBUUTRUUQSYFUYCRUU - QSWRYHVRUVFUWQUYNRUMUWSUWQUUQUUQUUGUUQUUDUUEVRXOUVFUYDUYOUYMUVCNUYFUYPU - USRUVBYRXPYSUUHUUF $. + UTUUSNZUUTPNUWQUVAPNUUJUYEUUTUUTUURYLYMUUIUWQAUWRUOUUTUUQYNYQYOYPXCZXCU + YBUUSUVHUVBYRXKYSUVFDUVCNUUNYTUVFUURUVBTZDUVCUVFUYGUURUUQSQZDUVFUURUUSN + ZUYGUYHUMUVFUURUWTNZUYIUVFUURVHUWTUVFUURUYAXGXSXTZRUURUUAVQZEUURUVAUYHU + USUVBUUTUURUUQSYEUYCUURUUQSWSYHVQUVFDUUQUVFDUWPYIUVFUUQUWSXMUVFUUQUWSXN + YJXOUVFUYDUYIUYGUVCNUYFUYLUUSUURUVBYRXKYSXCUUBUVFRUVBTZRUVCUVFUYMRUUQSQ + ZRUVFRUUSNZUYMUYNUMUVFUYJUYOUYKRUURUUCVQZERUVAUYNUUSUVBUUTRUUQSYEUYCRUU + QSWSYHVQUVFUWQUYNRUMUWSUWQUUQUUQUUGUUQUUDUUEVQXOUVFUYDUYOUYMUVCNUYFUYPU + USRUVBYRXKYSUUHUUF $. $( Lemma for ~ mbfi1fseq . This lemma is not as interesting as it is long - it is simply checking that ` G ` is in fact a sequence of @@ -345393,89 +345388,89 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wn citg1 reex mptex fnmpti a1i cfz mbfi1fseqlem3 elfznn0 nn0red nndivre syl2anr fmpttd fssd wfo fzfid dffn4 fofi ssfid cin mbfi1fseqlem2 fveq1d vex c0ex eldifsni syl6 simpllr cxp cpnf ffvelcdm syl2an sselid ad2antrl - cico nndivred ralrimivva fovrn syl3an1 3expa lemin letri3d eqeq2d leidd - fmpo min2 biantrud elrege0 simpld nngt0d lemuldiv nnmulcl syl2anc2 nnzd - ffvelcdmda flge simpl ovmpoa eleqtrrd wrex abeq2i elfzelz zcnd divcan1d - rnmpt oveq1 eleq1d rexlimdva syl5bi imp syldan flbi eqtr3d eqbrtrrd jca + cico nndivred ralrimivva fmpo fovcdm syl3an1 3expa lemin letri3d eqeq2d + min2 leidd biantrud ffvelcdmda elrege0 simpld lemuldiv nnmulcl syl2anc2 + nngt0d nnzd flge simpl ovmpoa eleqtrrd wrex rnmpt elfzelz zcnd divcan1d + abeq2i oveq1 eleq1d rexlimdva syl5bi imp syldan flbi eqtr3d eqbrtrrd ex impbid1 eldifi nnre nn0ge0d ifboth eqbrtrd ralrn mpbird r19.21bi sylan2 - sstrd divmul3d ifnefalse eleq2d nnrecred ltmul1 recid2d joinlmuladdmuld - ex readdcld anbi12d biancomd 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Verify that ` G ` describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, @@ -345492,60 +345487,60 @@ or are almost disjoint (the interiors are disjoint). 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YLUWP $. + mpbi nndivred fmpo fovcdm syl3an1 3expa mbfi1fseqlem1 ad2antrr peano2nn + nnre fovcdmd min1 2cn eqeltrrd 2cnd nnne0d divcan1d oveq1d 3eqtr2d flle + expp1 2re 2pos pm3.2i lemul1 syl3anc eqtr4d flcld zmulcl flge syl112anc + lemuldiv min2 lep1d iftrue renegcld lenegd iccss sselda iftrued iffalse + 2z wn simprd pm2.61dan inidm jca ) ADMNZUAZUBDGUCZOUDZPZUWSDUJUEQZGUCZU + WTPZUWRUXARBUKZDUFZDUGQZNZDUXEHQZDOPZUXIDUHZRUHZOPZBSUIUWRUXMBSUWRUXESN + ZUAZRUXKOPZRROPZUXMUXORUXIOPZRDOPZUXPUXORUXEFUCZTDULQZUMQZUNUCZUYAUOQZU + XIOUXOUYCSNZRUYCOPUYASNZRUYAUPPRUYDOPUXOUYCUXOUYBSNZRUYBOPZUYCUQNUXOUXT + UYAUXOUXTSNZRUXTOPZUXOUXTRURUSQZNUYIUYJUAZUWRSUYKUXEFASUYKFUTZUWQJVAVBU + XTVCVDZVEZUXOUYAUWQUYAMNZAUXNUWQTMNZDUQNZUYPVFDVGZTDVHVIVJZVNZVKZUXOUYL + UYFRUYAOPUYHUYNVUAUXOUYAUXOUYAUYTVLVMUXTUYAVOVPUYBVQWDZVRZUXOUYCVUCVMVU + AUXOUYAUYTVSUYCUYAVTWAUWQUXNUXIUYDWBAECDUXEMSCUKZFUCZTEUKZULQZUMQZUNUCZ + VUHUOQZUYDHVUGDWBZVUEUXEWBZUAZVUJUYCVUHUYAUOVUNVUIUYBUNVUNVUFUXTVUHUYAU + MVUNVUEUXEFVULVUMWCWEVUNVUGDTULVULVUMWFWGZWHWEVUOWHKUYCUYAUOWIWJWKZWLUW + QUXSAUXNUWQDUYSVMVJUXJUXRUXSUXPUXIDUXIUXKROWNDUXKROWNWMWDWOUXHUXPUXQUXM + UXKRUXKUXLROWNRUXLROWNWMWPWQUWRBWRSRUXLOSUBUWSWSWSUBWRWTZUWRVUQWRRXAXFZ + WRWTZRSNZVUSXGWRRSXBXCWRUBVURXDXEXHXIUWRUWSXJXKZNSSUWSUTUWSSWTAMVVADGAB + CEFGHIJKLXLZVBUWSXMSSUWSXNXOZWRWSNUWRXPXISWSNUWRUUAXIZSWRXQWRSUUBSWBUUC + SWRUUDUUJUXEWRNUXEUBUCRWBUWRUXEUUEXRUXOUXEUWSUCZUXEBSUXLXSZUCZUXLUWQVVE + VVGWBAUXNUWQUXEUWSVVFABCDEFGHIJKLXTYAVJUXOUXNUXLSNZVVGUXLWBUWRUXNWCZUXO + UXKSNVUTVVHUXOUXJUXIDSAUWQUXNUXISNZAMSXFZSHUTZUWQUXNVVJAVUKSNZCSUIEMUIV + VLAVVMECMSAVUGMNZVUESNZUAZUAZVUJVUHVVQVUISNVUJSNVVQVUFVUHVVQUYKSVUFUUFA + UYMVVOVUFUYKNVVPJVVNVVOWCSUYKVUEFYBYCUUGVVQVUHVVNVUHMNZAVVOVVNUYQVUGUQN + VVRVFVUGVGTVUGVHVIUUHZVNVKVUIYDYIVVSUUKUUIECMSVUKSHKUULVDDUXESMSHUUMUUN + UUOZUWQDSNZAUXNDUUSVJZYEZXGUXHUXKRSYFWPBSUXLSVVFVVFYGYHWDYJZYKYLUWRUXDU + XLUXEUXBUFZUXBUGQZNZUXBUXEHQZUXBOPZVWHUXBUHZRUHZOPZBSUIUWRVWLBSUXOUXHVW + LUXOUXHUAZUXKVWJUXLVWKOUXOUXKVWJOPZUXHUXOUXKVWHOPZUXKUXBOPZVWNUXOUXKUXI + VWHVWCVVTUXOVWHSNZRVWHOPZUXOVWHUYKNVWQVWRUAUXOUXBUXEUYKMSHAVVKUYKHUTUWQ + UXNACEFHIJKUUPUUQUWQUXBMNZAUXNDUURZVJZVVIUUTVWHVCVDZVEZUXOVVJVWAUXKUXIO + PVVTVWBUXIDUVAWDUXOUYDUXTTUXBULQZUMQZUNUCZVXDUOQZUXIVWHOUXOUYDVXDUMQZVX + FOPZUYDVXGOPZUXOVXHUYCTUMQZVXFOUXOVXHUYDUYATUMQZUMQUYDUYAUMQZTUMQVXKUXO + VXDVXLUYDUMUXOTWRNUYRVXDVXLWBUVBUWQUYRAUXNUYSVJTDUVJVIZWGUXOUYDUYATUXOU + YDUXOUXIUYDSVUPVVTUVCZYMUXOUYAVUAYMZUXOUVDZYNUXOVXMUYCTUMUXOUYCUYAUXOUY + CVUDYMVXPUXOUYAUYTUVEUVFUVGUVHUXOVXKVXEOPZVXKVXFOPZUXOVXKUYBTUMQZVXEOUX + OUYCUYBOPZVXKVXTOPZUXOUYGVYAVUBUYBUVIYIUXOUYEUYGTSNZRTUPPZUAZVYAVYBYOVU + DVUBVYEUXOVYCVYDUVKUVLUVMXIUYCUYBTUVNUVOYPUXOVXEUXTVXLUMQVXTUXOVXDVXLUX + TUMVXNWGUXOUXTUYATUXOUXTUYOYMVXPVXQYNUVPWLUXOVXESNZVXKYTNZVXRVXSYOUXOUX + TVXDUYOUXOVXDUXOUYQUXBUQNVXDMNVFUXOUXBVXAVLZTUXBVHVIZVNZVKZUXOUYCYTNTYT + NVYGUXOUYBVUBUVQUWKUYCTUVRWPVXEVXKUVSWDYPYQUXOUYDSNVXFSNZVXDSNRVXDUPPVX + IVXJYOVXOUXOVYFVYLVYKVXEYDYIVYJUXOVXDVYIVSUYDVXFVXDUWAUVTYPVUPUXOVWSUXN + VWHVXGWBVXAVVIECUXBUXEMSVUKVXGHVUGUXBWBZVUMUAZVUJVXFVUHVXDUOVYNVUIVXEUN + VYNVUFUXTVUHVXDUMVYNVUEUXEFVYMVUMWCWEVYNVUGUXBTULVYMVUMWFWGZWHWEVYOWHKV + XFVXDUOWIWJWDYRYSUXOUXKDUXBVWCVWBUXOUXBVXAVNZUXOVVJVWAUXKDOPVVTVWBUXIDU + WBWDUXODVWBUWCZYSVWIVWOVWPVWNVWHUXBVWHVWJUXKOWNUXBVWJUXKOWNWMWDVAUXHUXL + UXKWBUXOUXHUXKRUWDXRVWMVWGVWJRUXOUXGVWFUXEUXOVWESNUXBSNVWEUXFOPZDUXBOPZ + UXGVWFXQUXOUXBVYPUWEVYPUXOVYSVYRVYQUXODUXBVWBVYPUWFYPVYQVWEUXBUXFDUWGWA + UWHUWIYRUXOUXHUWLZUAUXLRVWKOVYTUXLRWBUXOUXHUXKRUWJXRUXORVWKOPZVYTUXORVW + JOPZUXQWUAUXOVWRRUXBOPZWUBUXOVWQVWRVXBUWMUXOUXBVYHVMVWIVWRWUCWUBVWHUXBV + WHVWJROWNUXBVWJROWNWMWDWOVWGWUBUXQWUAVWJRVWJVWKROWNRVWKROWNWMWPVAYQUWNW + QUWRBSSUXLVWKOSUWSUXCWSWSVVCUWRUXCVVANZSSUXCUTUXCSWTAMVVAGUTVWSWUDUWQVV + BVWTMVVAUXBGYBYCUXCXMSSUXCXNXOVVDVVDSUWOVWDUXOUXEUXCUCZUXEBSVWKXSZUCZVW + KUXOVWSWUEWUGWBVXAVWSUXEUXCWUFABCUXBEFGHIJKLXTYAYIUXOUXNVWKSNZWUGVWKWBV + VIUXOVWJSNVUTWUHUXOVWIVWHUXBSVXCVYPYEXGVWGVWJRSYFWPBSVWKSWUFWUFYGYHWDYJ + YKYLUWP $. $( Lemma for ~ mbfi1fseq . Verify that ` G ` converges pointwise to ` F ` , and wrap up the existential quantifier. (Contributed by Mario @@ -348105,23 +348100,23 @@ or are almost disjoint (the interiors are disjoint). (Contributed by S. A C _d x = S. B C _d x ) ) $= ( vy cmpt cibl wcel wceq cc0 wa cc adantr syl wf wb citg cif csb nfcv nfv cv nfcsb1v nfif eleq1w csbeq1a ifbieq1d cbvmpt wss cvol cdm cmbf mpteq2ia - iftrue eqtr4i simpr eqeltrrid iblmbf sselda syldan fmpttd feq1i fvmptelrn - sylib mbfdm2 cdif cun undif id cr covol cfv ssdifssd nulmbl syl2anc unmbl - syl2anr eqeltrrd wn eldifn adantl iffalsed iblss2 eqeltrid eqtr3i sylancl - ifcl dfss4 difmbl iblss impbida eleq2d biimpa itgeqa simpld itgss2 simprd - 0cn bitrd eqtrd jca ) ABCEKZLMZBDEKLMZUABCEUBZBDEUBZNAXHBDBUGZCMZEOUCZKZL - MZXIAXHXPAXHPZXOJDJUGZCMZBXREUDZOUCZKZLBJDXNYAJXNUEZXSBXTOXSBUFBXREUHZBOU - EUIZXLXRNXMXSEXTOBJCUJBXREUKZULZUMZXQJCDYAQACDUNZXHFRAXHCUOUPZMZDYJMZXQJC - YAQXQJCYAKZLMYMUQMXQYMXGLXGJCXTKYMBJCEXTJEUEYDYFUMJCYAXTXSXTOUSURUTZAXHVA - VBZYMVCSXQJCYAQXQCQXGTZCQYMTAYPXHABCEQAXMXLDMZEQMZACDXLFVDIVEVFRCQXGYMYNV - GVIVHZVJAYKPCDCVKZVLZDYJAUUADNZYKAYIUUBFCDVMVIRYKYKYTYJMZUUAYJMAYKVNAYTVO - UNYTVPVQONUUCADVOCGVRZHYTVSVTZCYTWAWBWCVEYSXQXRYTMZPXSXTOUUFXSWDXQXRDCWEW - FWGYOWHWIAXPPZXGYMLBCXNKXGYMBCXNEXMEOUSZURBJCXNYAYCYEYGUMWJUUGJCDYAQAYIXP - FRAXPYLYKUUGJDYAQUUGYBLMYBUQMUUGYBXOLYHAXPVAVBZYBVCSUUGJDYAQADQYBTZXPADQX - OTUUJABDXNQAYQPYROQMXNQMIXCXMEOQWLWKZVFDQXOYBYHVGVIRVHZVJAYLPDYTVKZCYJAUU - MCNZYLAYIUUNFCDWMVIZRYLYLUUCUUMYJMAYLVNUUEDYTWNWBWCVEUULUUIWOWIWPAXPXIUAZ - BDXNUBZXKNZABYTDXNEUUKIUUDHAXLUUMMZPXMXNENAUUSXMAUUMCXLUUOWQWRUUHSWSZWTXD - AXJUUQXKAYIXJUUQNFBCDEXASAUUPUURUUTXBXEXF $. + iftrue eqtr4i simpr eqeltrrid iblmbf sselda syldan feq1i sylib fvmptelcdm + fmpttd mbfdm2 cdif cun undif id covol cfv ssdifssd nulmbl syl2anc syl2anr + unmbl eqeltrrd eldifn adantl iffalsed iblss2 eqeltrid eqtr3i ifcl sylancl + cr wn dfss4 difmbl iblss impbida eleq2d biimpa itgeqa simpld bitrd itgss2 + 0cn simprd eqtrd jca ) ABCEKZLMZBDEKLMZUABCEUBZBDEUBZNAXHBDBUGZCMZEOUCZKZ + LMZXIAXHXPAXHPZXOJDJUGZCMZBXREUDZOUCZKZLBJDXNYAJXNUEZXSBXTOXSBUFBXREUHZBO + UEUIZXLXRNXMXSEXTOBJCUJBXREUKZULZUMZXQJCDYAQACDUNZXHFRAXHCUOUPZMZDYJMZXQJ + CYAQXQJCYAKZLMYMUQMXQYMXGLXGJCXTKYMBJCEXTJEUEYDYFUMJCYAXTXSXTOUSURUTZAXHV + AVBZYMVCSXQJCYAQXQCQXGTZCQYMTAYPXHABCEQAXMXLDMZEQMZACDXLFVDIVEVIRCQXGYMYN + VFVGVHZVJAYKPCDCVKZVLZDYJAUUADNZYKAYIUUBFCDVMVGRYKYKYTYJMZUUAYJMAYKVNAYTW + KUNYTVOVPONUUCADWKCGVQZHYTVRVSZCYTWAVTWBVEYSXQXRYTMZPXSXTOUUFXSWLXQXRDCWC + WDWEYOWFWGAXPPZXGYMLBCXNKXGYMBCXNEXMEOUSZURBJCXNYAYCYEYGUMWHUUGJCDYAQAYIX + PFRAXPYLYKUUGJDYAQUUGYBLMYBUQMUUGYBXOLYHAXPVAVBZYBVCSUUGJDYAQADQYBTZXPADQ + XOTUUJABDXNQAYQPYROQMXNQMIXCXMEOQWIWJZVIDQXOYBYHVFVGRVHZVJAYLPDYTVKZCYJAU + UMCNZYLAYIUUNFCDWMVGZRYLYLUUCUUMYJMAYLVNUUEDYTWNVTWBVEUULUUIWOWGWPAXPXIUA + ZBDXNUBZXKNZABYTDXNEUUKIUUDHAXLUUMMZPXMXNENAUUSXMAUUMCXLUUOWQWRUUHSWSZWTX + AAXJUUQXKAYIXJUUQNFBCDEXBSAUUPUURUUTXDXEXF $. $} ${ @@ -349895,29 +349890,29 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by 11-Feb-2015.) $) dvmptcl $p |- ( ( ph /\ x e. X ) -> B e. CC ) $= ( cc cmpt cdv co wf cdm wcel syl mpbid cpr dvfg dmeqd wral wceq ralrimiva - cr dmmptg eqtrd feq2d feq1d fvmptelrn ) ABGDLAGLEBGCMZNOZPZGLBGDMZPAUNQZL - UNPZUOAEUGLUARURHEUMUBSAUQGLUNAUQUPQZGAUNUPKUCADFRZBGUDUSGUEAUTBGJUFBGDFU - HSUIUJTAGLUNUPKUKTUL $. + cr dmmptg eqtrd feq2d feq1d fvmptelcdm ) ABGDLAGLEBGCMZNOZPZGLBGDMZPAUNQZ + LUNPZUOAEUGLUARURHEUMUBSAUQGLUNAUQUPQZGAUNUPKUCADFRZBGUDUSGUEAUTBGJUFBGDF + UHSUIUJTAGLUNUPKUKTUL $. ${ dvmptadd.c $e |- ( ( ph /\ x e. X ) -> C e. CC ) $. @@ -351710,13 +351705,13 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( S _D ( x e. X |-> E ) ) = ( x e. X |-> ( F x. B ) ) ) $= ( cmpt ccom cdv co cmul cof cc fmpttd cdm wcel wral wceq ralrimiva dmmptg dmeqd syl eqtrd dvcof eqidd fmptco oveq2d cvv ovex dmex eqeltrrdi dvmptcl - wf fmpt3d fco syl2anc feq1d mpbid fvmptelrn offval2 3eqtr3d ) AHCOFUFZBND - UFZUGZUHUIIWAUHUIZWBUGZHWBUHUIZUJUKUIHBNJUFZUHUIBNKEUJUIUFAIHWAWBONQPACOF - ULTUMABNDORUMZAWDUNCOGUFZUNZOAWDWIUCUTAGMUOZCOUPWJOUQAWKCOUAURCOGMUSVAVBA - WFUNZBNEUFZUNZNAWFWMUBUTAELUOZBNUPWNNUQAWOBNSURBNELUSVAVBZVCAWCWGHUHABCNO - DFJWBWARAWBVDZAWAVDUDVEVFABNKEUJWEWFVGULLANWLVGWPWFHWBUHVHVIVJABNKULANULW - EVLZNULBNKUFZVLAOULWDVLNOWBVLWRACOGULWDUCACFGIMOQTUAUCVKVMWHNOULWDWBVNVOA - NULWEWSABCNODGKWBWDRWQUCUEVEZVPVQVRSWTUBVSVT $. + wf fmpt3d fco syl2anc feq1d mpbid fvmptelcdm offval2 3eqtr3d ) AHCOFUFZBN + DUFZUGZUHUIIWAUHUIZWBUGZHWBUHUIZUJUKUIHBNJUFZUHUIBNKEUJUIUFAIHWAWBONQPACO + FULTUMABNDORUMZAWDUNCOGUFZUNZOAWDWIUCUTAGMUOZCOUPWJOUQAWKCOUAURCOGMUSVAVB + AWFUNZBNEUFZUNZNAWFWMUBUTAELUOZBNUPWNNUQAWOBNSURBNELUSVAVBZVCAWCWGHUHABCN + ODFJWBWARAWBVDZAWAVDUDVEVFABNKEUJWEWFVGULLANWLVGWPWFHWBUHVHVIVJABNKULANUL + WEVLZNULBNKUFZVLAOULWDVLNOWBVLWRACOGULWDUCACFGIMOQTUAUCVKVMWHNOULWDWBVNVO + ANULWEWSABCNODGKWBWDRWQUCUEVEZVPVQVRSWTUBVSVT $. $} ${ @@ -352365,61 +352360,61 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by cvv ovex mpbird simpld fvres wi ad2antrr cc0 fveq2 oveq1d simprl simprr weq sylib exp32 wo clt eliooord cxr w3a elioo2 mpbir3and eqeltrd simprd c0 rexrd xrltled raleqdv fveq2d letri3d mpbir2and tgioo2 iooretop dmeqd - dvmptres dvmptsub eqtrid dmmptg mprg eqtrdi dvcn syl31anc sylc cncffvrn - rescncf sylancr breqan12rd ralbidva adantl ioossicc ssralv ax-mp syl6bi - evthicc syldan fveq1d fvmpt sstrid breq1d cbvralvw subeq0d vex elpr wne - dvferm ne0i ndmioo necon1ai 3syl iooss2 dvferm1 eqbrtrrd suble0d elicc2 - simp3d breqtrrd simp2d eqbrtrd iooss1 dvferm2 breqtrd subge0d jaod elun - syl5bi cun prunioo syl3anc eleq2d bitr3id biimpar mpjaod syld reximdva + dvmptres dvmptsub eqtrid dmmptg mprg dvcn syl31anc rescncf sylc cncfcdm + eqtrdi sylancr evthicc breqan12rd ralbidva adantl ioossicc ssralv ax-mp + syl6bi syldan fveq1d sstrid breq1d cbvralvw dvferm subeq0d vex elpr wne + ne0i ndmioo necon1ai 3syl iooss2 dvferm1 eqbrtrrd suble0d elicc2 simp3d + fvmpt breqtrrd simp2d eqbrtrd iooss1 dvferm2 breqtrd subge0d syl5bi cun + jaod elun prunioo syl3anc eleq2d bitr3id biimpar mpjaod syld reximdva mpd ) AUAUCZHIJUDRZUEZUFZBUCZUXGUFZUGUHZUAUXFUIZBUXFUJZUXISGUNRZUFZFUKZ BUXFUJAUXMUXJUXHUGUHUAUXFUIBUXFUJABUABUAIJUXGADEULRZSIDEUMZKUOZAUXQSJUX RLUOZAIJUXSUXTOUPZAUXGUXFSUQRTZUXFSUXGURZAUXQSUXFHACUXQCUCZGUFZFUYDUSRZ @@ -353301,29 +353296,29 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by YCYPDIEUWHVLXPWUEUXIUXQVUSVWBVWPWUDVWRVNAVUTVWAWUDVVTXPVIWUEVXGUBIUXIUL RZUIVXHWUEVWGVXHVWBVXKVWGYAVXIYCWUEVXGUBWUMVWFWUEUXIJIULWUFWNYQXLUWIVWB VXBWUDVXDVNUWJWUEUXOFVWBVWQWUDVWSVNZAUYMVWAWUDVUEXPZUWKWLWUEUXOFWUNWUOY - SYTYDUWLUWNAVWJVWMYEZVWAWUPUXIVWFVWLUWOZTAVWAUXIVWFVWLUWMAWUQUXFUXIAWUJ + SYTYDUWNUWLAVWJVWMYEZVWAWUPUXIVWFVWLUWMZTAVWAUXIVWFVWLUWOAWUQUXFUXIAWUJ VYEIJUGUHWUQUXFUKWUKVYJUYAIJUWPUWQUWRUWSUWTUXAUXBUXCUXD $. $( Lemma for ~ dvivth . (Contributed by Mario Carneiro, 20-Feb-2015.) $) @@ -353352,37 +353347,37 @@ or are almost disjoint (the interiors are disjoint). 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If ` F ` and ` G ` @@ -354028,11 +354023,11 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by ( cr cmin cfzo csu cle wbr cneg cvv cicc cmpt cc0 df-neg mpteq2i ccnfld co ccncf ctopn cfv eqid subcn wcel cc wss 0red cuz eluzel2 zred eluzelz cz syl iccssre syl2anc ax-resscn sstrdi a1i cncfmptc syl3anc cncfmpt2ss - resubcl eqeltrid cv wa negex cpr reelprrecn ioossicc sseli wf fvmptelrn - cioo cncff sylan2 recnd dvmptneg negeqd renegcld caddc cxr adantr rexrd - wceq elfzole1 adantl iooss1 cfz fzofzp1 elfzle2 iooss2 sstrd sselda cdv - c1 cdm adantlr fmpttd ioossre dvfre sylancl dmeqd wral ralrimiva dmmptg - eqtrd feq12d mpbid syldan anasss adantrr lenegd dvfsumle fsumneg eleq1d - cfn fzofi fmpt sylibr eluzle ubicc2 rspcdva lbicc2 neg2subd negsubdi2d + resubcl eqeltrid cv cioo negex cpr reelprrecn ioossicc sseli fvmptelcdm + wa wf cncff sylan2 recnd dvmptneg wceq negeqd renegcld caddc cxr adantr + c1 rexrd elfzole1 adantl iooss1 cfz fzofzp1 elfzle2 iooss2 sstrd sselda + cdv cdm adantlr fmpttd ioossre dvfre sylancl dmeqd wral ralrimiva eqtrd + dmmptg feq12d mpbid syldan anasss adantrr lenegd dvfsumle fzofi fsumneg + cfn eleq1d fmpt sylibr eluzle ubicc2 rspcdva lbicc2 neg2subd negsubdi2d eqtr4d 3brtr3d resubcld fsumrecl mpbird ) AFEUAUNZHIUBUNZKGUCZUDUEUUIUF ZUUGUFZUDUEAUUHKUFZGUCFUFZEUFZUAUNZUUJUUKUDABCUFZDUFZUUNUUMGHIUGUULLABH IUHUNZUUPUIBUURUJCUAUNZUIUURTUOUNZBUURUUPUUSCUKULABUJCTUAUMUPUQZUURUVAU RZUVAUVBUSAUJTUTUURVAVBTVAVBZBUURUJUIUUTUTAVCAUURTVAAHTUTZITUTZUURTVBAH AIHVDUQUTZHVHUTLHIVEVIVFZAIAUVFIVHUTLHIVGVIVFZHIVJVKVLVMUVCAVLVNBUJUURT - VOVPMVLUJCVRVQVSUUQUGUTABVTZHIWIUNZUTZWAZDWBVNABCDTJUVJTTVAWCUTAWDVNUVL - CUVKAUVIUURUTZCTUTZUVJUURUVIHIWEWFZABUURCTABUURCUIZUUTUTUURTUVPWGZMUURT - UVPWJVIZWHZWKWLNOWMUVIHWTZCEPWNUVIIWTZCFQWNAGVTZUUHUTZWAZKRWOAUWCUVIUWB - UWBXKWPUNZWIUNZUTZWAWAZDKUDUEUULUUQUDUESUWHDKAUWCUWGDTUTZUWDUWGUVKUWIUW - DUWFUVJUVIUWDUWFHUWEWIUNZUVJUWDHWQUTZHUWBUDUEZUWFUWJVBUWDHAUVDUWCUVGWRW - SUWCUWLAUWBHIXAXBHUWBUWEXCVKUWDIWQUTZUWEIUDUEZUWJUVJVBUWDIAUVEUWCUVHWRW - SUWDUWEHIXDUNUTZUWNUWCUWOAHIUWBXEXBUWEHIXFVIHUWEIXGVKXHXIUWDBUVJDTUWDTB - UVJCUIZXJUNZXLZTUWQWGZUVJTBUVJDUIZWGUWDUVJTUWPWGUVJTVBUWSUWDBUVJCTUVKUW - DUVMUVNUVOAUVMUVNUWCUVSXMWKXNHIXOUVJUWPXPXQUWDUWRUVJTUWQUWTAUWQUWTWTUWC - OWRZUWDUWRUWTXLZUVJUWDUWQUWTUXAXRUWDDJUTZBUVJXSUXBUVJWTUWDUXCBUVJAUVKUX - CUWCNXMXTBUVJDJYAVIYBYCYDWHYEYFAUWCKTUTUWGRYGYHYDYIAUUHKGUUHYLUTAHIYMVN - ZUWDKRWLYJAUUOEFUAUNUUKAFEAFAUVNFTUTBUURIUWACFTQYKAUVQUVNBUURXSUVRBUURT - CUVPUVPURYNYOZAUWKUWMHIUDUEZIUURUTAHUVGWSZAIUVHWSZAUVFUXFLHIYPVIZHIYQVP - YRZWLZAEAUVNETUTBUURHUVTCETPYKUXEAUWKUWMUXFHUURUTUXGUXHUXIHIYSVPYRZWLZY + VOVPMVLUJCVRVQVSUUQUGUTABVTZHIWAUNZUTZWHZDWBVNABCDTJUVJTTVAWCUTAWDVNUVL + CUVKAUVIUURUTZCTUTZUVJUURUVIHIWEWFZABUURCTABUURCUIZUUTUTUURTUVPWIZMUURT + UVPWJVIZWGZWKWLNOWMUVIHWNZCEPWOUVIIWNZCFQWOAGVTZUUHUTZWHZKRWPAUWCUVIUWB + UWBWTWQUNZWAUNZUTZWHWHZDKUDUEUULUUQUDUESUWHDKAUWCUWGDTUTZUWDUWGUVKUWIUW + DUWFUVJUVIUWDUWFHUWEWAUNZUVJUWDHWRUTZHUWBUDUEZUWFUWJVBUWDHAUVDUWCUVGWSX + AUWCUWLAUWBHIXBXCHUWBUWEXDVKUWDIWRUTZUWEIUDUEZUWJUVJVBUWDIAUVEUWCUVHWSX + AUWDUWEHIXEUNUTZUWNUWCUWOAHIUWBXFXCUWEHIXGVIHUWEIXHVKXIXJUWDBUVJDTUWDTB + UVJCUIZXKUNZXLZTUWQWIZUVJTBUVJDUIZWIUWDUVJTUWPWIUVJTVBUWSUWDBUVJCTUVKUW + DUVMUVNUVOAUVMUVNUWCUVSXMWKXNHIXOUVJUWPXPXQUWDUWRUVJTUWQUWTAUWQUWTWNUWC + OWSZUWDUWRUWTXLZUVJUWDUWQUWTUXAXRUWDDJUTZBUVJXSUXBUVJWNUWDUXCBUVJAUVKUX + CUWCNXMXTBUVJDJYBVIYAYCYDWGYEYFAUWCKTUTUWGRYGYHYDYIAUUHKGUUHYLUTAHIYJVN + ZUWDKRWLYKAUUOEFUAUNUUKAFEAFAUVNFTUTBUURIUWACFTQYMAUVQUVNBUURXSUVRBUURT + CUVPUVPURYNYOZAUWKUWMHIUDUEZIUURUTAHUVGXAZAIUVHXAZAUVFUXFLHIYPVIZHIYQVP + YRZWLZAEAUVNETUTBUURHUVTCETPYMUXEAUWKUWMUXFHUURUTUXGUXHUXIHIYSVPYRZWLZY TAFEUXKUXMUUAUUBUUCAUUGUUIAFEUXJUXLUUDAUUHKGUXDRUUEYHUUF $. $} $} @@ -354414,9 +354409,9 @@ or are almost disjoint (the interiors are disjoint). (Contributed by 18-May-2016.) $) dvmptrecl $p |- ( ( ph /\ x e. S ) -> B e. RR ) $= ( cr cmpt cdv co cdm wf wss fmpttd dvfre syl2anc wcel wral wceq ralrimiva - dmeqd dmmptg syl eqtrd feq12d mpbid fvmptelrn ) ABEDKAKBECLZMNZOZKUMPZEKB - EDLZPAEKULPEKQUOABECKHRGEULSTAUNEKUMUPJAUNUPOZEAUMUPJUEADFUAZBEUBUQEUCAUR - BEIUDBEDFUFUGUHUIUJUK $. + dmeqd dmmptg syl eqtrd feq12d mpbid fvmptelcdm ) ABEDKAKBECLZMNZOZKUMPZEK + BEDLZPAEKULPEKQUOABECKHRGEULSTAUNEKUMUPJAUNUPOZEAUMUPJUEADFUAZBEUBUQEUCAU + RBEIUDBEDFUFUGUHUIUJUK $. $} ${ @@ -354529,105 +354524,105 @@ or are almost disjoint (the interiors are disjoint). 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(Contributed by Mario Carneiro, @@ -355469,37 +355464,37 @@ or are almost disjoint (the interiors are disjoint). (Contributed by 3-Sep-2014.) $) itgparts $p |- ( ph -> S. ( X (,) Y ) ( A x. D ) _d x = ( ( F - E ) - S. ( X (,) Y ) ( B x. C ) _d x ) ) $= - ( vt cioo co cmul citg caddc cmin cc cv wcel wa ccncf cncff syl fvmptelrn - cmpt wf cicc ioossicc sseli sylan2 mulcld itgcl pncan2d itgadd cr cdv cfv - fveq2 nfcv nfmpt1 nfov nffv cbvitg crn ctg ccnfld ctopn wss ax-resscn a1i - iccssre syl2anc tgioo2 cnt iccntr dvmptntr cpr reelprrecn eqtr3d dvmptmul - eqid wceq mulcomd oveq2d mpteq2dva 3eqtrd ctx ccn addcn cres resmpt ax-mp - rescncf mpsyl eqeltrrid mulcncf cncfmpt2f eqeltrd cibl ibladd ftc2 eqtrid - fveq1d adantr cvv simpr ovex sylancl eqtrd itgeq2dv csb cxr cle wbr rexrd - fvmpt2 ubicc2 syl3anc csbex fvmpts csbied lbicc2 oveq12d 3eqtr3d oveq1d ) - ABIJUEUFZDEUGUFZUHZBYTCFUGUFZUHZUIUFZUUBUJUFUUDHGUJUFZUUBUJUFAUUBUUDABYTU - UAUKABULZYTUMZUNZDEABYTDUKABYTDUSZYTUKUOUFZUMYTUKUUJUTPYTUKUUJUPUQURZUUHA - UUGIJVAUFZUMZEUKUMYTUUMUUGIJVBZVCZABUUMEUKABUUMEUSZUUMUKUOUFZUMZUUMUKUUQU - TOUUMUKUUQUPUQURZVDZVEZSVFABYTUUCUKUUICFUUHAUUNCUKUMUUPABUUMCUKABUUMCUSZU - URUMZUUMUKUVCUTNUUMUKUVCUPUQURZVDZABYTFUKABYTFUSZUUKUMYTUKUVGUTQYTUKUVGUP - UQURZVEZRVFVGAUUEUUFUUBUJABYTUUAUUCUIUFZUHZUUEUUFABYTUUAUUCUKUVBSUVIRVHAB - YTUUGVIBUUMCEUGUFZUSZVJUFZVKZUHZJUVMVKZIUVMVKZUJUFZUVKUUFAUVPUDYTUDULZUVN - VKZUHUVSBUDYTUVOUWAUUGUVTUVNVLUDUVOVMBUVTUVNBVIUVMVJBVIVMBVJVMBUUMUVLVNVO - BUVTVMVPVQAUDIJUVMKLMAUVNBYTUVJUSZUUKAUVNVIBYTUVLUSVJUFBYTUUAFCUGUFZUIUFZ - USUWBABUVLVIUEVRVSVKZVTWAVKZUUMYTVIUKWBAWCWDZAIVIUMZJVIUMZUUMVIWBKLIJWEWF - ZAUUNUNCEUVEUUTVEUWFUWFWOZWGZUWKAUWHUWIUUMUWEWHVKVKYTWPKLIJWIWFZWJABCDEFV - IUKUKYTVIVIUKWKUMAWLWDUVFUULAVIUVCVJUFVIBYTCUSZVJUFUUJABCVIUWEUWFUUMYTUWG - UWJUVEUWLUWKUWMWJTWMUVAUVHAVIUUQVJUFVIBYTEUSZVJUFUVGABEVIUWEUWFUUMYTUWGUW - JUUTUWLUWKUWMWJUAWMWNABYTUWDUVJUUIUWCUUCUUAUIUUIFCUVHUVFWQWRWSWTZABUUAUUC - UIUWFYTUWKUIUWFUWFXAUFUWFXBUFUMAUWFUWKXCWDABDEYTPAUWOUUQYTXDZUUKYTUUMWBZU - WQUWOWPUUOBUUMYTEXEXFUWRAUUSUWQUUKUMUUOOUUMUKYTUUQXGXHXIXJABCFYTAUWNUVCYT - XDZUUKUWRUWSUWNWPUUOBUUMYTCXEXFUWRAUVDUWSUUKUMUUONUUMUKYTUVCXGXHXIQXJXKXL - AUVNUWBXMUWPABYTUUAUUCUKUVBSUVIRXNXLABCEUUMNOXJXOXPABYTUVOUVJUUIUVOUUGUWB - VKZUVJAUVOUWTWPUUHAUUGUVNUWBUWPXQXRUUIUUHUVJXSUMUWTUVJWPAUUHXTUUAUUCUIYAB - YTUVJXSUWBUWBWOYJYBYCYDAUVQHUVRGUJAUVQBJUVLYEZHAJUUMUMZUXAXSUMUVQUXAWPAIY - FUMZJYFUMZIJYGYHZUXBAIKYIZAJLYIZMIJYKYLBJUVLCEUGYAZYMBJUVLUUMUVMXSUVMWOZY - NYBABJUVLHVILUCYOYCAUVRBIUVLYEZGAIUUMUMZUXJXSUMUVRUXJWPAUXCUXDUXEUXKUXFUX - GMIJYPYLBIUVLUXHYMBIUVLUUMUVMXSUXIYNYBABIUVLGVIKUBYOYCYQYRWMYSWM $. + ( vt cioo co cmul citg caddc cmin cc cv wcel wa ccncf wf cncff fvmptelcdm + cmpt syl cicc ioossicc sseli sylan2 mulcld itgcl pncan2d itgadd cdv fveq2 + cr cfv nfcv nfmpt1 nfov cbvitg crn ctg ccnfld ctopn wss ax-resscn iccssre + nffv a1i syl2anc eqid tgioo2 cnt wceq iccntr dvmptntr reelprrecn dvmptmul + cpr eqtr3d mulcomd oveq2d mpteq2dva 3eqtrd ctx addcn resmpt ax-mp rescncf + ccn cres mpsyl eqeltrrid mulcncf cncfmpt2f cibl ibladd ftc2 eqtrid fveq1d + eqeltrd adantr cvv simpr ovex fvmpt2 sylancl eqtrd itgeq2dv csb cxr rexrd + cle wbr ubicc2 syl3anc csbex fvmpts csbied lbicc2 oveq12d 3eqtr3d oveq1d + ) ABIJUEUFZDEUGUFZUHZBYTCFUGUFZUHZUIUFZUUBUJUFUUDHGUJUFZUUBUJUFAUUBUUDABY + TUUAUKABULZYTUMZUNZDEABYTDUKABYTDUSZYTUKUOUFZUMYTUKUUJUPPYTUKUUJUQUTURZUU + HAUUGIJVAUFZUMZEUKUMYTUUMUUGIJVBZVCZABUUMEUKABUUMEUSZUUMUKUOUFZUMZUUMUKUU + QUPOUUMUKUUQUQUTURZVDZVEZSVFABYTUUCUKUUICFUUHAUUNCUKUMUUPABUUMCUKABUUMCUS + ZUURUMZUUMUKUVCUPNUUMUKUVCUQUTURZVDZABYTFUKABYTFUSZUUKUMYTUKUVGUPQYTUKUVG + UQUTURZVEZRVFVGAUUEUUFUUBUJABYTUUAUUCUIUFZUHZUUEUUFABYTUUAUUCUKUVBSUVIRVH + ABYTUUGVKBUUMCEUGUFZUSZVIUFZVLZUHZJUVMVLZIUVMVLZUJUFZUVKUUFAUVPUDYTUDULZU + VNVLZUHUVSBUDYTUVOUWAUUGUVTUVNVJUDUVOVMBUVTUVNBVKUVMVIBVKVMBVIVMBUUMUVLVN + VOBUVTVMWDVPAUDIJUVMKLMAUVNBYTUVJUSZUUKAUVNVKBYTUVLUSVIUFBYTUUAFCUGUFZUIU + FZUSUWBABUVLVKUEVQVRVLZVSVTVLZUUMYTVKUKWAAWBWEZAIVKUMZJVKUMZUUMVKWAKLIJWC + WFZAUUNUNCEUVEUUTVEUWFUWFWGZWHZUWKAUWHUWIUUMUWEWIVLVLYTWJKLIJWKWFZWLABCDE + FVKUKUKYTVKVKUKWOUMAWMWEUVFUULAVKUVCVIUFVKBYTCUSZVIUFUUJABCVKUWEUWFUUMYTU + WGUWJUVEUWLUWKUWMWLTWPUVAUVHAVKUUQVIUFVKBYTEUSZVIUFUVGABEVKUWEUWFUUMYTUWG + UWJUUTUWLUWKUWMWLUAWPWNABYTUWDUVJUUIUWCUUCUUAUIUUIFCUVHUVFWQWRWSWTZABUUAU + UCUIUWFYTUWKUIUWFUWFXAUFUWFXFUFUMAUWFUWKXBWEABDEYTPAUWOUUQYTXGZUUKYTUUMWA + ZUWQUWOWJUUOBUUMYTEXCXDUWRAUUSUWQUUKUMUUOOUUMUKYTUUQXEXHXIXJABCFYTAUWNUVC + YTXGZUUKUWRUWSUWNWJUUOBUUMYTCXCXDUWRAUVDUWSUUKUMUUONUUMUKYTUVCXEXHXIQXJXK + XQAUVNUWBXLUWPABYTUUAUUCUKUVBSUVIRXMXQABCEUUMNOXJXNXOABYTUVOUVJUUIUVOUUGU + WBVLZUVJAUVOUWTWJUUHAUUGUVNUWBUWPXPXRUUIUUHUVJXSUMUWTUVJWJAUUHXTUUAUUCUIY + ABYTUVJXSUWBUWBWGYBYCYDYEAUVQHUVRGUJAUVQBJUVLYFZHAJUUMUMZUXAXSUMUVQUXAWJA + IYGUMZJYGUMZIJYIYJZUXBAIKYHZAJLYHZMIJYKYLBJUVLCEUGYAZYMBJUVLUUMUVMXSUVMWG + ZYNYCABJUVLHVKLUCYOYDAUVRBIUVLYFZGAIUUMUMZUXJXSUMUVRUXJWJAUXCUXDUXEUXKUXF + UXGMIJYPYLBIUVLUXHYMBIUVLUUMUVMXSUXIYNYCABIUVLGVKKUBYOYDYQYRWPYSWP $. $} ${ @@ -355535,12 +355530,12 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vt vv vz vy cmul co cdit cioo citg ditgpos cv cr cicc cmpt cdv cfv cc cmin ccncf csb wss ax-resscn a1i wcel syl2anc wf ccom eqidd wceq fmptco syl fmpttd wb ioossicc cxr clt wbr wa eliooord simpld simprd sstrdi cdm - sseli cle sselid rexrd adantr w3a elicc2 sselda fvmptelrn syldan ioombl - cncff cibl cres rescncf eqeltrrd syl3anc iblss sylan2 fveq2 nfcv cbvitg - cniccibl fvmpt2 itgeq2dv eqtrid mpteq2dva wn c0 lbicc2 mpd resmpt ax-mp - eqid mpsyl eqeltrrid iccntr dvmptntr dmmptd cncfco eqtr3d oveq1d 3eqtrd - wral eqeltrd cabs fveq1d crn ccnfld ctopn iccssre oveq2 itgeq1 iccssioo - ctg syl22anc sstrid ioossre sstrd cncffvrn mpbird biimpa simp3d adantlr + sseli cle sselid rexrd adantr w3a elicc2 sselda cncff fvmptelcdm syldan + ioombl cibl rescncf eqeltrrd cniccibl syl3anc iblss sylan2 fveq2 cbvitg + cres nfcv eqid fvmpt2 itgeq2dv eqtrid mpteq2dva wn c0 lbicc2 mpd resmpt + ax-mp eqeltrrid iccntr dvmptntr dmmptd cncfco wral eqtr3d oveq1d 3eqtrd + mpsyl eqeltrd cabs fveq1d crn ctg ctopn iccssre oveq2 iccssioo syl22anc + ccnfld itgeq1 sstrid ioossre sstrd cncfcdm mpbird biimpa simp3d adantlr iooss2 cvol resmptd sylc nffvmpt1 oveq2d n0i feq3 syl5ibcom f00 simprbi itgcl syl6 mtod mtbid letrid ord ftc1cn tgioo2 cnt 3eqtr3rd dmeqd eqtrd ioo0 dvcn syl31anc cpr reelprrecn cin elin sylib nfcsb1v csbeq1a cbvmpt @@ -355552,65 +355547,65 @@ or are almost disjoint (the interiors are disjoint). 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VFVXEBIYNUEYOCVWOIHFUVFYPUVGYQWDAVXCUWSVBVWLAVXCBKLCDFYEZEVAUMZUTZUWSBUGK + LVYHVXBVYDVYGVWTEVXAVAVYDCDVWSFVXHUVHVXOUVIUGVYHYJBVWTVXAVABCVWSFVXGBFYJU + VJBVAYJVXNUVKYMABVXJVYHYRBVXJUWRYRVYIUWSABVXJVYHUWRAVWEVXJVOZUSZVYGGEVAVY + KVWJVYGGVBVYJAVWHVWJVXJUWFVWEKLUVLUVMVWKUVNCDFGUXAVWJCGYNUCYOWCUVOUVPABKL + VYHPYSABKLUWRPYSUVQYQWDUVRUVSYTYTUVTUWBUWAUWCUWD $. $} ${ @@ -363665,28 +363660,28 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the nfeq breq1d ralimi ralbi sylibrd sylan2 anassrs ralimdva reximdva ralimdv 3syl cc cmap wf cz ulmf cdm fdm dmmptss eqsstrrdi uzid adantl ssel eleq2s eluzel2 syl6 syl2imc rexlimdva mpd wfn crn ralrimiva fnmpt rexlimivw df-f - frn sylanbrc eqidd ulmcl ulmscl ulm2 fvmptelrn elmapi fssresd cnex elmapg - ssexd sylancr mpbird fmpttd 3imtr4d ) ABICUDZFDUEPUFZBICEUGZUDZFEUGZEUEPU - FZMANUHZOUHZYRPZPZUUDFPZUIULUJPZUAUHZUKUFZNDQZOUBUHZUMPZQZUBIUNZUAUOQUUDU - UEUUAPZPZUUHUIULUJPZUUJUKUFZNEQZOUUNQZUBIUNZUAUOQYSUUCAUUPUVCUAUOAUUOUVBU - BIAUUMIRZSUULUVAOUUNAUVDUUEUUNRZUULUVAUPZUVDUVESAUUEIRZUVFGUUEUUMIJUQAUVG - SZUULUUKNEQZUVAUVHEDURZUULUVIUPAUVJUVGKUSUUKNEDUTVFUVHUURUUGVAZNEQZUUTUUK - VBZNEQUVAUVIVBAUVLOIAUUDBUHZUUAPZPZUUDUVNYRPZPZVAZNEQZBIQUVLOIQAUVSBNIEAU - VNIRZUUDERZSSZUUDYTPZUUDCPZUVPUVRUWBUWDUWEVAAUWAUUDECVCVDUWCUUDUVOYTUWCUW - AYTTRUVOYTVAAUWAUWBVEZUWCCEHAUWACHRZUWBLVGZVHBIYTTUUAUUAVIVJVKVLUWCUUDUVQ - CUWCUWAUWGUVQCVAUWFUWHBICHYRYRVIZVJVKVLVMVNUVTUVLBOIUVTOVOUVKBNEBEVPBUURU - UGBUUDUUQBIYTUUEVQBUUDVPZVRBUUDUUFBICUUEVQUWJVRWGVSUVNUUEVAZUVSUVKNEUWKUV - PUURUVRUUGUWKUUDUVOUUQUVNUUEUUAVTVLUWKUUDUVQUUFUVNUUEYRVTVLWAWBWCWDWEUVKU - VMNEUVKUUSUUIUUJUKUURUUGUUHUJUIWFWHWIUUTUUKNEWJWQWKWLWMWNWOWPAUANUUHUUGDU - BOYRFGTIJAUCUHZUMPZWRDWSULZYRWTZUCXAUNZGXARZAYSUWPMDUCYRFXBVFZAUWOUWQUCXA - UWOUWMIURZAUWLXARZSUWLUWMRZUWQUWOUWMYRXCIUWMUWNYRXDBICYRUWIXEXFUWTUXAAUWL - XGXHUWSUXAUWLIRUWQUWMIUWLXIUWQUWLGUMPIGUWLXKJXJXLXMXNXOZAYRIXPZYRXQUWNURZ - IUWNYRWTAUWGBIQUXCAUWGBILXRBICYRHUWIXSVFAUWPUXDUWRUWOUXDUCXAUWMUWNYRYBXTV - FIUWNYRYAYCZAUVGUUDDRZSSUUGYDAUXFSUUHYDAYSDWRFWTMDYRFYEVFZAYSDTRMDYRFYFVF - ZYGAUANUUHUUREUBOUUAUUBGTIJUXBABIYTWREWSULZAUWASZYTUXIRZEWRYTWTZUXJDWRECU - XJCUWNRDWRCWTABICUWNUXEYHCWRDYIVFAUVJUWAKUSYJUXJWRTRETRZUXKUXLVBYKAUXMUWA - AEDTUXHKYMZUSWREYTTTYLYNYOYPAUVGUWBSSUURYDUWBUUDUUBPUUHVAAUUDEFVCXHADWREF - UXGKYJUXNYGYQXO $. + frn sylanbrc eqidd ulmcl ulmscl ulm2 fvmptelcdm elmapi fssresd cnex ssexd + elmapg sylancr mpbird fmpttd 3imtr4d ) ABICUDZFDUEPUFZBICEUGZUDZFEUGZEUEP + UFZMANUHZOUHZYRPZPZUUDFPZUIULUJPZUAUHZUKUFZNDQZOUBUHZUMPZQZUBIUNZUAUOQUUD + UUEUUAPZPZUUHUIULUJPZUUJUKUFZNEQZOUUNQZUBIUNZUAUOQYSUUCAUUPUVCUAUOAUUOUVB + UBIAUUMIRZSUULUVAOUUNAUVDUUEUUNRZUULUVAUPZUVDUVESAUUEIRZUVFGUUEUUMIJUQAUV + GSZUULUUKNEQZUVAUVHEDURZUULUVIUPAUVJUVGKUSUUKNEDUTVFUVHUURUUGVAZNEQZUUTUU + KVBZNEQUVAUVIVBAUVLOIAUUDBUHZUUAPZPZUUDUVNYRPZPZVAZNEQZBIQUVLOIQAUVSBNIEA + UVNIRZUUDERZSSZUUDYTPZUUDCPZUVPUVRUWBUWDUWEVAAUWAUUDECVCVDUWCUUDUVOYTUWCU + WAYTTRUVOYTVAAUWAUWBVEZUWCCEHAUWACHRZUWBLVGZVHBIYTTUUAUUAVIVJVKVLUWCUUDUV + QCUWCUWAUWGUVQCVAUWFUWHBICHYRYRVIZVJVKVLVMVNUVTUVLBOIUVTOVOUVKBNEBEVPBUUR + UUGBUUDUUQBIYTUUEVQBUUDVPZVRBUUDUUFBICUUEVQUWJVRWGVSUVNUUEVAZUVSUVKNEUWKU + VPUURUVRUUGUWKUUDUVOUUQUVNUUEUUAVTVLUWKUUDUVQUUFUVNUUEYRVTVLWAWBWCWDWEUVK + UVMNEUVKUUSUUIUUJUKUURUUGUUHUJUIWFWHWIUUTUUKNEWJWQWKWLWMWNWOWPAUANUUHUUGD + UBOYRFGTIJAUCUHZUMPZWRDWSULZYRWTZUCXAUNZGXARZAYSUWPMDUCYRFXBVFZAUWOUWQUCX + AUWOUWMIURZAUWLXARZSUWLUWMRZUWQUWOUWMYRXCIUWMUWNYRXDBICYRUWIXEXFUWTUXAAUW + LXGXHUWSUXAUWLIRUWQUWMIUWLXIUWQUWLGUMPIGUWLXKJXJXLXMXNXOZAYRIXPZYRXQUWNUR + ZIUWNYRWTAUWGBIQUXCAUWGBILXRBICYRHUWIXSVFAUWPUXDUWRUWOUXDUCXAUWMUWNYRYBXT + VFIUWNYRYAYCZAUVGUUDDRZSSUUGYDAUXFSUUHYDAYSDWRFWTMDYRFYEVFZAYSDTRMDYRFYFV + FZYGAUANUUHUUREUBOUUAUUBGTIJUXBABIYTWREWSULZAUWASZYTUXIRZEWRYTWTZUXJDWREC + UXJCUWNRDWRCWTABICUWNUXEYHCWRDYIVFAUVJUWAKUSYJUXJWRTRETRZUXKUXLVBYKAUXMUW + AAEDTUXHKYLZUSWREYTTTYMYNYOYPAUVGUWBSSUURYDUWBUUDUUBPUUHVAAUUDEFVCXHADWRE + FUXGKYJUXNYGYQXO $. $} ${ @@ -363907,9 +363902,9 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the $( Lemma for ~ ulmdv . (Contributed by Mario Carneiro, 8-May-2015.) $) ulmdvlem2 $p |- ( ( ph /\ k e. Z ) -> dom ( S _D ( F ` k ) ) = X ) $= ( wcel cfv cc cv wa cdv co cmap cdm wceq cmpt wfn culm wbr cvv wral rgenw - wf ovex eqid fnmpt mp1i ulmf2 syl2anc fvmptelrn elmapi fdm 3syl ) ADUAZJR - UBCVFESZUCUDZTIUEUDZRITVHUOVHUFIUGADJVHVIADJVHUHZJUIZVJGIUJSUKJVIVJUOVHUL - RZDJUMVKAVLDJCVGUCUPUNDJVHVJULVJUQURUSQIVJGJUTVAVBVHTIVCITVHVDVE $. + wf ovex eqid fnmpt mp1i ulmf2 syl2anc fvmptelcdm elmapi fdm 3syl ) ADUAZJ + RUBCVFESZUCUDZTIUEUDZRITVHUOVHUFIUGADJVHVIADJVHUHZJUIZVJGIUJSUKJVIVJUOVHU + LRZDJUMVKAVLDJCVGUCUPUNDJVHVJULVJUQURUSQIVJGJUTVAVBVHTIVCITVHVDVE $. $( Lemma for ~ ulmdv . (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.) $) @@ -364248,18 +364243,18 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ( vz cmpt wcel citg cfv nfcv cc vn cibl cli wbr fmpttd iblulm cv nffvmpt1 itgulm nffv nfitg fveq2 nfmpt1 nfmpt cbvitg fveq1d adantr itgeq2dv eqtrid wceq cbvmpt wa cvv simplr culm ulmscl mptexg 3syl ad2antrr fvmpt2 syl2anc - eqid simpr cmap co wf wfn wral ralrimivw fnmpt syl ulmf2 fvmptelrn elmapi - eqtrd mpteq2dva ulmcl 3brtr3d jca ) ABEDOZUBPFHBECQZOZBEDQZUCUDAEFHBECOZO - ZWJGHIJAFHWNUBKUEZLMUFAUAHNENUGZUAUGZWORZRZQZOZNEWQWJRZQZWLWMUCANEUAWOWJG - HIJWPLMUIAXBFHBEBUGZFUGZWORZRZQZOWLUAFHXAXINFEWTFESFWQWSFHWNWRUHFWQSUJUKU - AXISWRXFUTZXABEXEWSRZQXINBEWTXKWQXEWSULBWQWSBWRWOBFHWNBHSBECUMUNBWRSUJBWQ - SUJNXKSUOXJBEXKXHXJXKXHUTXEEPZXJXEWSXGWRXFWOULUPUQURUSVAAFHXIWKAXFHPZVBZB - EXHCXNXLVBZXHXEWNRZCXOXEXGWNXOXMWNVCPZXGWNUTAXMXLVDAXQXMXLAWOWJEVERUDZEVC - PXQLEWOWJVFBECVCVGVHZVIFHWNVCWOWOVLZVJVKUPXOXLCTPXPCUTXNXLVMXNBECTXNWNTEV - NVOZPETWNVPAFHWNYAAWOHVQZXRHYAWOVPAXQFHVRYBAXQFHXSVSFHWNWOVCXTVTWALEWOWJH - WBVKWCWNTEWDWAWCBECTWNWNVLVJVKWEURWFUSAXDBEXEWJRZQWMNBEXCYCWQXEWJULBEDWQU - HNYCSUOABEYCDAXLVBXLDTPYCDUTAXLVMABEDTAXRETWJVPLEWOWJWGWAWCBEDTWJWJVLVJVK - URUSWHWI $. + eqid simpr cmap co wf wfn wral ralrimivw fnmpt syl ulmf2 fvmptelcdm eqtrd + elmapi mpteq2dva ulmcl 3brtr3d jca ) ABEDOZUBPFHBECQZOZBEDQZUCUDAEFHBECOZ + OZWJGHIJAFHWNUBKUEZLMUFAUAHNENUGZUAUGZWORZRZQZOZNEWQWJRZQZWLWMUCANEUAWOWJ + GHIJWPLMUIAXBFHBEBUGZFUGZWORZRZQZOWLUAFHXAXINFEWTFESFWQWSFHWNWRUHFWQSUJUK + UAXISWRXFUTZXABEXEWSRZQXINBEWTXKWQXEWSULBWQWSBWRWOBFHWNBHSBECUMUNBWRSUJBW + QSUJNXKSUOXJBEXKXHXJXKXHUTXEEPZXJXEWSXGWRXFWOULUPUQURUSVAAFHXIWKAXFHPZVBZ + BEXHCXNXLVBZXHXEWNRZCXOXEXGWNXOXMWNVCPZXGWNUTAXMXLVDAXQXMXLAWOWJEVERUDZEV + CPXQLEWOWJVFBECVCVGVHZVIFHWNVCWOWOVLZVJVKUPXOXLCTPXPCUTXNXLVMXNBECTXNWNTE + VNVOZPETWNVPAFHWNYAAWOHVQZXRHYAWOVPAXQFHVRYBAXQFHXSVSFHWNWOVCXTVTWALEWOWJ + HWBVKWCWNTEWEWAWCBECTWNWNVLVJVKWDURWFUSAXDBEXEWJRZQWMNBEXCYCWQXEWJULBEDWQ + UHNYCSUOABEYCDAXLVBXLDTPYCDUTAXLVMABEDTAXRETWJVPLEWOWJWGWAWCBEDTWJWJVLVJV + KURUSWHWI $. $} @@ -365628,27 +365623,27 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the ( cr wcel clt c1 co cmul caddc ce cres cfv wf wss crp cc ax-resscn cdv wceq wbr w3a cioo wa cmin cicc cima simpl1 simpl2 simpl3 ccncf wf1o reeff1o f1of cc0 ax-mp rpssre fss iccssre syl2anc fssres2 sylancr wb sstrdi efcn rescncf - mp2an mpisyl cncffvrn mpbird wiso reefiso a1i ioossre eqidd isores3 syl3anc - crn ctg cnt ssid ccnfld ctopn eqid tgioo2 dvres mpanl12 resabs1d oveq2d cpr - cdm reelprrecn dvef dmeqi fdmi eqtri sseqtrri dvres3 mp4an reseq1i reseq12d - iccntr 3eqtr3d isoeq1 simpr dvcvx ax-1cn sselid recnd nncan oveq1d ioossicc - eff syl iirev lincmb01cmp syldan eqeltrrd fvresd cxr cle rexrd ltled lbicc2 - ubicc2 oveq12d 3brtr3d ) ADEZBDEZABFUAZUBZCUOGUCHZEZUDZCAIHZGCUEHZBIHZJHZKA - BUFHZLZMCAYTMZIHZYPBYTMZIHZJHYRKMCAKMZIHZYPBKMZIHZJHFYNABYRCYTKDLZABUCHZUGZ - YHYIYJYMUHZYHYIYJYMUIZYHYIYJYMUJZYNYTYSDUKHEZYSDYTNZYNDDUUINZYSDOZUUPDPUUIN - ZPDOUUQDPUUIULUUSUMDPUUIUNUPUQDPDUUIURVGZYNYHYIUURUULUUMABUSUTZDDYSKVAVBYND - QOZYTYSQUKHEZUUOUUPVCRYNYSQOKQQUKHEUVCYNYSDQUVARVDVEQQYSKVFVHYSQDYTVIVBVJYN - UUJUUKFFDYTSHZVKZUUJUUKFFUUIUUJLZVKZYNDPFFUUIVKZUUJDOZUUKUUKTUVGUVHYNVLVMUV - IYNABVNVMYNUUKVODPFFUUIUUJUUKVPVQYNUVDUVFTUVEUVGVCYNDUUIYSLZSHZDUUISHZYSUCV - RVSMZVTMMZLZUVDUVFYNDDOZUURUVKUVOTZDWAUVAUVBDQUUINZUVPUURUDUVQRUUQUVBUVRUUT - RDDQUUIURVGDYSDUVMUUIWBWCMZUVSWDZUVSUVTWEWFWGVBYNUVJYTDSYNKYSDUVAWHWIYNUVLU - UIUVNUUJUVLUUITYNUVLQKSHZDLZUUIDDQWJEQQKNQQODUWAWKZOUVLUWBTWLXMQWADQUWCRUWC - KWKQUWAKWMWNQQKXMWOWPWQQDKWRWSUWAKDWMWTWPVMYNYHYIUVNUUJTUULUUMABXBUTXAXCUUJ - UUKFFUVFUVDXDXNVJYKYMXEZYRWDXFYNYRYSKYNGYPUEHZAIHZYQJHZYRYSYNUWFYOYQJYNUWEC - AIYNGQECQEUWECTXGYNCYNYLDCUOGVNUWDXHXIGCXJVBXKXKYKYMYPUOGUFHZEZUWGYSEYNCUWH - EUWIYNYLUWHCUOGXLUWDXHCXOXNABYPXPXQXRXSYNUUBUUFUUDUUHJYNUUAUUECIYNAYSKYNAXT - EZBXTEZABYAUAZAYSEYNAUULYBZYNBUUMYBZYNABUULUUMUUNYCZABYDVQXSWIYNUUCUUGYPIYN - BYSKYNUWJUWKUWLBYSEUWMUWNUWOABYEVQXSWIYFYG $. + mp2an mpisyl cncfcdm wiso reefiso a1i ioossre eqidd isores3 syl3anc crn ctg + mpbird cnt ssid ccnfld ctopn eqid tgioo2 mpanl12 resabs1d oveq2d reelprrecn + dvres cpr cdm dvef dmeqi fdmi eqtri sseqtrri dvres3 reseq1i iccntr reseq12d + eff mp4an 3eqtr3d isoeq1 syl simpr dvcvx ax-1cn sselid recnd nncan ioossicc + oveq1d iirev lincmb01cmp syldan eqeltrrd fvresd cxr cle rexrd lbicc2 ubicc2 + ltled oveq12d 3brtr3d ) ADEZBDEZABFUAZUBZCUOGUCHZEZUDZCAIHZGCUEHZBIHZJHZKAB + UFHZLZMCAYTMZIHZYPBYTMZIHZJHYRKMCAKMZIHZYPBKMZIHZJHFYNABYRCYTKDLZABUCHZUGZY + HYIYJYMUHZYHYIYJYMUIZYHYIYJYMUJZYNYTYSDUKHEZYSDYTNZYNDDUUINZYSDOZUUPDPUUINZ + PDOUUQDPUUIULUUSUMDPUUIUNUPUQDPDUUIURVGZYNYHYIUURUULUUMABUSUTZDDYSKVAVBYNDQ + OZYTYSQUKHEZUUOUUPVCRYNYSQOKQQUKHEUVCYNYSDQUVARVDVEQQYSKVFVHYSQDYTVIVBVSYNU + UJUUKFFDYTSHZVJZUUJUUKFFUUIUUJLZVJZYNDPFFUUIVJZUUJDOZUUKUUKTUVGUVHYNVKVLUVI + YNABVMVLYNUUKVNDPFFUUIUUJUUKVOVPYNUVDUVFTUVEUVGVCYNDUUIYSLZSHZDUUISHZYSUCVQ + VRMZVTMMZLZUVDUVFYNDDOZUURUVKUVOTZDWAUVAUVBDQUUINZUVPUURUDUVQRUUQUVBUVRUUTR + DDQUUIURVGDYSDUVMUUIWBWCMZUVSWDZUVSUVTWEWJWFVBYNUVJYTDSYNKYSDUVAWGWHYNUVLUU + IUVNUUJUVLUUITYNUVLQKSHZDLZUUIDDQWKEQQKNQQODUWAWLZOUVLUWBTWIXBQWADQUWCRUWCK + WLQUWAKWMWNQQKXBWOWPWQQDKWRXCUWAKDWMWSWPVLYNYHYIUVNUUJTUULUUMABWTUTXAXDUUJU + UKFFUVFUVDXEXFVSYKYMXGZYRWDXHYNYRYSKYNGYPUEHZAIHZYQJHZYRYSYNUWFYOYQJYNUWECA + IYNGQECQEUWECTXIYNCYNYLDCUOGVMUWDXJXKGCXLVBXNXNYKYMYPUOGUFHZEZUWGYSEYNCUWHE + UWIYNYLUWHCUOGXMUWDXJCXOXFABYPXPXQXRXSYNUUBUUFUUDUUHJYNUUAUUECIYNAYSKYNAXTE + ZBXTEZABYAUAZAYSEYNAUULYBZYNBUUMYBZYNABUULUUMUUNYEZABYCVPXSWHYNUUCUUGYPIYNB + YSKYNUWJUWKUWLBYSEUWMUWNUWOABYDVPXSWHYFYG $. ${ $d x y P $. @@ -368370,25 +368365,25 @@ imaginary part lies in the interval (-pi, pi]. See dvrelog $p |- ( RR _D ( log |` RR+ ) ) = ( x e. RR+ |-> ( 1 / x ) ) $= ( cr crp cres cdv co ce c1 cdiv cfv wceq wtru ccncf wcel wss reeff1o cc a1i cdm eqtri clog ccnv cv cmpt dfrelog oveq2i wf1o f1of ax-mp rpssre fss mp2an - wf ax-resscn efcn rescncf mp2 cncffvrn mpbir cpr reelprrecn ssid dvef dmeqi - wb eff fdmi sseqtrri dvres3 mp4an reseq1i cc0 crn 0nrp rneqi wfo f1ofo forn - wn mp2b eleq2i mtbir dvcnvre mptru fveq1i f1ocnvfv2 eqtrid oveq2d mpteq2ia - mpan ) BUACDZEFBGBDZUBZEFZACHAUCZIFZUDZWKWMBEUEUFWNACHWOWMJZBWLEFZJZIFZUDZW - QWNXBKLAWLBCWLBBMFNZLXCBBWLUMZBCWLUMZCBOXDBCWLUGZXEPBCWLUHUIZUJBCBWLUKULBQO - ZWLBQMFNZXCXDVEUNXHGQQMFNXIUNUOQQBGUPUQBQBWLURULUSRWSSZBKLXJWLSBWSWLWSQGEFZ - BDZWLBBQUTNQQGUMQQOBXKSZOWSXLKVAVFQVBBQXMUNXMGSQXKGVCVDQQGVFVGTVHQBGVIVJXKG - BVCVKTZVDBCWLXGVGTRVLWSVMZNZVSLXPVLCNVNXOCVLXOWLVMZCWSWLXNVOXFBCWLVPXQCKPBC - WLVQBCWLVRVTTWAWBRXFLPRWCWDACXAWPWOCNZWTWOHIXRWTWRWLJZWOWRWSWLXNWEXFXRXSWOK - PBCWOWLWFWJWGWHWITT $. + wf wb ax-resscn efcn rescncf mp2 cncfcdm mpbir cpr reelprrecn eff ssid dvef + dmeqi fdmi sseqtrri dvres3 mp4an reseq1i cc0 crn 0nrp rneqi f1ofo forn mp2b + wfo eleq2i mtbir dvcnvre mptru fveq1i f1ocnvfv2 mpan eqtrid oveq2d mpteq2ia + wn ) BUACDZEFBGBDZUBZEFZACHAUCZIFZUDZWKWMBEUEUFWNACHWOWMJZBWLEFZJZIFZUDZWQW + NXBKLAWLBCWLBBMFNZLXCBBWLUMZBCWLUMZCBOXDBCWLUGZXEPBCWLUHUIZUJBCBWLUKULBQOZW + LBQMFNZXCXDUNUOXHGQQMFNXIUOUPQQBGUQURBQBWLUSULUTRWSSZBKLXJWLSBWSWLWSQGEFZBD + ZWLBBQVANQQGUMQQOBXKSZOWSXLKVBVCQVDBQXMUOXMGSQXKGVEVFQQGVCVGTVHQBGVIVJXKGBV + EVKTZVFBCWLXGVGTRVLWSVMZNZWJLXPVLCNVNXOCVLXOWLVMZCWSWLXNVOXFBCWLVSXQCKPBCWL + VPBCWLVQVRTVTWARXFLPRWBWCACXAWPWOCNZWTWOHIXRWTWRWLJZWOWRWSWLXNWDXFXRXSWOKPB + CWOWLWEWFWGWHWITT $. $( The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.) $) relogcn $p |- ( log |` RR+ ) e. ( RR+ -cn-> RR ) $= ( vx clog crp cres cr ccncf co wcel wf wf1o relogf1o ax-mp cc wss ax-resscn f1of wb mp2an c1 cdiv fss rpssre w3a cdv cdm wceq ovex dvrelog dmmpti mpan2 - cv dvcn mp3an cncffvrn mpbir ) BCDZCEFGHZCEUPIZCEUPJURKCEUPPLZEMNZUPCMFGHZU - QURQOUTCMUPIZCENZVAOURUTVBUSOCEMUPUARUBUTVBVCUCEUPUDGZUECUFVAACSAUKZTGVDSVE - TUGAUHUICEUPULUJUMCMEUPUNRUO $. + cv dvcn mp3an cncfcdm mpbir ) BCDZCEFGHZCEUPIZCEUPJURKCEUPPLZEMNZUPCMFGHZUQ + URQOUTCMUPIZCENZVAOURUTVBUSOCEMUPUARUBUTVBVCUCEUPUDGZUECUFVAACSAUKZTGVDSVET + UGAUHUICEUPULUJUMCMEUPUNRUO $. ${ $d w x y z D $. @@ -368586,26 +368581,26 @@ imaginary part lies in the interval (-pi, pi]. See cre wne relog syl2anc absrpcld fvresd eqtr4d oveq1d 3eqtrd mpteq2ia eqtri ccnfld ctopn eqid ctx ccn addcn a1i crest cnfldtopon ssriv resttopon absf ctopon feqmptd eqtrdi wral eqeltrd rgen ffnfv mpbir2an wb ax-resscn sstri - rpssre abscncf rescncf mp2 cncffvrn eqeltrrdi cncfcn eleqtrdi ssid cncfss - mpbir sselii toponrestid cnmpt11f eleqtrrdi ccom imcld recnd adantl eqidd - relogcn oveq2 fmptco logcnlem5 ax-icn mulc1cncf cncfco eqeltrrd cncfmpt2f - mp1i mptru eqeltri ) DAUBZCACUCZEFZDGUBZFZHYNDFZUDFZUEIZJIZKZALMIZYMCAYNY - MFZKZUUBYMAUFZYMUUENADUGZYMSZUUFLUHUOUIZUUGDSZAUUIOUUHUUIUUGDUJUUJUKUUIUU - GDULUMABUNUUIUUGADUPPAUUGYMUQUMCAYMURUSCAUUDUUAYNAQZUUDYRYRVGFZYTJIUUAYNA - DUTUUKYRUUKYNUUKYNLQZYNTQYNGQVAYNABVBVCZYNABVDZVEZVFUUKUULYQYTJUUKUULYODF - ZYQUUKUUMYNUHVHUULUUQNUUNUUOYNVIVJUUKYOGDUUKYNUUNUUOVKZVLVMVNVOVPVQUUBUUC - QRCYQYTJVRVSFZAUUSVTZJUUSUUSWAIUUSWBIQRUUSUUTWCWDRCAYQKUUSAWEIZUUSWBIZUUC - RCYOYPUVAUUSGWEIZUUSAUVAAWJFQZRUUSLWJFQALOZUVDUUSUUTWFZCALUUNWGZAUUSLWHPW - DRCAYOKZAGMIZUVAUVCWBIZRUVHEAUBZUVIRUVKCAYNUVKFZKUVHRCATUVKATUVKSZRLTESUV - EUVMWIUVGLTAEUPPZWDWKCAUVLYOYNAEUTZVPWLUVKUVIQZAGUVKSZUVQUVKAUFZUVLGQZCAW - MUVMUVRUVNATUVKUQUMUVSCAUUKUVLYOGUVOUURWNWOCAGUVKWPWQGLOZUVKATMIZQZUVPUVQ - WRGTLXAWSWTZUVEELTMIQUWBUVGXBLTAEXCXDATGUVKXEPXKXFUVEUVTUVIUVJNUVGUWCAGUU - SUVAUVCUUTUVAVTZUVCVTZXGPXHRYPGLMIZUVCUUSWBIZYPUWFQRGTMIZUWFYPTLOZLLOZUWH - UWFOWSLXIZGTLXJPYAXLWDUVTUWJUWFUWGNUWCUWKGLUUSUVCUUSUUTUWEUUSLUVFXMZXGPXH - XNUVEUWJUUCUVBNUVGUWKALUUSUVAUUSUUTUWDUWLXGPXORUALHUAUCZUEIZKZCAYSKZXPCAY - TKUUCRCUAALYSUWNYTUWPUWOUUKYSLQRUUKYSUUKYRUUPXQXRXSRUWPXTRUWOXTUWMYSHUEYB - YCRALLUWPUWOUWPUUCQRUWAUUCUWPUWIUWJUWAUUCOWSUWKATLXJPCABYDXLWDHLQUWOLLMIQ - RYEUAHUWOUWOVTYFYJYGYHYIYKYL $. + rpssre abscncf rescncf mp2 cncfcdm mpbir eqeltrrdi cncfcn eleqtrdi cncfss + ssid relogcn sselii toponrestid cnmpt11f eleqtrrdi ccom imcld recnd eqidd + adantl oveq2 fmptco logcnlem5 ax-icn mulc1cncf mp1i cncfco eqeltrrd mptru + cncfmpt2f eqeltri ) DAUBZCACUCZEFZDGUBZFZHYNDFZUDFZUEIZJIZKZALMIZYMCAYNYM + FZKZUUBYMAUFZYMUUENADUGZYMSZUUFLUHUOUIZUUGDSZAUUIOUUHUUIUUGDUJUUJUKUUIUUG + DULUMABUNUUIUUGADUPPAUUGYMUQUMCAYMURUSCAUUDUUAYNAQZUUDYRYRVGFZYTJIUUAYNAD + UTUUKYRUUKYNUUKYNLQZYNTQYNGQVAYNABVBVCZYNABVDZVEZVFUUKUULYQYTJUUKUULYODFZ + YQUUKUUMYNUHVHUULUUQNUUNUUOYNVIVJUUKYOGDUUKYNUUNUUOVKZVLVMVNVOVPVQUUBUUCQ + RCYQYTJVRVSFZAUUSVTZJUUSUUSWAIUUSWBIQRUUSUUTWCWDRCAYQKUUSAWEIZUUSWBIZUUCR + CYOYPUVAUUSGWEIZUUSAUVAAWJFQZRUUSLWJFQALOZUVDUUSUUTWFZCALUUNWGZAUUSLWHPWD + RCAYOKZAGMIZUVAUVCWBIZRUVHEAUBZUVIRUVKCAYNUVKFZKUVHRCATUVKATUVKSZRLTESUVE + UVMWIUVGLTAEUPPZWDWKCAUVLYOYNAEUTZVPWLUVKUVIQZAGUVKSZUVQUVKAUFZUVLGQZCAWM + UVMUVRUVNATUVKUQUMUVSCAUUKUVLYOGUVOUURWNWOCAGUVKWPWQGLOZUVKATMIZQZUVPUVQW + RGTLXAWSWTZUVEELTMIQUWBUVGXBLTAEXCXDATGUVKXEPXFXGUVEUVTUVIUVJNUVGUWCAGUUS + UVAUVCUUTUVAVTZUVCVTZXHPXIRYPGLMIZUVCUUSWBIZYPUWFQRGTMIZUWFYPTLOZLLOZUWHU + WFOWSLXKZGTLXJPXLXMWDUVTUWJUWFUWGNUWCUWKGLUUSUVCUUSUUTUWEUUSLUVFXNZXHPXIX + OUVEUWJUUCUVBNUVGUWKALUUSUVAUUSUUTUWDUWLXHPXPRUALHUAUCZUEIZKZCAYSKZXQCAYT + KUUCRCUAALYSUWNYTUWPUWOUUKYSLQRUUKYSUUKYRUUPXRXSYARUWPXTRUWOXTUWMYSHUEYBY + CRALLUWPUWOUWPUUCQRUWAUUCUWPUWIUWJUWAUUCOWSUWKATLXJPCABYDXMWDHLQUWOLLMIQR + YEUAHUWOUWOVTYFYGYHYIYKYJYL $. $( Lemma for ~ dvlog . (Contributed by Mario Carneiro, 24-Feb-2015.) $) dvloglem $p |- ( log " D ) e. ( TopOpen ` CCfld ) $= @@ -368706,27 +368701,27 @@ imaginary part lies in the interval (-pi, pi]. See cdif cim cpi cneg cioc df-log reseq1i cnveqi wf wfun eff cnvimass sseqtri imf fdmi fssres ffun funcnvres2 mp2b resabs1 3eqtri imaeq1i eqtr4i f1oeq1 reseq2i mpbi ccncf wrel relres dfrel2 f1of mp1i imassrn ssriv sstri logcn - eqtr3i logrncn cncffvrn sylibr eqeltrid cin cnt ssid dvres mp4an cnfldtop - dvef dvloglem isopn3i reseq12i eqtri dmeqi dmres sseqtrri df-ss wne neirr - ctop wn resss eqsstri rnssi eff2 frn sseli eldifsn sylib simprd mto dvcnv - wa mptru oveq2i fveq1i f1ocnvfv2 mpan eqtrid oveq2d mpteq2ia 3eqtr3i ) DE - FBUAZGZHZITZABUBAUCZUUHJZDUUGITZJZUDTZUEZDFBGZITABUBUUJUDTZUEUUIUUOKLADUU - GUFUGJZUURUUFBUURULZUURDUURUUSUHUIZDUJDUMMLUKNBUURMLBCUNNUUFBUUGOZLUUFBUU - PHZOZUVABUUFUUPOZUVCDPUOVEZFUPZFUQZBUVEQUVDUVEUVFFOUVGURUVEUVFFUSRBCUTUVE - UVFBFVAVBZBUUFUUPVCRUVBUUGKUVCUVAVDUVBEEVFHVGVHVGVITZUAZGZHZBUAZGZUUGUVBU - VLBGZHZUVKUVMGZUVNUUPUVOFUVLBVJVKVLUVJDUVKVMZUVKVNUVPUVQKDDEVMZUVJDQUVRVO - UVJVFSDVFUVIVPDUJVFVRVSVQDDUVJEVTVBZUVJDUVKWABUVKWBWCUVMUVJQUVQUVNKUVMUVK - SUVJUVKBVPUVJDUVKUVTVSVQEUVMUVJWDRWEUUFUVMEFUVLBVJWFWIWGZUUFBUVBUUGWHRWJZ - NLUUHUUPBUUFWKTZUVBHZUUHUUPUVBUUGUWAVLUUPWLUWDUUPKFBWMUUPWNWJXAZLBUUFUUPV - MZUUPUWCMZUVDUWFLUVHBUUFUUPWOWPUUFDQZUUPBDWKTMUWGUWFVDUUFUVFDFBWQAUVFDUUJ - XBWRWSZBCWTBDUUFUUPXCVBXDXEUULSZUUFKLUWJUUGSUUFESZXFZUUFUULUUGUULDEITZUUF - UURXGJJZGZUUGDDQZUVSUWPUWHUULUWOKDXHZVOUWQUWIDUUFDUUREUURUUSUUTXIXJZUWMEU - WNUUFXLUURYCMUUFUURMUWNUUFKUURUUSXKBCXMUUFUURXNVBXOXPZXQEUUFXRUUFUWKQUWLU - UFKUUFDUWKUWIDDEVOVSXSUUFUWKXTWJWENPUULUPZMZYDLUXAPPYAZPYBUXAPDMZUXBUXAPU - VEMUXCUXBYPUWTUVEPUWTEUPZUVEUULEUULUWMEUULUWOUWMUWRUWMUWNYEYFXLVQYGDUVEEV - MUXDUVEQYHDUVEEYIRWSYJPDPYKYLYMYNNYOYQUUHUUPDIUWEYRABUUNUUQUUJBMZUUMUUJUB - UDUXEUUMUUKUUGJZUUJUUKUULUUGUWSYSUVAUXEUXFUUJKUWBUUFBUUJUUGYTUUAUUBUUCUUD - UUE $. + eqtr3i logrncn cncfcdm sylibr eqeltrid cin cnt ssid dvres mp4an dvef ctop + cnfldtop dvloglem isopn3i reseq12i eqtri dmeqi dmres sseqtrri df-ss neirr + wn wne wa resss eqsstri rnssi eff2 sseli eldifsn sylib simprd dvcnv mptru + frn mto oveq2i fveq1i f1ocnvfv2 mpan eqtrid oveq2d mpteq2ia 3eqtr3i ) DEF + BUAZGZHZITZABUBAUCZUUHJZDUUGITZJZUDTZUEZDFBGZITABUBUUJUDTZUEUUIUUOKLADUUG + UFUGJZUURUUFBUURULZUURDUURUUSUHUIZDUJDUMMLUKNBUURMLBCUNNUUFBUUGOZLUUFBUUP + HZOZUVABUUFUUPOZUVCDPUOVEZFUPZFUQZBUVEQUVDUVEUVFFOUVGURUVEUVFFUSRBCUTUVEU + VFBFVAVBZBUUFUUPVCRUVBUUGKUVCUVAVDUVBEEVFHVGVHVGVITZUAZGZHZBUAZGZUUGUVBUV + LBGZHZUVKUVMGZUVNUUPUVOFUVLBVJVKVLUVJDUVKVMZUVKVNUVPUVQKDDEVMZUVJDQUVRVOU + VJVFSDVFUVIVPDUJVFVRVSVQDDUVJEVTVBZUVJDUVKWABUVKWBWCUVMUVJQUVQUVNKUVMUVKS + UVJUVKBVPUVJDUVKUVTVSVQEUVMUVJWDRWEUUFUVMEFUVLBVJWFWIWGZUUFBUVBUUGWHRWJZN + LUUHUUPBUUFWKTZUVBHZUUHUUPUVBUUGUWAVLUUPWLUWDUUPKFBWMUUPWNWJXAZLBUUFUUPVM + ZUUPUWCMZUVDUWFLUVHBUUFUUPWOWPUUFDQZUUPBDWKTMUWGUWFVDUUFUVFDFBWQAUVFDUUJX + BWRWSZBCWTBDUUFUUPXCVBXDXEUULSZUUFKLUWJUUGSUUFESZXFZUUFUULUUGUULDEITZUUFU + URXGJJZGZUUGDDQZUVSUWPUWHUULUWOKDXHZVOUWQUWIDUUFDUUREUURUUSUUTXIXJZUWMEUW + NUUFXKUURXLMUUFUURMUWNUUFKUURUUSXMBCXNUUFUURXOVBXPXQZXREUUFXSUUFUWKQUWLUU + FKUUFDUWKUWIDDEVOVSXTUUFUWKYAWJWENPUULUPZMZYCLUXAPPYDZPYBUXAPDMZUXBUXAPUV + EMUXCUXBYEUWTUVEPUWTEUPZUVEUULEUULUWMEUULUWOUWMUWRUWMUWNYFYGXKVQYHDUVEEVM + UXDUVEQYIDUVEEYPRWSYJPDPYKYLYMYQNYNYOUUHUUPDIUWEYRABUUNUUQUUJBMZUUMUUJUBU + DUXEUUMUUKUUGJZUUJUUKUULUUGUWSYSUVAUXEUXFUUJKUWBUUFBUUJUUGYTUUAUUBUUCUUDU + UE $. $} ${ @@ -369013,112 +369008,112 @@ imaginary part lies in the interval (-pi, pi]. See peano2re cpnf cicc iccssxr ifex mpteq2i nn0cn neqne reccl seqeq3d rabbiia recn eleq1d supeq1i radcnvcl sselid 1re avglt1 0red lelttrd absidd avglt2 ltled syl2anc eqeltrd radcnvle xrltletrd w3a 0re elico2 mpbir3and wfn ffn - absf elpreima mp2b sylanbrc cnvimass fdmi sseqtri eqtr2d psercn fvmptelrn - ccncf cncff cpr cnelprrecn nncan elbl cmnf cioc dvlog2lem eldifad logdmn0 - cdif ccnfld ctopn 1cnd dvmptc dvmptid dvmptsub df-neg eqtr4di toponrestid - cnfldtopon cnfldtopn blopn dvmptres cres crn csn wf1o logf1o f1of logdmss - sstri fssres feqmptd dvlog2 eqtr3di dvmptco dvmptneg reccld mulcom mulm1d - mp2an fvres negnegd dmeqd dmmptg sumex cbvsumv pserdv2 ssriv nnnn0 neneqd - mprg nnne0 iffalsed nncn recidd nnm1nn0 mulid2d nnuz 1e0p1 eqtri isumshft - fveq2i 1zzd pncan2 sumeq2i geoisum 3eqtrd crp 1rp blcntr log1 neg0 eqeq1d - 1m0e1 simpll eqeltrdi simplr expcld mul02d 0expd mul01d ifbothda cfn orci - eqimssi dv11cn negex sumeq2sdv 3eqtr3d breqtrrd seqex elnnuz sylbi addid2 - sumz 1eluzge0 1nn0 ffvelcdm elfz1eq oveq2i eleq2s iftrue eqtrid sylan9eqr - 1m1e0 0nn0 seqid divrec2d id 3eqtr4d sylan2br seqfeq climeq ) AFGZAHIZJUE - UFZKZUGBLBUHZMNZMJVYJOTZUIZAVYJPTZQTZRZMUJZJAUKTZULIZUMZUNUFUGBURVYNVYJOT - ZRZJUJZVYTUNUFVYIVYQLCUHZMNZMJWUDOTZUIZAWUDPTZQTZCUOZVYTUNVYIWUICVYPMLUPV - YIUQWUDLGZWUDVYPIZWUINZVYIBWUDVYOWUILVYPBCUSZVYMWUGVYNWUHQWUNVYKWUEVYLWUF - MVYJWUDMUTVYJWUDJOVAVBVYJWUDAPVAZVCVYPVDZWUGWUHQVEVFVGZVYIWUKKZWUGWUHWURW - UEMWUFFWURWUEKVKWURWUEVHZKZWUFWUTWUDWURWUSWUDURGZWURWVAWUEWURWVAWUEWURWUK - WVAWUEVLZVYIWUKVIWUDUUAZVJVMVNVOVPVQVRVYFWUKWUHFGZVYHAWUDWCVSZVTABWAUUBVY - IAMJHUKUUDZUUEIZTZGZVYTWUJNVYIWVIAMWVFTZJUEUFZVYIWVJVYGJUEVYIWVJAMUKTZHIZ - VYGVYIVYFMFGZWVJWVMNVYFVYHUUCZWBAMWVFWVFVDZWDWEVYIWVLAHVYFWVLANVYHAUUFWFW - GWHVYFVYHVIWIVYIWVNVYFWVIWVKWJZWBWVOWVFFUUGIGZJWKGZWVNVYFKWVQWLWNAWVFMJFW - OWMWPWQWVIADWVHJDUHZUKTZULIZUMZRZIZADWVHLWUGWVTWUDPTZQTZCUOZRZIZVYTWUJSWW - EWWJNWVIWRSAWWDWWISMMJWWDWWIWVHWVHVDSVKWVSSWNWSSDWVHWWCFWVTWVHGZWWCFGSWWK - 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(Contributed by Mario Carneiro, 31-Mar-2015.) $) @@ -369180,50 +369175,50 @@ imaginary part lies in the interval (-pi, pi]. 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eqeltrrid cncffvrn mpbir ) DUAUBUCEZULZAXMAFZDGZHZXMIUDEZXNXQJKXNALXPHZXM - ULZXQKDXSXMKALLDLLDSKUEMUFUGXMLNZXTXQJKAXMLXOXMOZXOYBXOIOZUAXOUHUMZXOUIZU - JUKZUNZALXMXPUOVEUPUQXQXROZXMIXQSZAXMIXPXQXQPYBYCYDURXPIOYEXOUSUTVAILNXQX - MLUDEZOZYHYIVBVCYKKXQAXMXOTVDVFEZVGEZHZYJAXMYMXPYBXOLOYMXPJYFXOVHVIVJKYNV - KVLGZXMWHEZYOVMEZYJKABCXOYLBFZCFZVGEZYMYPYPYOQVNRVOZWHEZYOXMXMUUAKYOLVQGO - ZYAYPXMVQGOUUCKYOYOPZVPZMZYGXMYOLVRVSZKAYPXMUUGVTKAYLYPUUBXMUUAUUGKUUCUUA - LNUUBUUAVQGOUUFUUAQWALQRWBLIQWCWDWEUUAYOLVRVSZYLUUAOZKUUIYLLOZYLQGZROZWFU - UKYLRYLROZYLIOUUKYLJTROUUMWGTWIWJZYLWKYLWLWMUUNWNLIQSQLWSUUIUUJUULURVBWCL - IQWTLYLRQWOWMWPMWQUUGUUHBCXMUUAYTWRYPUUBXAEYOVMEOKBCUUAYOYPUUBUUAPUUDYPPZ - UUBPXBMYRXOYSYLVGXCXDYALLNYJYQJYGLXEXMLYOYPYOUUDUUOYOLUUEXFXGXHXIXJUQXMLI - XQXKXHXLWN $. + eqeltrrid cncfcdm mpbir ) DUAUBUCEZULZAXMAFZDGZHZXMIUDEZXNXQJKXNALXPHZXMU + LZXQKDXSXMKALLDLLDSKUEMUFUGXMLNZXTXQJKAXMLXOXMOZXOYBXOIOZUAXOUHUMZXOUIZUJ + UKZUNZALXMXPUOVEUPUQXQXROZXMIXQSZAXMIXPXQXQPYBYCYDURXPIOYEXOUSUTVAILNXQXM + LUDEZOZYHYIVBVCYKKXQAXMXOTVDVFEZVGEZHZYJAXMYMXPYBXOLOYMXPJYFXOVHVIVJKYNVK + VLGZXMWHEZYOVMEZYJKABCXOYLBFZCFZVGEZYMYPYPYOQVNRVOZWHEZYOXMXMUUAKYOLVQGOZ + YAYPXMVQGOUUCKYOYOPZVPZMZYGXMYOLVRVSZKAYPXMUUGVTKAYLYPUUBXMUUAUUGKUUCUUAL + NUUBUUAVQGOUUFUUAQWALQRWBLIQWCWDWEUUAYOLVRVSZYLUUAOZKUUIYLLOZYLQGZROZWFUU + KYLRYLROZYLIOUUKYLJTROUUMWGTWIWJZYLWKYLWLWMUUNWNLIQSQLWSUUIUUJUULURVBWCLI + QWTLYLRQWOWMWPMWQUUGUUHBCXMUUAYTWRYPUUBXAEYOVMEOKBCUUAYOYPUUBUUAPUUDYPPZU + UBPXBMYRXOYSYLVGXCXDYALLNYJYQJYGLXEXMLYOYPYOUUDUUOYOLUUEXFXGXHXIXJUQXMLIX + QXKXHXLWN $. sqrcn.d $e |- D = ( CC \ ( -oo (,] 0 ) ) $. $( Continuity of the square root function. (Contributed by Mario Carneiro, @@ -370600,12 +370595,12 @@ the complex square root function (the branch cut is in the same place for cmpt cv cdiv c2 cmul 0re simpl cicc cres w3a elicc2 sylancr biimpa simp1d ccncf simp2d ge0p1rpd fvresd mpteq2dva ccnfld ctopn crest ccn ctopon eqid wb wss cnfldtopon ex ssrdv ax-resscn sstrdi resttopon fmpttd rpssre sstri - ctx addcn ssid cncfmptid sylancl 1cnd cncfmptc syl3anc cncfmpt2f cncffvrn - wf mpbird cncfcn eleqtrd relogcn mp2an eleqtri cnmpt11f eleqtrrd eqeltrrd - wceq crn ctg cpr reelprrecn simpr 1rp rpaddcl relogcld recnd rpreccld cdv - cioo relogcl adantl rpreccl peano2re sselda dvmptid dvmptc dvmptadd 1p0e1 - 0cnd mpteq2i eqtrdi tgioo2 cpnf ioorp iooretop eqeltrri dvmptres relogf1o - wf1o fveq2 rpcnd eqtrd clt cexp rpred oveq1d mpbid cxr 0xr mp3an1 sylan + wf ctx addcn ssid cncfmptid sylancl 1cnd syl3anc cncfmpt2f cncfcdm mpbird + cncfmptc wceq cncfcn eleqtrd relogcn mp2an eleqtri cnmpt11f eleqtrrd cioo + eqeltrrd crn ctg cpr reelprrecn simpr 1rp rpaddcl relogcld recnd rpreccld + relogcl adantl rpreccl peano2re sselda dvmptid 0cnd dvmptc dvmptadd 1p0e1 + mpteq2i eqtrdi tgioo2 cpnf ioorp iooretop eqeltrri dvmptres wf1o relogf1o + cdv fveq2 rpcnd eqtrd clt cexp rpred oveq1d mpbid cxr 0xr mp3an1 sylan f1of feqmptd fvres mpteq2ia oveq2d dvrelog eqtr3di oveq2 dvmptco ioossicc mp1i mulid1d sseli sylan2 eliooord simpld elrpd elrege0 sylanbrc resabs1d cico sqrtf feqresmpt resqrtcn rescncf mpisyl sqrtcld 2rp rpsqrtcl rpmulcl @@ -370618,35 +370613,35 @@ the complex square root function (the branch cut is in the same place for LZUMZUXRUXSUXTUYAUYCUOZUPZUQURUWRUXOUSUTNZUXLVALZUYGDVALZVBLZUXPUWRBUXCUX MUYHUYGRVALZUYIUXLUWRUYGSVCNEUXLSVFZUYHUXLVCNEUYGUYGVDZVGUWRUXLDSUWRBUXLD UWRUXQUXSUYDVHVIVJVKZUXLUYGSVLUKUWRBUXLUXCTZUXLRUNLZUYHUYKVBLZUWRUYOUYPEZ - 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Lemma for ~ logrec . @@ -377628,114 +377623,113 @@ group sum in the additive group (i.e. the sum of the elements). 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(Contributed by Mario Carneiro, 3-Jul-2017.) $) @@ -404870,7 +404864,7 @@ then any point of the half line ( ` R A ` ) also lies opposite to $( Closure of the midpoint. (Contributed by Thierry Arnoux, 1-Dec-2019.) $) midcl $p |- ( ph -> ( A ( midG ` G ) B ) e. P ) $= - ( cmid cfv midf fovrnd ) ABCDDDEOPADEFGHIJKLQMNR $. + ( cmid cfv midf fovcdmd ) ABCDDDEOPADEFGHIJKLQMNR $. ${ $d A a b m $. $d B a b m $. $d M m $. $d S a b g m $. @@ -421724,10 +421718,10 @@ segment of the walk (of length ` N ` ) forms a walk on the subgraph $( Lemma 1 for ~ wlkd . (Contributed by AV, 7-Feb-2021.) $) wlkdlem1 $p |- ( ph -> P : ( 0 ... ( # ` F ) ) --> V ) $= ( cc0 chash cfv cfzo co cvv wf cfz cword wcel syl wrdf c1 caddc cz wceq - oveq2d cn0 lencl nn0zd fzval3 eqtr4d feq2d wss cv wb ssv frnssb sylancr - wral bitrd mpbid ) AJBKLZMNZOBPZJDKLZQNZEBPZABORZSVDFOBUATAVDVFOBPZVGAV - CVFOBAVCJVEUBUCNZMNZVFAVBVJJMHUFAVEUDSVFVKUEAVEADVHSVEUGSGODUHTUIJVEUJT - UKULAEOUMCUNBLESCVFUSVIVGUOEUPIVFCBEOUQURUTVA $. + oveq2d cn0 lencl nn0zd fzval3 eqtr4d feq2d wss wral ssv fcdmssb sylancr + cv wb bitrd mpbid ) AJBKLZMNZOBPZJDKLZQNZEBPZABORZSVDFOBUATAVDVFOBPZVGA + VCVFOBAVCJVEUBUCNZMNZVFAVBVJJMHUFAVEUDSVFVKUEAVEADVHSVEUGSGODUHTUIJVEUJ + TUKULAEOUMCURBLESCVFUNVIVGUSEUOIVFCBEOUPUQUTVA $. $} $d I k $. $d ph k $. @@ -439284,8 +439278,8 @@ Additional material on group theory (deprecated) $( Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) $) grpocl $p |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G B ) e. X ) $= - ( cgr wcel cxp wf co wfo grpofo fof syl fovrn syl3an1 ) CFGZDDHZDCIZADGBD - GABCJDGQRDCKSCDELRDCMNABDDDCOP $. + ( cgr wcel cxp wf co wfo grpofo fof syl fovcdm syl3an1 ) CFGZDDHZDCIZADGB + DGABCJDGQRDCKSCDELRDCMNABDDDCOP $. $( A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) @@ -439774,8 +439768,8 @@ Additional material on group theory (deprecated) $( Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) $) grpodivcl $p |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) e. X ) $= - ( cgr wcel cxp wf co grpodivf fovrn syl3an1 ) DHIEEJECKAEIBEIABCLEICDEFGM - ABEEECNO $. + ( cgr wcel cxp wf co grpodivf fovcdm syl3an1 ) DHIEEJECKAEIBEIABCLEICDEFG + MABEEECNO $. $( Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.) $) @@ -440058,8 +440052,8 @@ which is an Abelian group (i.e. the vectors, with the operation of $( Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) $) vccl $p |- ( ( W e. CVecOLD /\ A e. CC /\ B e. X ) -> ( A S B ) e. X ) $= - ( cvc wcel cc cxp wf co vcsm fovrn syl3an1 ) EJKLFMFCNALKBFKABCOFKCDEFGHI - PABFLFCQR $. + ( cvc wcel cc cxp wf co vcsm fovcdm syl3an1 ) EJKLFMFCNALKBFKABCOFKCDEFGH + IPABFLFCQR $. $( Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use ~ clmvs1 @@ -440991,8 +440985,8 @@ which is an Abelian group (i.e. the vectors, with the operation of vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) $) nvmcl $p |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) e. X ) $= - ( cnv wcel cxp wf co nvmf fovrn syl3an1 ) CHIEEJEDKAEIBEIABDLEICDEFGMABEE - EDNO $. + ( cnv wcel cxp wf co nvmf fovcdm syl3an1 ) CHIEEJEDKAEIBEIABDLEICDEFGMABE + EEDNO $. $( Cancellation law for vector subtraction. ( ~ nnncan1 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) $) @@ -472004,26 +471998,26 @@ Class abstractions (a.k.a. class builders) foresf1o $p |- ( ( A e. V /\ F : A -onto-> B ) -> E. x e. ~P A ( F |` x ) : x -1-1-onto-> B ) $= ( vg vy vz wcel wa cv wf cfv wrex wceq fveq2d nfv nfan syl2anc eqtrd ccnv - wfo csn cima wral cres wf1o cpw cvv wex fornex imp foelrn wi wfn fniniseg - fofn eqcom biimpar anassrs sylan2br sylanl1 reximdva adantr mpd ralrimiva - ex adantll eleq1 ac6sg sylc crn wss frn ad2antrl vex rnex elpw sylibr fof - ad2antlr fssresd dffn3 sylib fvres adantl nfra1 ad5antlr simplrr ad2antrr - ffn simpr simplr simplbda eqtr3d fvelrnb biimpa sylan r19.29af ffvelcdmda - rspa ad3antlr ralrimi 2fvidf1od reseq2 id eqidd f1oeq123d rspcev exlimddv - syl ) BEIZBCDUBZJZCBFKZLZGKZXOMZDUAXQUCUDZIZGCUEZJZAKZCDYCUFZUGZABUHZNZFX - NCUIIZHKZXSIZHBNZGCUEYBFUJXLXMYHBCEDUKULXNYKGCXMXQCIZYKXLXMYLJXQYIDMZOZHB - NZYKHBCXQDUMXMYOYKUNYLXMYNYJHBXMYIBIZJYNYJXMDBUOZYPYNYJBCDUQZYNYQYPJYMXQO - ZYJYMXQURYQYPYSYJYQYJYPYSJBXQYIDUPUSUTVAVBVGVCVDVEVHVFYJXTGHCBFUIYIXRXSVI - VJVKXNYBJZXOVLZYFIZUUACDUUAUFZUGZYGYTUUABVMZUUBXPUUEXNYACBXOVNVOZUUABXOFV - PVQVRVSYTUUACUUCXOHGYTBCUUADXMBCDLXLYBBCDVTWAUUFWBYTXOCUOZCUUAXOLXPUUGXNY - ACBXOWKVOZCXOWCWDZYTYIUUCMZXOMZYIOHUUAYTYIUUAIZJZUUKYMXOMZYIUUMUUJYMXOUUL - UUJYMOYTYIUUADWEWFPUUMXRYIOZUUNYIOGCYTUULGXNYBGXNGQXPYAGXPGQXTGCWGRRZUULG - QRUUMYLJZUUOJZUUNXRYIUURYMXQXOUURXRDMZYMXQUURXRYIDUUQUUOWLZPUURYQXTUUSXQO - ZXMYQXLYBUULYLUUOYRWHUURYAYLXTUUMYAYLUUOXNXPYAUULWIWJUUMYLUUOWMXTGCXAZSYQ - XTXRBIUVABXQXRDUPWNZSWOPUUTTYTUUGUULUUOGCNZUUHUUGUULUVDGCYIXOWPWQWRWSTVFY - TXRUUCMZXQOZGCUUPYTYLUVFYTYLJZUVEUUSXQUVGXRUUAIUVEUUSOYTCUUAXQXOUUIWTXRUU - ADWEXKUVGYQXTUVAXMYQXLYBYLYRXBUVGYAYLXTXNXPYAYLWIYTYLWLUVBSUVCSTVGXCXDYEU - UDAUUAYFYCUUAOZYCUUACCYDUUCYCUUADXEUVHXFUVHCXGXHXISXJ $. + wfo csn cima wral cres wf1o cpw cvv wex focdmex imp foelrn wfn fofn eqcom + wi fniniseg biimpar anassrs sylan2br sylanl1 ex reximdva adantr ralrimiva + mpd adantll eleq1 sylc crn wss frn ad2antrl vex rnex elpw sylibr ad2antlr + ac6sg fof fssresd dffn3 sylib fvres nfra1 simpr ad5antlr simplrr ad2antrr + ffn adantl simplr rspa simplbda eqtr3d fvelrnb biimpa r19.29af ffvelcdmda + sylan syl ad3antlr ralrimi 2fvidf1od reseq2 id f1oeq123d rspcev exlimddv + eqidd ) BEIZBCDUBZJZCBFKZLZGKZXOMZDUAXQUCUDZIZGCUEZJZAKZCDYCUFZUGZABUHZNZ + FXNCUIIZHKZXSIZHBNZGCUEYBFUJXLXMYHBCEDUKULXNYKGCXMXQCIZYKXLXMYLJXQYIDMZOZ + HBNZYKHBCXQDUMXMYOYKUQYLXMYNYJHBXMYIBIZJYNYJXMDBUNZYPYNYJBCDUOZYNYQYPJYMX + QOZYJYMXQUPYQYPYSYJYQYJYPYSJBXQYIDURUSUTVAVBVCVDVEVGVHVFYJXTGHCBFUIYIXRXS + VIVTVJXNYBJZXOVKZYFIZUUACDUUAUFZUGZYGYTUUABVLZUUBXPUUEXNYACBXOVMVNZUUABXO + FVOVPVQVRYTUUACUUCXOHGYTBCUUADXMBCDLXLYBBCDWAVSUUFWBYTXOCUNZCUUAXOLXPUUGX + NYACBXOWKVNZCXOWCWDZYTYIUUCMZXOMZYIOHUUAYTYIUUAIZJZUUKYMXOMZYIUUMUUJYMXOU + ULUUJYMOYTYIUUADWEWLPUUMXRYIOZUUNYIOGCYTUULGXNYBGXNGQXPYAGXPGQXTGCWFRRZUU + LGQRUUMYLJZUUOJZUUNXRYIUURYMXQXOUURXRDMZYMXQUURXRYIDUUQUUOWGZPUURYQXTUUSX + QOZXMYQXLYBUULYLUUOYRWHUURYAYLXTUUMYAYLUUOXNXPYAUULWIWJUUMYLUUOWMXTGCWNZS + YQXTXRBIUVABXQXRDURWOZSWPPUUTTYTUUGUULUUOGCNZUUHUUGUULUVDGCYIXOWQWRXAWSTV + FYTXRUUCMZXQOZGCUUPYTYLUVFYTYLJZUVEUUSXQUVGXRUUAIUVEUUSOYTCUUAXQXOUUIWTXR + UUADWEXBUVGYQXTUVAXMYQXLYBYLYRXCUVGYAYLXTXNXPYAYLWIYTYLWGUVBSUVCSTVCXDXEY + EUUDAUUAYFYCUUAOZYCUUACCYDUUCYCUUADXFUVHXGUVHCXKXHXISXJ $. $} ${ @@ -473768,8 +473762,8 @@ Class abstractions (a.k.a. class builders) $( The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.) $) ofrn $p |- ( ph -> ran ( F oF .+ G ) C_ C ) $= - ( vx vy cof co cv fovrnda inidm off frnd ) ABDFGEOPAMNBBBECCDFGHHAMQNQDCC - EKRIJLLBSTUA $. + ( vx vy cof co cv fovcdmda inidm off frnd ) ABDFGEOPAMNBBBECCDFGHHAMQNQDC + CEKRIJLLBSTUA $. $d a z A $. $d z B $. $d a z F $. $d a x z G $. $d a z .+ $. $d a z ph $. $d x y z a $. @@ -475538,12 +475532,12 @@ its graph has a given second element (that is, function value). ${ ffs2.1 $e |- C = ( B \ { Z } ) $. - $( Rewrite a function's support based with its range rather than the - universal class. See also ~ frnsuppeq . (Contributed by Thierry - Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) $) + $( Rewrite a function's support based with its codomain rather than the + universal class. See also ~ fsuppeq . (Contributed by Thierry Arnoux, + 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) $) ffs2 $p |- ( ( A e. V /\ Z e. W /\ F : A --> B ) -> ( F supp Z ) = ( `' F " C ) ) $= - ( wcel wf w3a csupp co ccnv csn cdif cima wceq frnsuppeq 3impia imaeq2i + ( wcel wf w3a csupp co ccnv csn cdif cima wceq fsuppeq 3impia imaeq2i eqtr4di ) AEIZGFIZABDJZKDGLMZDNZBGOPZQZUGCQUCUDUEUFUIRBDAEFGSTCUHUGHUAUB $. $} @@ -489169,144 +489163,144 @@ such that every prime ideal contains a prime element (this is a islss4 biimpar lbslinds sselid resssca sraaddg eqeltrd lmodvscl 3expb ressplusg ressvsca ressmulr lindspropd lsslinds syl31anc islinds5 wfn sylan cfrlm xpexd frlmelbas elmapd ffnd cmnd drngring ringmnd subg0cl - mpbid crg ress0g breqtrrd fsuppcurry1 breqtrd fovrnda anassrs fvmpt2d - wfun csupp cabl lmodabl 3eltr4d mptscmfsupp0 gsumsubgcl ffund cdm cop - subrg0 wrex fconstmpt eqeq2i bitri necon3abii df-ov eqcomi necon3abid - rexbidva rexnal bitr2di bitrid rabbidva fveq2 dmrab eqtr4di suppvalfn - neeq1d fsuppimpd dmfi cdif nfv nfcv nfmpt1 nfrabw nfdif nfcri eldifad - nfne nfan eldifbd oveq1 mpteq2dv elrab sylnib mpnanrd 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( A ( ~Met ` D ) B /\ E ( ~Met ` D ) F ) ) -> ( A D E ) = ( B D F ) ) $= ( wcel wbr wa co wceq cle cxr wss syl syl2anc sseldd psmetsym cxad cc0 dmss cpsmet cfv cmetid simpl cdm metidss dmxpid sseqtrdi wrel xpss sstrdi df-rel - cxp cvv sylibr simprl releldm simprr syl3anc psmetf fovrnda syl12anc rnxpid - eqeltrd crn rnss relelrn eqeltrrd psmettri2 syl13anc metidv biimpa syl21anc - eqtr3d oveq1d xaddid2 eqtrd breqtrd oveq2d xaddid1 xrletrd 3eqtr4d xrletri3 - jca 3eqtrd wb mpbir2and ) CFUBUCGZABCUDUCZHZDEWJHZIZIZADCJZBECJZKZWOWPLHZWP - WOLHZWNWOBDCJZWPWNWODACJZMWNWIAFGZDFGZWOXAKWIWMUEZWNWJUFZFAWNWIXEFNXDWIXEFF - UNZUFZFWIWJXFNZXEXGNCFUGZWJXFUAOFUHUIOZWNWJUJZWKAXEGWNWIXKXDWIWJUOUOUNZNXKW - IWJXFXLXIFFUKULWJUMUPOZWIWKWLUQZABWJURPQZWNXEFDXJWNXKWLDXEGXMWIWKWLUSZDEWJU - RPQZADCFRUTZWNWIXCXBXAMGZXDXQXOWIDAMFFCCFVAZVBVCZVEZWNWIBFGZXCWTMGZXDWNWJVF - ZFBWNWIYEFNXDWIYEXFVFZFWIXHYEYFNXIWJXFVGOFVDUIOZWNXKWKBYEGXMXNABWJVHPQZXQWI - BDMFFCXTVBVCZWNEBCJZWPMWNWIEFGZYCYJWPKXDWNYEFEYGWNXKWLEYEGXMXPDEWJVHPQZYHEB - CFRUTZWNWIYKYCYJMGZXDYLYHWIEBMFFCXTVBVCZVIZWNWOBACJZWTSJZWTLWNWIYCXBXCWOYRL - HXDYHXOXQADBCFVJVKWNYRTWTSJZWTWNYQTWTSWNABCJZYQTWNWIXBYCYTYQKXDXOYHABCFRUTW - NWIXBYCIZWKYTTKZXDWNXBYCXOYHWEXNWIUUAIWKUUBABCFVLVMVNZVOVPWNYDYSWTKYIWTVQOV - RVSWNWTYJEDCJZSJZWPLWNWIYKYCXCWTUUELHXDYLYHXQBDECFVJVKWNUUEYJTSJZYJWPWNUUDT - YJSWNUUDDECJZTWNWIYKXCUUDUUGKXDYLXQEDCFRUTWNWIXCYKIZWLUUGTKZXDWNXCYKXQYLWEX - PWIUUHIWLUUIDECFVLVMVNZVRVTWNYNUUFYJKYOYJWAOYMWFVSWBWNWPAECJZWOYPWNWIXBYKUU - KMGZXDXOYLWIAEMFFCXTVBVCZYBWNWPYTUUKSJZUUKLWNWIXBYCYKWPUUNLHXDXOYHYLBEACFVJ - VKWNUUNTUUKSJZUUKWNYTTUUKSUUCVPWNUULUUOUUKKUUMUUKVQOVRVSWNUUKXAUUGSJZWOLWNW - IXCXBYKUUKUUPLHXDXQXOYLAEDCFVJVKWNXATSJZXAUUPWOWNXSUUQXAKYAXAWAOWNUUGTXASUU - JVTXRWCVSWBWNWOMGWPMGWQWRWSIWGYBYPWOWPWDPWH $. + cxp cvv sylibr simprl releldm simprr syl3anc fovcdmda syl12anc eqeltrd rnss + psmetf crn rnxpid eqeltrrd psmettri2 syl13anc metidv biimpa syl21anc eqtr3d + relelrn jca oveq1d xaddid2 breqtrd oveq2d xaddid1 3eqtrd xrletrd 3eqtr4d wb + eqtrd xrletri3 mpbir2and ) CFUBUCGZABCUDUCZHZDEWJHZIZIZADCJZBECJZKZWOWPLHZW + PWOLHZWNWOBDCJZWPWNWODACJZMWNWIAFGZDFGZWOXAKWIWMUEZWNWJUFZFAWNWIXEFNXDWIXEF + FUNZUFZFWIWJXFNZXEXGNCFUGZWJXFUAOFUHUIOZWNWJUJZWKAXEGWNWIXKXDWIWJUOUOUNZNXK + WIWJXFXLXIFFUKULWJUMUPOZWIWKWLUQZABWJURPQZWNXEFDXJWNXKWLDXEGXMWIWKWLUSZDEWJ + URPQZADCFRUTZWNWIXCXBXAMGZXDXQXOWIDAMFFCCFVEZVAVBZVCZWNWIBFGZXCWTMGZXDWNWJV + FZFBWNWIYEFNXDWIYEXFVFZFWIXHYEYFNXIWJXFVDOFVGUIOZWNXKWKBYEGXMXNABWJVOPQZXQW + IBDMFFCXTVAVBZWNEBCJZWPMWNWIEFGZYCYJWPKXDWNYEFEYGWNXKWLEYEGXMXPDEWJVOPQZYHE + BCFRUTZWNWIYKYCYJMGZXDYLYHWIEBMFFCXTVAVBZVHZWNWOBACJZWTSJZWTLWNWIYCXBXCWOYR + LHXDYHXOXQADBCFVIVJWNYRTWTSJZWTWNYQTWTSWNABCJZYQTWNWIXBYCYTYQKXDXOYHABCFRUT + WNWIXBYCIZWKYTTKZXDWNXBYCXOYHVPXNWIUUAIWKUUBABCFVKVLVMZVNVQWNYDYSWTKYIWTVRO + WFVSWNWTYJEDCJZSJZWPLWNWIYKYCXCWTUUELHXDYLYHXQBDECFVIVJWNUUEYJTSJZYJWPWNUUD + TYJSWNUUDDECJZTWNWIYKXCUUDUUGKXDYLXQEDCFRUTWNWIXCYKIZWLUUGTKZXDWNXCYKXQYLVP + XPWIUUHIWLUUIDECFVKVLVMZWFVTWNYNUUFYJKYOYJWAOYMWBVSWCWNWPAECJZWOYPWNWIXBYKU + UKMGZXDXOYLWIAEMFFCXTVAVBZYBWNWPYTUUKSJZUUKLWNWIXBYCYKWPUUNLHXDXOYHYLBEACFV + IVJWNUUNTUUKSJZUUKWNYTTUUKSUUCVQWNUULUUOUUKKUUMUUKVROWFVSWNUUKXAUUGSJZWOLWN + WIXCXBYKUUKUUPLHXDXQXOYLAEDCFVIVJWNXATSJZXAUUPWOWNXSUUQXAKYAXAWAOWNUUGTXASU + UJVTXRWDVSWCWNWOMGWPMGWQWRWSIWEYBYPWOWPWGPWH $. ${ $d x y z D $. $d x y z X $. @@ -492361,36 +492355,36 @@ closed sets (see for example ~ zarcls0 for the definition of ` V ` ). 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(Contributed by Thierry eqidd offval cres ffvelcdmda xmstps tpsuni ad2antrr eleqtrrd ovres syl2anc ctps cxmet xmsxmet xmet0 eqtrd xrge0base xrge00 xrge0tps a1i cmnd xrge0cmn - ccmn ax-mp cslmd ct1 cha haust1 mp1i biimpar simprl ofresid cmpo fovrnda + ccmn ax-mp cslmd ct1 cha haust1 mp1i biimpar simprl ofresid cmpo fovcdmda feq3 fnov simprr sylibr ralrimivva rnmposs df-f eqeltrd sibfof cxmu ovex jca off mp2an eqtri 0xr wb adantlr ralrimiva offn ofval mpteq2dva eqtr4di eqtr3d fconstmpt fveq2d cordt crest xrge0topn eqcomi xrge0slmod xrge0haus @@ -502915,32 +502909,32 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry c0 wf wf1o bitsf1o f1of ax-mp cn wss w3a eulerpartlemt0 biimpi simp1d nn0ex nnex elmap sylib c2 cv cdvds wbr wn crab ssrab2 eqsstri sylancl fssres fco2 sylancr pwex inex1 ssexi sylibr cc0 csupp wceq suppimacnv - simp2d 0nn0 mpan2 wi frnsuppeq mp2an syl eqtr3d eleq1d imaeq2i eleq1i - dfn2 bitr4di mpbird resss cnvss imass1 mp2b cnvco imaeq1i imaco eqtri - ssfi wfun ffun funres 3syl ssv ssdif cz bitsf difpreima wf1 bitsf1 0z - snssi eqtr3i wa f1imacnv cfv wfn 0bits sneqi difeq2i 3sstr4i sspreima - ffn fnsnfv eqsstrid syl2anc oveq1 elrab2 wb zex fex resexg 0ex anbi2d - coexg bitrid mpbir2and ) DHGULZUMZUNDQUOZUPZPUMZUVGUQURZUSULZQUTVAZUM - ZUVGVBZVCVGVDZVEZVFZUSUMZUVEQUVJUVGVHZUVLUVEUQUVJUNUQUOZVHZQUQUVFVHZU - VRUQUVJUVSVIUVTVJUQUVJUVSVKVLUVEVMUQDVHZQVMVNUWAUVEDUQVMUTVAUMZUWBUVE - UWCDVBZVMVFZUSUMZUWEQVNZUVEUWCUWFUWGVOABCDEFGHIJKLNPQRSTUAUBUCUDUEUFU - GUHUIUJVPVQZVRUQVMDVSVTWAWBZQWCCWDWEWFWGZCVMWHVMUEUWJCVMWIWJZVMUQQDWL - WKQUQUVJUNUVFWMWNUVJQUVGUVIUSUQVSWOWPQVMVTUWKWQWAWRUVEUVFVBZVCWSVDZVE - ZVFZUSUMZUVPUWOVNUVQUVEUWDUWNVFZUSUMZUWOUWQVNZUWPUVEUWRUWFUVEUWCUWFUW - GUWHXCUVEUWRUWDUQUWMVEZVFZUSUMUWFUVEUWQUXAUSUVEDWSWTVAZUWQUXAUVEWSUQU - MZUXBUWQXAXDDUVDUQWSXBXEUVEUWBUXBUXAXAZUWIVMVCUMUXCUWBUXDXFVTXDUQDVMV - CUQWSXGXHXIXJXKUWEUXAUSVMUWTUWDXNXLXMXOXPUVFDVNUWLUWDVNUWSDQXQUVFDXRU - WLUWDUWNXSXTUWQUWOYEWKUVEUVPUWLUNVBZUVOVFZVFZUWOUVPUWLUXEUPZUVOVFUXGU - VMUXHUVOUNUVFYAYBUWLUXEUVOYCYDUVEUVFYFZUXFUWNVNUXGUWOVNUVEUWBDYFUXIUW - IVMUQDYGQDYHYIUXEVCVFZUXEUVNVFZVEZVCUXKVEZUXFUWNUXJVCVNUXLUXMVNUXJYJU - XJVCUXKYKVLYLUVIUNVHZUNYFUXFUXLXAYMYLUVIUNYGVCUVNUNYNXTUWMUXKVCUXEUNU - WMVFZVFZUWMUXKYLUVIUNYOUWMYLVNZUXPUWMXAYPWSYLUMZUXQYQWSYLYRVLYLUVIUWM - UNUUAXHUXOUVNUXEWSUNUUBZVDZUXOUVNUNYLUUCZUXRUXTUXOXAUXNUYAYMYLUVIUNUU - IVLYQYLWSUNUUJXHUXSVGUUDUUEYSXLYSUUFUUGUXFUWNUVFUUHWKUUKUWOUVPYEUULUV - HUVLUVGVGWTVAZUSUMZYTZUVEUVLUVQYTZUAWDZVGWTVAZUSUMUYCUAUVGUVKPUYFUVGX - AUYGUYBUSUYFUVGVGWTUUMXKUGUUNUVEUVGVCUMZUYDUYEUUOUVEUNVCUMZUVFVCUMUYH - UXNYLVCUMUYIYMUUPYLUVIVCUNUUQXHDQUVDUURUNUVFVCVCUVAWNUYHUYCUVQUVLUYHU - YBUVPUSUYHVGVCUMUYBUVPXAUUSUVGVCVCVGXBXEXKUUTXIUVBUVC $. + simp2d 0nn0 mpan2 fsuppeq mp2an syl eqtr3d eleq1d dfn2 imaeq2i eleq1i + bitr4di mpbird resss cnvss imass1 mp2b ssfi cnvco imaeq1i imaco eqtri + wi wfun ffun funres 3syl ssv ssdif cz bitsf difpreima bitsf1 0z snssi + wf1 eqtr3i wa f1imacnv cfv wfn ffn fnsnfv 0bits sneqi difeq2i 3sstr4i + sspreima eqsstrid syl2anc oveq1 elrab2 wb zex fex resexg coexg anbi2d + 0ex bitrid mpbir2and ) DHGULZUMZUNDQUOZUPZPUMZUVGUQURZUSULZQUTVAZUMZU + VGVBZVCVGVDZVEZVFZUSUMZUVEQUVJUVGVHZUVLUVEUQUVJUNUQUOZVHZQUQUVFVHZUVR + UQUVJUVSVIUVTVJUQUVJUVSVKVLUVEVMUQDVHZQVMVNUWAUVEDUQVMUTVAUMZUWBUVEUW + CDVBZVMVFZUSUMZUWEQVNZUVEUWCUWFUWGVOABCDEFGHIJKLNPQRSTUAUBUCUDUEUFUGU + HUIUJVPVQZVRUQVMDVSVTWAWBZQWCCWDWEWFWGZCVMWHVMUEUWJCVMWIWJZVMUQQDWLWK + QUQUVJUNUVFWMWNUVJQUVGUVIUSUQVSWOWPQVMVTUWKWQWAWRUVEUVFVBZVCWSVDZVEZV + FZUSUMZUVPUWOVNUVQUVEUWDUWNVFZUSUMZUWOUWQVNZUWPUVEUWRUWFUVEUWCUWFUWGU + WHXCUVEUWRUWDUQUWMVEZVFZUSUMUWFUVEUWQUXAUSUVEDWSWTVAZUWQUXAUVEWSUQUMZ + UXBUWQXAXDDUVDUQWSXBXEUVEUWBUXBUXAXAZUWIVMVCUMUXCUWBUXDYEVTXDUQDVMVCU + QWSXFXGXHXIXJUWEUXAUSVMUWTUWDXKXLXMXNXOUVFDVNUWLUWDVNUWSDQXPUVFDXQUWL + UWDUWNXRXSUWQUWOXTWKUVEUVPUWLUNVBZUVOVFZVFZUWOUVPUWLUXEUPZUVOVFUXGUVM + UXHUVOUNUVFYAYBUWLUXEUVOYCYDUVEUVFYFZUXFUWNVNUXGUWOVNUVEUWBDYFUXIUWIV + MUQDYGQDYHYIUXEVCVFZUXEUVNVFZVEZVCUXKVEZUXFUWNUXJVCVNUXLUXMVNUXJYJUXJ + VCUXKYKVLYLUVIUNVHZUNYFUXFUXLXAYMYLUVIUNYGVCUVNUNYNXSUWMUXKVCUXEUNUWM + VFZVFZUWMUXKYLUVIUNYRUWMYLVNZUXPUWMXAYOWSYLUMZUXQYPWSYLYQVLYLUVIUWMUN + UUAXGUXOUVNUXEWSUNUUBZVDZUXOUVNUNYLUUCZUXRUXTUXOXAUXNUYAYMYLUVIUNUUDV + LYPYLWSUNUUEXGUXSVGUUFUUGYSXLYSUUHUUIUXFUWNUVFUUJWKUUKUWOUVPXTUULUVHU + VLUVGVGWTVAZUSUMZYTZUVEUVLUVQYTZUAWDZVGWTVAZUSUMUYCUAUVGUVKPUYFUVGXAU + YGUYBUSUYFUVGVGWTUUMXJUGUUNUVEUVGVCUMZUYDUYEUUOUVEUNVCUMZUVFVCUMUYHUX + NYLVCUMUYIYMUUPYLUVIVCUNUUQXGDQUVDUURUNUVFVCVCUUSWNUYHUYCUVQUVLUYHUYB + UVPUSUYHVGVCUMUYBUVPXAUVAUVGVCVCVGXBXEXJUUTXHUVBUVC $. $d t n w x y A $. $d n t x y B $. $d n t x y F $. $d t w x y J $. $d r H $. $d n t x y M $. $d n r t w x y R $. $d n r t w x y T $. @@ -508229,10 +508223,10 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ( vx csqrt crp cres ccncf co wcel wf cv cdm cc cr wceq sqrtf ax-mp wb mpbir wss cc0 cpnf cfv wa wral rpssre ax-resscn sstri fdm sseqtrri sseli rpsqrtcl rgen wfun ffun ffvresb cico cioo ioossico eqsstrri resabs1 resqrtcn rescncf - jca ioorp mp2 eqeltrri cncffvrn mp2an ) BCDZCCEFGZCCVHHZVJAIZBJZGZVKBUACGZU - BZACUCZVOACVKCGVMVNCVLVKCKVLCLKUDUEUFZKKBHZVLKMNKKBUGOUHUIVKUJVBUKBULZVJVPP - VRVSNKKBUMOACCBUNOQCKRVHCLEFZGVIVJPVQBSTUOFZDZCDZVHVTCWARZWCVHMCSTUPFWAVCST - UQURZBCWAUSOWDWBWALEFGWCVTGWEUTWALCWBVAVDVECLCVHVFVGQ $. + jca ioorp mp2 eqeltrri cncfcdm mp2an ) BCDZCCEFGZCCVHHZVJAIZBJZGZVKBUACGZUB + ZACUCZVOACVKCGVMVNCVLVKCKVLCLKUDUEUFZKKBHZVLKMNKKBUGOUHUIVKUJVBUKBULZVJVPPV + RVSNKKBUMOACCBUNOQCKRVHCLEFZGVIVJPVQBSTUOFZDZCDZVHVTCWARZWCVHMCSTUPFWAVCSTU + QURZBCWAUSOWDWBWALEFGWCVTGWEUTWALCWBVAVDVECLCVHVFVGQ $. $( @{ @@ -508375,21 +508369,21 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry fdvneggt $p |- ( ph -> ( F ` B ) < ( F ` A ) ) $= ( vy cfv cr wcel cc clt wbr cneg cv cmpt ffvelcdmda renegcld fmpttd cdv wa co ccncf cvv cpr reelprrecn ax-resscn sselid fvexd feqmptd oveq2d wf - a1i cncff syl eqtr3d dvmptneg wss wb ssid cncfss eqid negfcncf cncffvrn - mp2an sylancr mpbird eqeltrd cioo adantr cicc ioossicc fct2relem sselda - sstrd ffvelcdmd lt0neg1d mpbid wceq fveq1d simpr fveq2d negeqd breqtrrd - cc0 fvmptd eqtrd fdvposlt eqidd 3brtr3d ltnegd ) ADHQZCHQZUAUBXBUCZXAUC - ZUAUBACPGPUDZHQZUCZUEZQDXHQXCXDUAABCDEFGXHIJKAPGXGRAXEGSUJZXFAGRXEHLUFZ - UGUHARXHUIUKZPGXERHUIUKZQZUCZUEZGRULUKZAPXFXMRUMGRRTUNSAUOVBXIRTXFUPXJU - QXIXEXLURAXLRPGXFUEZUIUKPGXMUEAHXQRUIAPGRHLUSUTAPGRXLAXLXPSGRXLVAZMGRXL - VCVDZUSVEVFZAXOXPSZGRXOVAZAPGXNRXIXMAGRXEXLXSUFUGUHARTVGZXOGTULUKZSZYAY - BVHUPAXLYDSYEAXPYDXLYCTTVGXPYDVGUPTVIGRTVJVNMUQPGXLXOXOVKZVLVDGTRXOVMVO - VPVQNABUDZCDVRUKZSZUJZWNYGXLQZUCZYGXKQZUAYJYKWNUAUBWNYLUAUBOYJYKYJGRYGX - LAXRYIXSVSAYHGYGAYHCDVTUKZGYHYNVGACDWAVBACDEFGIJKWBWDWCZWEZWFWGYJYMYGXO - QYLYJYGXKXOAXKXOWHYIXTVSWIYJPYGXNYLGXORXOXOWHYJYFVBYJXEYGWHZUJZXMYKYRXE - YGXLYJYQWJWKWLYOYJYKYPUGWOWPWMWQAPCXGXCGXHRAXHWRZAXECWHZUJZXFXBUUAXECHA - YTWJWKWLJAXBAGRCHLJWEZUGWOAPDXGXDGXHRYSAXEDWHZUJZXFXAUUDXEDHAUUCWJWKWLK - AXAAGRDHLKWEZUGWOWSAXAXBUUEUUBWTVP $. + a1i cncff syl eqtr3d dvmptneg wss wb ssid cncfss mp2an negfcncf cncfcdm + eqid sylancr mpbird eqeltrd cioo cc0 adantr cicc fct2relem sstrd sselda + ioossicc ffvelcdmd lt0neg1d mpbid wceq fveq1d simpr fveq2d negeqd eqtrd + fvmptd breqtrrd fdvposlt eqidd 3brtr3d ltnegd ) ADHQZCHQZUAUBXBUCZXAUCZ + UAUBACPGPUDZHQZUCZUEZQDXHQXCXDUAABCDEFGXHIJKAPGXGRAXEGSUJZXFAGRXEHLUFZU + GUHARXHUIUKZPGXERHUIUKZQZUCZUEZGRULUKZAPXFXMRUMGRRTUNSAUOVBXIRTXFUPXJUQ + XIXEXLURAXLRPGXFUEZUIUKPGXMUEAHXQRUIAPGRHLUSUTAPGRXLAXLXPSGRXLVAZMGRXLV + CVDZUSVEVFZAXOXPSZGRXOVAZAPGXNRXIXMAGRXEXLXSUFUGUHARTVGZXOGTULUKZSZYAYB + VHUPAXLYDSYEAXPYDXLYCTTVGXPYDVGUPTVIGRTVJVKMUQPGXLXOXOVNZVLVDGTRXOVMVOV + PVQNABUDZCDVRUKZSZUJZVSYGXLQZUCZYGXKQZUAYJYKVSUAUBVSYLUAUBOYJYKYJGRYGXL + AXRYIXSVTAYHGYGAYHCDWAUKZGYHYNVGACDWEVBACDEFGIJKWBWCWDZWFZWGWHYJYMYGXOQ + YLYJYGXKXOAXKXOWIYIXTVTWJYJPYGXNYLGXORXOXOWIYJYFVBYJXEYGWIZUJZXMYKYRXEY + GXLYJYQWKWLWMYOYJYKYPUGWOWNWPWQAPCXGXCGXHRAXHWRZAXECWIZUJZXFXBUUAXECHAY + TWKWLWMJAXBAGRCHLJWFZUGWOAPDXGXDGXHRYSAXEDWIZUJZXFXAUUDXEDHAUUCWKWLWMKA + XAAGRDHLKWFZUGWOWSAXAXBUUEUUBWTVP $. $} ${ @@ -508423,21 +508417,21 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry fdvnegge $p |- ( ph -> ( F ` B ) <_ ( F ` A ) ) $= ( vy cfv cr wcel cc cle wbr cneg cv cmpt ffvelcdmda renegcld fmpttd cdv wa co ccncf cvv cpr reelprrecn ax-resscn sselid fvexd feqmptd oveq2d wf - a1i cncff syl eqtr3d dvmptneg wss wb ssid cncfss eqid negfcncf cncffvrn - mp2an sylancr mpbird eqeltrd cioo adantr cicc ioossicc fct2relem sselda - sstrd ffvelcdmd le0neg1d mpbid wceq fveq1d simpr fveq2d negeqd breqtrrd - cc0 fvmptd eqtrd fdvposle eqidd 3brtr3d lenegd ) ADHQZCHQZUAUBXBUCZXAUC - ZUAUBACPGPUDZHQZUCZUEZQDXHQXCXDUAABCDEFGXHIJKAPGXGRAXEGSUJZXFAGRXEHLUFZ - UGUHARXHUIUKZPGXERHUIUKZQZUCZUEZGRULUKZAPXFXMRUMGRRTUNSAUOVBXIRTXFUPXJU - QXIXEXLURAXLRPGXFUEZUIUKPGXMUEAHXQRUIAPGRHLUSUTAPGRXLAXLXPSGRXLVAZMGRXL - VCVDZUSVEVFZAXOXPSZGRXOVAZAPGXNRXIXMAGRXEXLXSUFUGUHARTVGZXOGTULUKZSZYAY - BVHUPAXLYDSYEAXPYDXLYCTTVGXPYDVGUPTVIGRTVJVNMUQPGXLXOXOVKZVLVDGTRXOVMVO - VPVQNABUDZCDVRUKZSZUJZWNYGXLQZUCZYGXKQZUAYJYKWNUAUBWNYLUAUBOYJYKYJGRYGX - LAXRYIXSVSAYHGYGAYHCDVTUKZGYHYNVGACDWAVBACDEFGIJKWBWDWCZWEZWFWGYJYMYGXO - QYLYJYGXKXOAXKXOWHYIXTVSWIYJPYGXNYLGXORXOXOWHYJYFVBYJXEYGWHZUJZXMYKYRXE - YGXLYJYQWJWKWLYOYJYKYPUGWOWPWMWQAPCXGXCGXHRAXHWRZAXECWHZUJZXFXBUUAXECHA - YTWJWKWLJAXBAGRCHLJWEZUGWOAPDXGXDGXHRYSAXEDWHZUJZXFXAUUDXEDHAUUCWJWKWLK - AXAAGRDHLKWEZUGWOWSAXAXBUUEUUBWTVP $. + a1i cncff syl eqtr3d dvmptneg wss wb ssid cncfss mp2an negfcncf cncfcdm + eqid sylancr mpbird eqeltrd cioo cc0 adantr cicc fct2relem sstrd sselda + ioossicc ffvelcdmd le0neg1d mpbid wceq fveq1d simpr fveq2d negeqd eqtrd + fvmptd breqtrrd fdvposle eqidd 3brtr3d lenegd ) ADHQZCHQZUAUBXBUCZXAUCZ + UAUBACPGPUDZHQZUCZUEZQDXHQXCXDUAABCDEFGXHIJKAPGXGRAXEGSUJZXFAGRXEHLUFZU + GUHARXHUIUKZPGXERHUIUKZQZUCZUEZGRULUKZAPXFXMRUMGRRTUNSAUOVBXIRTXFUPXJUQ + XIXEXLURAXLRPGXFUEZUIUKPGXMUEAHXQRUIAPGRHLUSUTAPGRXLAXLXPSGRXLVAZMGRXLV + CVDZUSVEVFZAXOXPSZGRXOVAZAPGXNRXIXMAGRXEXLXSUFUGUHARTVGZXOGTULUKZSZYAYB + VHUPAXLYDSYEAXPYDXLYCTTVGXPYDVGUPTVIGRTVJVKMUQPGXLXOXOVNZVLVDGTRXOVMVOV + PVQNABUDZCDVRUKZSZUJZVSYGXLQZUCZYGXKQZUAYJYKVSUAUBVSYLUAUBOYJYKYJGRYGXL + AXRYIXSVTAYHGYGAYHCDWAUKZGYHYNVGACDWEVBACDEFGIJKWBWCWDZWFZWGWHYJYMYGXOQ + YLYJYGXKXOAXKXOWIYIXTVTWJYJPYGXNYLGXORXOXOWIYJYFVBYJXEYGWIZUJZXMYKYRXEY + GXLYJYQWKWLWMYOYJYKYPUGWOWNWPWQAPCXGXCGXHRAXHWRZAXECWIZUJZXFXBUUAXECHAY + TWKWLWMJAXBAGRCHLJWFZUGWOAPDXGXDGXHRYSAXEDWIZUJZXFXAUUDXEDHAUUCWKWLWMKA + XAAGRDHLKWFZUGWOWSAXAXBUUEUUBWTVP $. $} $} @@ -509907,58 +509901,58 @@ strict in the case where the sets B(x) overlap. 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(Contributed by Thierry Arnoux, @@ -524265,90 +524259,90 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by cvmlift2lem9 $p |- ( ph -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) |`t ( U X. V ) ) Cn C ) ) $= ( vm vn cii ctx co cxp cop iitop ccn eqeltrd ctop wcel a1i wss cuni - iiuni elssuni syl sseldd opelxpi syl2anc wa fovrnd wceq df-ov sylib - cfv cima csn cres snidg ovres crest ccnv eqid cconn cvv snex adantr - wf txrest syl22anc cpw iitopon restsn2 sylancr c0 indisconn eqeltri - cpr pwsn eqeltrdi txconn xpss2 snssd restuni sseqtrd syl3anc crn wb - xpss1 df-ima imass2 3syl eqtr3id mpbird sstrd cnrest2 mpbid eleqtrd - eqsstrrid conncn feq2d cc0 c1 cicc txunii cvmlift2lem5 cvmlift2lem7 - ccom txtopi sseqtrrdi fvco3 fveq1d eqtr3d 3eqtr4g cvmsiota syl13anc - ccvm fveq2i eleq1i anbi2i xpss12 wral ad2antrl simprr ctopon sselda - cv adantrr cvmlift2lem6 syldan cnrest resabs1d ovex restabs 3eltr3d - xpex oveq1d cvmtop1 toptopon simprl imaco cnvco cnveqd imaeq1d wfun - sseqtrrd cdm ffund funimass3 cnvimass cvmcn cnf fdm sseqtrid cvmsss - fdmd simpld cvmsuni cvmsrcl cnima restopn2 mpbir2and cvmscld eqtr4d - ccld mpdan simprd eqeltrrd ralrimivva funimassov cvmlift2lem9a ) AR - 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( 0 [,] 1 ) ) $. @@ -524637,75 +524631,75 @@ property of an acyclic graph (see also ~ acycgr0v ). 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B ) ) /\ ( X e. A /\ Y e. B ) ) -> ta ) $= $( The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.) $) noprc $p |- -. No e. _V $= - ( csur cvv wcel con0 onprc cbday wfo bdayfo fornex mpi mto ) ABCZDBCZELADFG - MHADBFIJK $. + ( csur cvv wcel con0 onprc cbday wfo bdayfo focdmex mpi mto ) ABCZDBCZELADF + GMHADBFIJK $. $( @@ -549962,9 +549956,9 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= Ipsen, 25-Aug-2021.) $) knoppcn2 $p |- ( ph -> W e. 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RR /\ y e. RR ) /\ ( ( x ^ 2 ) + ( y ^ 2 ) ) <_ ( R ^ 2 ) ) } $. @@ -573702,10 +573696,10 @@ curry M LIndF ( R freeLMod I ) ) ) $= 12-Sep-2015.) $) cncfres $p |- G e. ( J Cn K ) $= ( ccncf co wcel cc ccn wf fmpti wss wb cmpt cres wceq resmpt ax-mp eqtr4i - eqeltrri rescncf mp2 eqeltri cncffvrn mp2an mpbir cabs cmin ccom cxp eqid - cncfmet eleqtri ) FBCQRZGHUARZFVFSZBCFUBZABCDFLMUCCTUDZFBTQRZSVHVIUEJFATD - UFZBUGZVKFABDUFZVMLBTUDZVMVNUHIATBDUIUJUKVOVLTTQRZSVMVKSIEVLVPKNULTTBVLUM - UNUOBTCFUPUQURVOVJVFVGUHIJBCUSUTVAZBBVBUGZVQCCVBUGZGHVRVCVSVCOPVDUQVE $. + eqeltrri rescncf mp2 eqeltri cncfcdm mp2an cabs cmin ccom cncfmet eleqtri + mpbir cxp eqid ) FBCQRZGHUARZFVFSZBCFUBZABCDFLMUCCTUDZFBTQRZSVHVIUEJFATDU + FZBUGZVKFABDUFZVMLBTUDZVMVNUHIATBDUIUJUKVOVLTTQRZSVMVKSIEVLVPKNULTTBVLUMU + NUOBTCFUPUQVCVOVJVFVGUHIJBCURUSUTZBBVDUGZVQCCVDUGZGHVRVEVSVEOPVAUQVB $. $} @@ -574068,38 +574062,38 @@ curry M LIndF ( R freeLMod I ) ) ) $= readdcld mettri2 syl13anc clt simplrr eleqtrd metxmet rpxr ad2antlr elbl2 cxr wb syl22anc mpbid lt2addd recnd 2timesd breqtrrd lelttrd ltled elicc2 w3a mpbir3and ralrimivva ffnov oveq2 feq3d rspcev syl2anc expr rexlimdvva - mpd pm2.61dane jca c1 simpllr met0 simplr fovrnd simp3d eqbrtrrd ge0p1rpd - simpr crab fovrn 3expa adantlll simp1d peano2re ltp1d rabid2 sylibr rexrd - blval eqtr4d rspceeqv isbnd r19.29an impbii ) BCUAUBGZBCUCUBGZCCUDZHAUEZU - FIZBUQZAJUGZKUUKUULUUQBCUHZUUKUUQCLUUKCLMZKZJLUIUUPAJNUUQHJUJUKUUTUUPAJUU - TUUNJGZKZBUUMULZBUMZUUOUNUUPUUKUVCUUSUVAUUKUULUVCUURUULUUMJBBCUOZUPZOURUV - BUVDLUUOUVBBUSZLMZUVDLMUUTUVHUVAUUKUUSUVGUUMLUUKUUMJBUUKUULUUMJBUQUURUVEO - UTUUSUUMCLUDLCLCVACVBVCVDPBVEVFUUOVIVGUUMUUOBVHVJVKUUPAJVLQUUKCLUIZKZCDUE - ZEUEZBVMUBZIZMZERUGZDCUGZUUQUVJBCVNUBGZUVQDBCEVOVPUUKUVQUUQVQZUVIUUKUULUV - SUURUULUVOUUQDECRUULUVKCGZUVLRGZKZUVOUUQUULUWBUVOKZKZVRUVLVSIZJGZUUMHUWEU - FIZBUQZUUQUWDVRJGUVLJGZUWFWAUWDUVLUULUVTUWAUVOVTWBZVRUVLWCQZUWDUVCUUNFUEZ - BIZUWGGZFCNACNUWHUULUVCUWCUVFPUWDUWNAFCCUWDUUNCGZUWLCGZKZKZUWNUWMJGZHUWMS - TZUWMUWESTZUWRUULUWOUWPUWSUULUWCUWQWDZUWDUWOUWPWEZUWDUWOUWPWFZUUNUWLBCWGW - HZUWRUULUWOUWPUWTUXBUXCUXDUUNUWLBCWIWHUWRUWMUWEUXEUWDUWFUWQUWKPZUWRUWMUVK - UUNBIZUVKUWLBIZWJIZUWEUXEUWRUXGUXHUWRUULUVTUWOUXGJGUXBUWDUVTUWQUULUVTUWAU - VOWKPZUXCUVKUUNBCWGWHZUWRUULUVTUWPUXHJGZUXBUXJUXDUVKUWLBCWGWHZWLUXFUWRUUL - 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(Contributed by Mario Carneiro, 22-Sep-2015.) $) @@ -574320,50 +574314,50 @@ counterexample is the discrete extended metric (assigning distinct syl2an sselii ne0i mp1i wor w3a ltso fisupcl mpan syl3anc sseldd simprl cle wbr simprr metcl cxr eleqtrd adantlr prdsbascl r19.21bi ad2ant2r wb 0re elicc2 sylancr suprub syl31anc sylanbrc metf ad2antrr metge0 oveqdr - adantrr 3syl prdsdsval3 eqtrd ffvelcdm ad2ant2lr fovrnd simp3d fimaxre2 - mpbid ssun1 ad2antlr fnfvelrn letrd expr ralimdva impr ovex rgenw breq1 - sselid ralrnmptw ax-mp sylibr a1i elsni breq1d syl5ibrcom ralunb fmpttd - ralrimiv frnd ressxr sstrdi rexrd supxrleub mpbird mpbir3and ralrimivva - eqbrtrd an32s ffnov rspcev exlimddv ) ADCUGUHZUIZCCUJZUKUAULZUMUNZDUOZU - AUSUPZDCUQUHUIAFBHBULZEUHZUTZURUNZVAUHZUWSVBUHZUGUHDUWIABUXAUWTUWQFGHIJ - UWSVCUWSVDZUXAVDZNOUWTVDZQRAUWPHUIZWDZUWPEVEUXFGIUQUHUIZGIUGUHUIZTGIVFV - GZVHADKVAUHUWTPAKUWSVAAKFEURUNUWSLAEUWRFURAEHVIEUWRVJSBHEVKVMVLVNZVOVNZ - ACUXAUGACKVBUHUXAMAKUWSVBUXJVOVNZVOVPZAHUSUBULZUOZIIUJZUKUWPUXNUHZUMUNZ - 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(New usage is discouraged.) (Proof modification is discouraged.) $) clmgmOLD $p |- ( ( G e. Magma /\ A e. X /\ B e. X ) -> ( A G B ) e. X ) $= - ( cmagm wcel co wi cxp wf ismgmOLD fovrn 3exp syl6bi pm2.43i 3imp ) CFGZA - DGZBDGZABCHDGZRSTUAIIZRRDDJDCKZUBFCDELUCSTUAABDDDCMNOPQ $. + ( cmagm wcel co wi cxp wf ismgmOLD fovcdm 3exp syl6bi pm2.43i 3imp ) CFGZ + ADGZBDGZABCHDGZRSTUAIIZRRDDJDCKZUBFCDELUCSTUAABDDDCMNOPQ $. $} ${ @@ -576955,8 +576949,8 @@ the next (since the empty set has a finite subcover, the $( Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) $) rngocl $p |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X ) $= - ( crngo wcel cxp wf co rngosm fovrn syl3an1 ) CJKFFLFEMAFKBFKABENFKCDEFGH - IOABFFFEPQ $. + ( crngo wcel cxp wf co rngosm fovcdm syl3an1 ) CJKFFLFEMAFKBFKABENFKCDEFG + HIOABFFFEPQ $. $( The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) 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(Contributed by Jeff Madsen, 10-Jun-2010.) $) @@ -578315,34 +578309,34 @@ the next (since the empty set has a finite subcover, the ( wcel co wceq wa vw cop crngo ccm2 ccring cv w3a id 3com13 eleq1 3anbi1d wi anbi2d oveq2 oveq12d eqeq12d imbi12d 3anbi3d oveq1 oveq1d cablo adantr crn simpr3 eleqtrd simpr2 eqid ablocom syl3anc simpr1 cgr ablogrpo grpocl - syl eleqtrrd jca chvarvv vtocl syldan 3adantr3 3adantr2 cxp fovrnd 3eqtrd - ovex wf 3eqtr2d sylan2 imbi2d an12s ex vtoclga mpcom eqtrd isrngod eleq2d - wral anbi12d biimpar ralrimivva cvv wb rnexg eqeltrd xpexd iscom2 syl2anc - fexd mpbird iscrngo sylanbrc ) AFGUBZUCQXLUDQZXLUEQABCDEFGHIJKLMBUFZHQZCU - FZHQZDUFZHQZUGZAXSXQXOUGZXNXPFRZXRGRZXNXRGRZXPXRGRZFRZSZXSXQXOYAYAUHUIAUA - UFZHQZXQXOUGZTZYBYHGRZXNYHGRZXPYHGRZFRZSZULZAYATZYGULUADYHXRSZYKYRYPYGYSY - JYAAYSYIXSXQXOYHXRHUJUKUMYSYLYCYOYFYHXRYBGUNYSYMYDYNYEFYHXRXNGUNYHXRXPGUN - UOUPUQAYIXQXSUGZTZXRXPFRZYHGRZXRYHGRZYNFRZSZULZYQDBXRXNSZUUAYKUUFYPUUHYTY - JAUUHXSXOYIXQXRXNHUJURUMUUHUUCYLUUEYOUUHUUBYBYHGXRXNXPFUSUTUUHUUDYMYNFXRX - NYHGUSUTUPUQAXTTZUUBXNGRZXRXNGRZXPXNGRZFRZSZULUUGBUAXNYHSZUUIUUAUUNUUFUUO - XTYTAUUOXOYIXQXSXNYHHUJUKUMUUOUUJUUCUUMUUEXNYHUUBGUNUUOUUKUUDUULYNFXNYHXR - GUNXNYHXPGUNUOUPUQUUIUUJXPXRFRZXNGRZXNUUPGRZUUMUUIUUBUUPXNGUUIFVAQZXRFVCZ - QZXPUUTQZUUBUUPSAUUSXTIVBZUUIXRHUUTAXOXQXSVDZAHUUTSXTJVBZVEZUUIXPHUUTAXOX - QXSVFZUVEVEZXRXPFUUTUUTVGZVHVIUTAXTXOUUPHQZTZUURUUQSZUUIXOUVJAXOXQXSVJZUU - IUUPUUTHUUIFVKQZUVBUVAUUPUUTQUUIUUSUVNUVCFVLVNUVHUVFXPXRFUUTUVIVMVIUVEVOV - PAXOYITZTZYMYHXNGRZSZULZAUVKTZUVLULUAUUPXPXRFWEYHUUPSZUVPUVTUVRUVLUWAUVOU - VKAUWAYIUVJXOYHUUPHUJUMUMUWAYMUURUVQUUQYHUUPXNGUNYHUUPXNGUSUPUQAXOXQTZTZX - NXPGRZUULSZULZUVSCUAXPYHSZUWCUVPUWEUVRUWGUWBUVOAUWGXQYIXOXPYHHUJUMUMUWGUW - DYMUULUVQXPYHXNGUNXPYHXNGUSUPUQPVQVRVSUUIUURUWDYDFRUULUUKFRZUUMMUUIUWDUUL - YDUUKFAXOXQUWEXSPVTAXOXSYDUUKSZXQUWFAXOXSTZTZUWIULCDXPXRSZUWCUWKUWEUWIUWL - UWBUWJAUWLXQXSXOXPXRHUJUMUMUWLUWDYDUULUUKXPXRXNGUNXPXRXNGUSUPUQPVQWAUOUUI - UUSUULUUTQUUKUUTQUWHUUMSUVCUUIUULHUUTUUIXPXNHHHGAHHWBZHGWFXTKVBZUVGUVMWCU - VEVEUUIUUKHUUTUUIXRXNHHHGUWNUVDUVMWCUVEVEUULUUKFUUTUVIVHVIWDWGVQVQVQWHNAX - QTZEXPGRZXPEGRZXPEHQZUWOUWPUWQSZAUWRXQNVBUWOUWEULUWOUWSULBEHXNESZUWEUWSUW - OUWTUWDUWPUULUWQXNEXPGUSXNEXPGUNUPWIXOUWOUWEAXOXQUWEPWJWKWLWMOWNOWOAXMUWE - CUUTWQBUUTWQZAUWEBCUUTUUTAXNUUTQZUVBTZUWBUWEAUWBUXCAXOUXBXQUVBAHUUTXNJWPA - HUUTXPJWPWRWSPVSWTAUUSGXAQXMUXAXBIAUWMHXAGKAHHXAXAAHUUTXAJAUUSUUTXAQIFVAX - CVNXDZUXDXEXHVAXAFGBCXFXGXIXLXJXK $. + syl eleqtrrd jca ovex chvarvv syldan 3adantr3 3adantr2 cxp fovcdmd 3eqtrd + vtocl wf 3eqtr2d sylan2 imbi2d an12s ex vtoclga mpcom isrngod wral eleq2d + eqtrd anbi12d biimpar ralrimivva cvv wb rnexg eqeltrd fexd iscom2 syl2anc + xpexd mpbird iscrngo sylanbrc ) AFGUBZUCQXLUDQZXLUEQABCDEFGHIJKLMBUFZHQZC + UFZHQZDUFZHQZUGZAXSXQXOUGZXNXPFRZXRGRZXNXRGRZXPXRGRZFRZSZXSXQXOYAYAUHUIAU + AUFZHQZXQXOUGZTZYBYHGRZXNYHGRZXPYHGRZFRZSZULZAYATZYGULUADYHXRSZYKYRYPYGYS + YJYAAYSYIXSXQXOYHXRHUJUKUMYSYLYCYOYFYHXRYBGUNYSYMYDYNYEFYHXRXNGUNYHXRXPGU + NUOUPUQAYIXQXSUGZTZXRXPFRZYHGRZXRYHGRZYNFRZSZULZYQDBXRXNSZUUAYKUUFYPUUHYT + YJAUUHXSXOYIXQXRXNHUJURUMUUHUUCYLUUEYOUUHUUBYBYHGXRXNXPFUSUTUUHUUDYMYNFXR + XNYHGUSUTUPUQAXTTZUUBXNGRZXRXNGRZXPXNGRZFRZSZULUUGBUAXNYHSZUUIUUAUUNUUFUU + OXTYTAUUOXOYIXQXSXNYHHUJUKUMUUOUUJUUCUUMUUEXNYHUUBGUNUUOUUKUUDUULYNFXNYHX + RGUNXNYHXPGUNUOUPUQUUIUUJXPXRFRZXNGRZXNUUPGRZUUMUUIUUBUUPXNGUUIFVAQZXRFVC + ZQZXPUUTQZUUBUUPSAUUSXTIVBZUUIXRHUUTAXOXQXSVDZAHUUTSXTJVBZVEZUUIXPHUUTAXO + XQXSVFZUVEVEZXRXPFUUTUUTVGZVHVIUTAXTXOUUPHQZTZUURUUQSZUUIXOUVJAXOXQXSVJZU + UIUUPUUTHUUIFVKQZUVBUVAUUPUUTQUUIUUSUVNUVCFVLVNUVHUVFXPXRFUUTUVIVMVIUVEVO + VPAXOYITZTZYMYHXNGRZSZULZAUVKTZUVLULUAUUPXPXRFVQYHUUPSZUVPUVTUVRUVLUWAUVO + UVKAUWAYIUVJXOYHUUPHUJUMUMUWAYMUURUVQUUQYHUUPXNGUNYHUUPXNGUSUPUQAXOXQTZTZ + XNXPGRZUULSZULZUVSCUAXPYHSZUWCUVPUWEUVRUWGUWBUVOAUWGXQYIXOXPYHHUJUMUMUWGU + WDYMUULUVQXPYHXNGUNXPYHXNGUSUPUQPVRWEVSUUIUURUWDYDFRUULUUKFRZUUMMUUIUWDUU + LYDUUKFAXOXQUWEXSPVTAXOXSYDUUKSZXQUWFAXOXSTZTZUWIULCDXPXRSZUWCUWKUWEUWIUW + LUWBUWJAUWLXQXSXOXPXRHUJUMUMUWLUWDYDUULUUKXPXRXNGUNXPXRXNGUSUPUQPVRWAUOUU + IUUSUULUUTQUUKUUTQUWHUUMSUVCUUIUULHUUTUUIXPXNHHHGAHHWBZHGWFXTKVBZUVGUVMWC + UVEVEUUIUUKHUUTUUIXRXNHHHGUWNUVDUVMWCUVEVEUULUUKFUUTUVIVHVIWDWGVRVRVRWHNA + XQTZEXPGRZXPEGRZXPEHQZUWOUWPUWQSZAUWRXQNVBUWOUWEULUWOUWSULBEHXNESZUWEUWSU + WOUWTUWDUWPUULUWQXNEXPGUSXNEXPGUNUPWIXOUWOUWEAXOXQUWEPWJWKWLWMOWQOWNAXMUW + ECUUTWOBUUTWOZAUWEBCUUTUUTAXNUUTQZUVBTZUWBUWEAUWBUXCAXOUXBXQUVBAHUUTXNJWP + AHUUTXPJWPWRWSPVSWTAUUSGXAQXMUXAXBIAUWMHXAGKAHHXAXAAHUUTXAJAUUSUUTXAQIFVA + XCVNXDZUXDXHXEVAXAFGBCXFXGXIXLXJXK $. $} $( A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) @@ -649727,8 +649721,8 @@ fixed reference functional determined by this vector (corresponding to intlewftc $p |- ( ph -> ( F ` B ) <_ ( G ` B ) ) $= ( vt cfv cmin co caddc cle wbr cicc cr ccncf wcel wf cncff syl leidd 3jca w3a wb elicc2 syl2anc mpbird ffvelcdmd resubcld cioo citg cibl cmpt mpbid - eleq1d feq1d fvmptelrn itgle cv itgmpt wceq fveq1d adantr eqcomd itgeq2dv - wa cdv wss ax-resscn a1i fss ssidd cncffvrn eqeltrrd ftc2 breq12d le2addd + eleq1d feq1d fvmptelcdm itgle cv itgmpt fveq1d adantr eqcomd itgeq2dv cdv + wa wceq wss ax-resscn a1i fss ssidd cncfcdm eqeltrrd ftc2 breq12d le2addd cc eqtrd sselid npcand ) ADIUGZCIUGZUHUIZXBUJUIZDJUGZCJUGZUHUIZXFUJUIZUKU LXAXEUKULAXCXBXGXFAXAXBACDUMUIZUNDIAIXIUNUOUIZUPZXIUNIUQZNXIUNIURUSZADXIU PZDUNUPZCDUKULZDDUKULZVBZAXOXPXQLMADLUTVAACUNUPZXOXNXRVCKLCDDVDVEVFZVGZAX @@ -649738,13 +649732,13 @@ fixed reference functional determined by this vector (corresponding to 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(Contributed by - 1 ) x. ( ( A rmY M ) x. ( A rmY N ) ) ) ) /\ ( A rmY ( M + N ) ) = ( ( ( A rmY M ) x. ( A rmX N ) ) + ( ( A rmX M ) x. ( A rmY N ) ) ) ) ) $= - ( wcel cz caddc co crmx cexp c1 crmy cmul wceq syl2anc cq cn0 fovrnd sselid - zssq qmulcl c2 cuz cfv w3a csqrt wa simp1 zaddcl 3adant1 rmxyval cc cc0 wne - cmin eluzelz 3ad2ant1 zcnd zq qsqcl 1z sselii a1i qsubcl qcn sqrtcld addcld - 3syl syl rmbaserp rpne0d simp2 simp3 expaddz syl22anc wf frmx nn0cnd mulcld - cxp frmy muladdd oveq12d eqtr3d mul4d msqsqrtd mulcomd oveq2d mul12d adddid - eqtrd addcomd oveq1d 3eqtr2d cdif wb rmspecsqrtnq nn0ssq qirropth syl122anc - qaddcl mpbid ) AUAUBUCZDZBEDZCEDZUDZABCFGZHGZAUAIGZJUNGZUEUCZAXGKGZLGFGZABH - GZACHGZLGZXJABKGZACKGZLGZLGZFGZXKXQXOLGZXNXRLGZFGZLGZFGZMZXHYAMXLYDMUFZXFXM - AXKFGZXGIGZYFXFXCXGEDZXMYJMXCXDXEUGZXDXEYKXCBCUHUIZAXGUJNXFYJYIBIGZYICIGZLG - ZXPXKXRLGZXKXQLGZLGZFGZXNYQLGZXOYRLGZFGZFGZYFXFYIUKDYIULUMZXDXEYJYPMXFAXKXF - AXCXDAEDZXEUAAUOUPZUQXFXJXFXJODZXJUKDXFXIODZJODZUUHXFUUFAODUUIUUGAURAUSVGUU - JXFEOJSUTVAVBXIJVCNZXJVDVHZVEZVFXCXDUUEXEXCYIAVIVJUPXCXDXEVKZXCXDXEVLZYIBCV - MVNXFXNYRFGZXOYQFGZLGUUDYPXFXNYRXOYQXFXNXFABPXBEHXBEVSZPHVOXFVPVBZYLUUNQZVQ - ZXFXKXQUUMXFXQXFABEXBEKUUREKVOXFVTVBZYLUUNQZUQZVRXFXOXFACPXBEHUUSYLUUOQZVQZ - XFXKXRUUMXFXRXFACEXBEKUVBYLUUOQZUQZVRWAXFUUPYNUUQYOLXFXCXDUUPYNMYLUUNABUJNX - FXCXEUUQYOMYLUUOACUJNWBWCXFYTYAUUCYEFXFYSXTXPFXFYSXKXKLGZXRXQLGZLGXTXFXKXRX - KXQUUMUVHUUMUVDWDXFUVIXJUVJXSLXFXJUULWEXFXRXQUVHUVDWFWBWJWGXFUUCXKYCLGZXKXO - XQLGZLGZFGXKYCUVLFGZLGYEXFUUAUVKUUBUVMFXFXNXKXRUVAUUMUVHWHXFXOXKXQUVFUUMUVD - WHWBXFXKYCUVLUUMXFXNXRUVAUVHVRZXFXOXQUVFUVDVRZWIXFUVNYDXKLXFUVNUVLYCFGYDXFY - CUVLUVOUVPWKXFUVLYBYCFXFXOXQUVFUVDWFWLWJWGWMWBWMWJXFXKUKOWNDZXHODXLODYAODZY - DODZYGYHWOXCXDUVQXEAWPUPXFPOXHWQXFAXGPXBEHUUSYLYMQRXFEOXLSXFAXGEXBEKUVBYLYM - QRXFXPODZXTODZUVRXFXNODZXOODZUVTXFPOXNWQUUTRZXFPOXOWQUVERZXNXOTNXFUUHXSODZU - WAUUKXFXQODZXRODZUWFXFEOXQSUVCRZXFEOXRSUVGRZXQXRTNXJXSTNXPXTWTNXFYBODZYCODZ - UVSXFUWGUWCUWKUWIUWEXQXOTNXFUWBUWHUWLUWDUWJXNXRTNYBYCWTNXKXHXLYAYDWRWSXA $. + ( wcel cz caddc co crmx cexp crmy cmul wceq syl2anc zssq cn0 fovcdmd sselid + c1 cq qmulcl c2 cuz cfv w3a cmin csqrt simp1 zaddcl 3adant1 rmxyval cc0 wne + wa cc eluzelz 3ad2ant1 zcnd zq qsqcl 3syl sselii a1i qsubcl qcn syl sqrtcld + 1z addcld rmbaserp rpne0d simp2 simp3 expaddz syl22anc cxp frmx nn0cnd frmy + wf mulcld muladdd oveq12d eqtr3d mul4d msqsqrtd mulcomd eqtrd oveq2d mul12d + adddid addcomd oveq1d 3eqtr2d rmspecsqrtnq nn0ssq qaddcl qirropth syl122anc + cdif wb mpbid ) AUAUBUCZDZBEDZCEDZUDZABCFGZHGZAUAIGZRUEGZUFUCZAXGJGZKGFGZAB + HGZACHGZKGZXJABJGZACJGZKGZKGZFGZXKXQXOKGZXNXRKGZFGZKGZFGZLZXHYALXLYDLUMZXFX + MAXKFGZXGIGZYFXFXCXGEDZXMYJLXCXDXEUGZXDXEYKXCBCUHUIZAXGUJMXFYJYIBIGZYICIGZK + GZXPXKXRKGZXKXQKGZKGZFGZXNYQKGZXOYRKGZFGZFGZYFXFYIUNDYIUKULZXDXEYJYPLXFAXKX + FAXCXDAEDZXEUAAUOUPZUQXFXJXFXJSDZXJUNDXFXISDZRSDZUUHXFUUFASDUUIUUGAURAUSUTU + UJXFESRNVGVAVBXIRVCMZXJVDVEZVFZVHXCXDUUEXEXCYIAVIVJUPXCXDXEVKZXCXDXEVLZYIBC + VMVNXFXNYRFGZXOYQFGZKGUUDYPXFXNYRXOYQXFXNXFABOXBEHXBEVOZOHVSXFVPVBZYLUUNPZV + QZXFXKXQUUMXFXQXFABEXBEJUUREJVSXFVRVBZYLUUNPZUQZVTXFXOXFACOXBEHUUSYLUUOPZVQ + ZXFXKXRUUMXFXRXFACEXBEJUVBYLUUOPZUQZVTWAXFUUPYNUUQYOKXFXCXDUUPYNLYLUUNABUJM + XFXCXEUUQYOLYLUUOACUJMWBWCXFYTYAUUCYEFXFYSXTXPFXFYSXKXKKGZXRXQKGZKGXTXFXKXR + XKXQUUMUVHUUMUVDWDXFUVIXJUVJXSKXFXJUULWEXFXRXQUVHUVDWFWBWGWHXFUUCXKYCKGZXKX + OXQKGZKGZFGXKYCUVLFGZKGYEXFUUAUVKUUBUVMFXFXNXKXRUVAUUMUVHWIXFXOXKXQUVFUUMUV + DWIWBXFXKYCUVLUUMXFXNXRUVAUVHVTZXFXOXQUVFUVDVTZWJXFUVNYDXKKXFUVNUVLYCFGYDXF + YCUVLUVOUVPWKXFUVLYBYCFXFXOXQUVFUVDWFWLWGWHWMWBWMWGXFXKUNSWSDZXHSDXLSDYASDZ + YDSDZYGYHWTXCXDUVQXEAWNUPXFOSXHWOXFAXGOXBEHUUSYLYMPQXFESXLNXFAXGEXBEJUVBYLY + MPQXFXPSDZXTSDZUVRXFXNSDZXOSDZUVTXFOSXNWOUUTQZXFOSXOWOUVEQZXNXOTMXFUUHXSSDZ + UWAUUKXFXQSDZXRSDZUWFXFESXQNUVCQZXFESXRNUVGQZXQXRTMXJXSTMXPXTWPMXFYBSDZYCSD + ZUVSXFUWGUWCUWKUWIUWEXQXOTMXFUWBUWHUWLUWDUWJXNXRTMYBYCWPMXKXHXLYAYDWQWRXA + $. $( Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.) $) @@ -669456,23 +669451,23 @@ element in each topology (which need not be in the closed set - if it ( wcel c2o ccnv c1o cima wf1o cfn cin wss syl c0 wceq cmap co cv csn cmpt cres cpw wf1 eqid pw2f1o2 f1of1 cfsupp crab ssrab2 eqsstri f1ores sylancl wbr wb wa csupp wfun cvv w3a elmapfun 0ex a1i 3jca adantl funisfsupp cdif - id wf anim2i elmapi frnsuppeq sylc csuc df-2o df-suc equncomi eqtri df1o2 - cun eqcomi difeq12i difun2 incom word 1on onordi orddisj ax-mp disj3 mpbi - eqtr4i imaeq2i eqtrdi eleq1d cnvimass fssdm 3bitrd elfpw bitr4di rabbidva - biantrurd cnveq imaeq1d cbvmptv mptpreima 3eqtr4g imaeq2d wfo f1ofo inss1 - foimacnv eqtrd f1oeq3 resmpt f1oeq1 mp1i bitrd mpbid ) CFIZDAJCUAUBZAUCZK - ZLUDZMZUEZDMZYJDUFZNZDCUGZOPZENZYDYEYNYJUHZDYEQZYMYDYEYNYJNZYQACYJFYJUIUJ - ZYEYNYJUKRDBUCZSULURZBYEUMZYEGUUBBYEUNUOZYEYNDYJUPUQYDYMDYOYLNZYPYDYKYOTY - MUUEUSYDYKYJYJKYOMZMZYOYDDUUFYJYDUUCUUAKZYHMZYOIZBYEUMDUUFYDUUBUUJBYEYDUU - AYEIZUTZUUBUUICQZUUIOIZUTZUUJUULUUBUUASVAUBZOIZUUNUUOUULUUAVBZUUKSVCIZVDZ - UUBUUQUSUUKUUTYDUUKUURUUKUUSUUAJCVEUUKVLUUSUUKVFVGZVHVIUUAYEVCSVJRUULUUPU - UIOUULUUPUUHJSUDZVKZMZUUIUULYDUUSUTCJUUAVMZUUPUVDTUUKUUSYDUVAVNUUKUVEYDUU - AJCVOVIZJUUACFVCSVPVQUVCYHUUHUVCYHLWDZLVKZYHJUVGUVBLJLVRZUVGVSUVILYHLVTWA - WBLUVBWCWEWFUVHYHLVKZYHYHLWGYHLPZSTYHUVJTUVKLYHPZSYHLWHLWIUVLSTLWJWKLWLWM - WBYHLWNWOWPWBWQWRWSUULUUMUUNUULCJUUIUUAUUAYHWTUVFXAXFXBUUICXCXDXEGBYEUUIY - OYJABYEYIUUIYFUUATYGUUHYHYFUUAXGXHXIXJXKXLYDYEYNYJXMZYOYNQUUGYOTYDYSUVMYT - YEYNYJXNRYNOXOYEYNYOYJXPUQXQYKYODYLXRRYLETUUEYPUSYDYLADYIUEZEYRYLUVNTUUDA - YEDYIXSWMHWPDYOYLEXTYAYBYC $. + id anim2i elmapi fsuppeq sylc cun csuc df-2o df-suc equncomi eqtri eqcomi + wf df1o2 difeq12i difun2 incom word 1on onordi orddisj ax-mp disj3 eqtr4i + mpbi imaeq2i eqtrdi eleq1d cnvimass fssdm biantrurd 3bitrd elfpw rabbidva + bitr4di cnveq imaeq1d cbvmptv mptpreima 3eqtr4g imaeq2d f1ofo inss1 eqtrd + wfo foimacnv f1oeq3 resmpt f1oeq1 mp1i bitrd mpbid ) CFIZDAJCUAUBZAUCZKZL + UDZMZUEZDMZYJDUFZNZDCUGZOPZENZYDYEYNYJUHZDYEQZYMYDYEYNYJNZYQACYJFYJUIUJZY + EYNYJUKRDBUCZSULURZBYEUMZYEGUUBBYEUNUOZYEYNDYJUPUQYDYMDYOYLNZYPYDYKYOTYMU + UEUSYDYKYJYJKYOMZMZYOYDDUUFYJYDUUCUUAKZYHMZYOIZBYEUMDUUFYDUUBUUJBYEYDUUAY + EIZUTZUUBUUICQZUUIOIZUTZUUJUULUUBUUASVAUBZOIZUUNUUOUULUUAVBZUUKSVCIZVDZUU + BUUQUSUUKUUTYDUUKUURUUKUUSUUAJCVEUUKVLUUSUUKVFVGZVHVIUUAYEVCSVJRUULUUPUUI + OUULUUPUUHJSUDZVKZMZUUIUULYDUUSUTCJUUAWDZUUPUVDTUUKUUSYDUVAVMUUKUVEYDUUAJ + CVNVIZJUUACFVCSVOVPUVCYHUUHUVCYHLVQZLVKZYHJUVGUVBLJLVRZUVGVSUVILYHLVTWAWB + LUVBWEWCWFUVHYHLVKZYHYHLWGYHLPZSTYHUVJTUVKLYHPZSYHLWHLWIUVLSTLWJWKLWLWMWB + YHLWNWPWOWBWQWRWSUULUUMUUNUULCJUUIUUAUUAYHWTUVFXAXBXCUUICXDXFXEGBYEUUIYOY + JABYEYIUUIYFUUATYGUUHYHYFUUAXGXHXIXJXKXLYDYEYNYJXPZYOYNQUUGYOTYDYSUVMYTYE + YNYJXMRYNOXNYEYNYOYJXQUQXOYKYODYLXRRYLETUUEYPUSYDYLADYIUEZEYRYLUVNTUUDAYE + DYIXSWMHWODYOYLEXTYAYBYC $. $} ${ @@ -699449,11 +699444,11 @@ not even needed (it can be any class). (Contributed by Glauco (Contributed by Glauco Siliprandi, 17-Aug-2020.) $) founiiun $p |- ( F : A -onto-> B -> U. B = U_ x e. A ( F ` x ) ) $= ( vy wfo cuni cv ciun cfv uniiun wss wrex wral wcel wa wceq syl ralrimiva - iunss2 foelrni eqimss2 reximi ffvelcdmda ssidd sseq2 rspcev syl2anc eqssd - fof eqtrid ) BCDFZCGECEHZIZABAHZDJZIZECKULUNUQULUMUPLZABMZECNUNUQLULUSECU - LUMCOPUPUMQZABMUSABCDUMUAUTURABUMUPUBUCRSEACBUMUPTRULUPUMLZECMZABNUQUNLUL - VBABULUOBOPZUPCOUPUPLZVBULBCUODBCDUJUDVCUPUEVAVDEUPCUMUPUPUFUGUHSAEBCUPUM - TRUIUK $. + iunss2 foelcdmi eqimss2 reximi fof ffvelcdmda ssidd rspcev syl2anc eqtrid + sseq2 eqssd ) BCDFZCGECEHZIZABAHZDJZIZECKULUNUQULUMUPLZABMZECNUNUQLULUSEC + ULUMCOPUPUMQZABMUSABCDUMUAUTURABUMUPUBUCRSEACBUMUPTRULUPUMLZECMZABNUQUNLU + LVBABULUOBOPZUPCOUPUPLZVBULBCUODBCDUDUEVCUPUFVAVDEUPCUMUPUPUJUGUHSAEBCUPU + MTRUKUI $. $} $( Distribution law for range of a restriction over a union. (Contributed by @@ -699700,19 +699695,19 @@ not even needed (it can be any class). (Contributed by Glauco founiiun0 $p |- ( F : A -onto-> ( B u. { (/) } ) -> U. B = U_ x e. A ( F ` x ) ) $= ( vy c0 cun wfo cuni cv ciun wss wrex wral wcel wceq syl ralrimiva adantl - wa csn cfv uniiun elun1 foelrni sylan2 eqimss2 reximi iunss2 simpl wb 0un - uneq1 eqtrdi foeq3 mpbid founiiun unisn0 eqtr3di 0ss eqsstrdi ssid rspcev - wn sseq2 mpan2 fof ffvelcdmda elunnel1 sylan elsni adantllr biimpi adantr - wex neq0 wi id anim1ci eximdv mpd df-rex sylibr ad4ant24 syldan pm2.61dan - ex eqssd eqtrid ) BCFUAZGZDHZCIECEJZKZABAJZDUBZKZECUCWLWNWQWLWMWPLZABMZEC - NWNWQLWLWSECWLWMCOZTWPWMPZABMZWSWTWLWMWKOXBWMCWJUDABWKDWMUEUFXAWRABWMWPUG - UHQREACBWMWPUIQWLCFPZWQWNLZWLXCTZBWJDHZXDXEWLXFWLXCUJXEWKWJPZWLXFUKXCXGWL - XCWKFWJGWJCFWJUMWJULUNSWKWJBDUOQUPXFWQFWNXFWJIWQFABWJDUQURUSWNUTVAQWLXCVD - ZTZWPWMLZECMZABNXDXIXKABXIWOBOZTZWPCOZXKXNXKXMXNWPWPLZXKWPVBXJXOEWPCWMWPW - PVEVCVFSXMXNVDZWPFPZXKWLXLXPXQXHWLXLTZXPTWPWJOZXQXRWPWKOXPXSWLBWKWODBWKDV - GVHWPCWJVIVJWPFVKQVLXHXQXKWLXLXHXQTZWTXJTZEVOZXKXTWTEVOZYBXHYCXQXHYCECVPV - MVNXTWTYAEXQWTYAVQXHXQWTYAXQXJWTXQWPFWMXQVRWMUTVAVSWGSVTWAXJECWBWCWDWEWFR - AEBCWPWMUIQWFWHWI $. + wa csn cfv uniiun elun1 foelcdmi sylan2 eqimss2 reximi iunss2 simpl uneq1 + wb 0un eqtrdi foeq3 mpbid founiiun unisn0 eqtr3di 0ss eqsstrdi ssid sseq2 + rspcev mpan2 fof ffvelcdmda elunnel1 sylan elsni adantllr wex neq0 biimpi + wn adantr wi id anim1ci ex eximdv df-rex sylibr ad4ant24 syldan pm2.61dan + mpd eqssd eqtrid ) BCFUAZGZDHZCIECEJZKZABAJZDUBZKZECUCWLWNWQWLWMWPLZABMZE + CNWNWQLWLWSECWLWMCOZTWPWMPZABMZWSWTWLWMWKOXBWMCWJUDABWKDWMUEUFXAWRABWMWPU + GUHQREACBWMWPUIQWLCFPZWQWNLZWLXCTZBWJDHZXDXEWLXFWLXCUJXEWKWJPZWLXFULXCXGW + LXCWKFWJGWJCFWJUKWJUMUNSWKWJBDUOQUPXFWQFWNXFWJIWQFABWJDUQURUSWNUTVAQWLXCV + OZTZWPWMLZECMZABNXDXIXKABXIWOBOZTZWPCOZXKXNXKXMXNWPWPLZXKWPVBXJXOEWPCWMWP + WPVCVDVESXMXNVOZWPFPZXKWLXLXPXQXHWLXLTZXPTWPWJOZXQXRWPWKOXPXSWLBWKWODBWKD + VFVGWPCWJVHVIWPFVJQVKXHXQXKWLXLXHXQTZWTXJTZEVLZXKXTWTEVLZYBXHYCXQXHYCECVM + VNVPXTWTYAEXQWTYAVQXHXQWTYAXQXJWTXQWPFWMXQVRWMUTVAVSVTSWAWGXJECWBWCWDWEWF + RAEBCWPWMUIQWFWHWI $. $} ${ @@ -712965,10 +712960,10 @@ distinct definitions for the same symbol (limit of a sequence). Glauco Siliprandi, 11-Dec-2019.) $) cncfmptssg $p |- ( ph -> ( x e. C |-> E ) e. ( C -cn-> D ) ) $= ( cmpt ccncf co wcel cc wss syl2anc wf fmpttd wb cncfrss2 syl sstrd cv wa - cfv wceq sselda fvmpt2 mpteq2dva nfmpt1 nfcxfr cncfmptss eqeltrrd mpbird - cncffvrn ) ABEGNZEFOPQZEFUTUAZABEGFMUBAFRSUTEDOPZQVAVBUCAFDRLAHCDOPQDRSJC - DHUDUEUFABEBUGZHUIZNUTVCABEVEGAVDEQUHVDCQGFQVEGUJAECVDKUKMBCGFHIULTUMABCD - EHBHBCGNIBCGUNUOJKUPUQEDFUTUSTUR $. + cfv wceq sselda fvmpt2 mpteq2dva nfmpt1 nfcxfr cncfmptss eqeltrrd cncfcdm + mpbird ) ABEGNZEFOPQZEFUTUAZABEGFMUBAFRSUTEDOPZQVAVBUCAFDRLAHCDOPQDRSJCDH + UDUEUFABEBUGZHUIZNUTVCABEVEGAVDEQUHVDCQGFQVEGUJAECVDKUKMBCGFHIULTUMABCDEH + BHBCGNIBCGUNUOJKUPUQEDFUTURTUS $. $} ${ @@ -713052,9 +713047,9 @@ distinct definitions for the same symbol (limit of a sequence). resincncf $p |- ( sin |` RR ) e. ( RR -cn-> RR ) $= ( vx csin cr cres ccncf co wcel wf wfn cv cfv wral wss sinf ax-mp ax-resscn cc ffn fnssres mp2an fvres resincl eqeltrd rgen ffnfv mpbir2an wb sincn mp2 - rescncf cncffvrn mpbir ) BCDZCCEFGZCCUMHZUOUMCIZAJZUMKZCGZACLBQIZCQMZUPQQBH - UTNQQBROPQCBSTUSACUQCGURUQBKCUQCBUAUQUBUCUDACCUMUEUFVAUMCQEFGZUNUOUGPVABQQE - FGVBPUHQQCBUJUICQCUMUKTUL $. + rescncf cncfcdm mpbir ) BCDZCCEFGZCCUMHZUOUMCIZAJZUMKZCGZACLBQIZCQMZUPQQBHU + TNQQBROPQCBSTUSACUQCGURUQBKCUQCBUAUQUBUCUDACCUMUEUFVAUMCQEFGZUNUOUGPVABQQEF + GVBPUHQQCBUJUICQCUMUKTUL $. ${ $d A x $. $d B x $. @@ -713193,12 +713188,12 @@ distinct definitions for the same symbol (limit of a sequence). $( Composition of continuous functions. A generalization of ~ cncfmpt1f to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.) $) cncfcompt $p |- ( ph -> ( x e. A |-> ( F ` B ) ) e. ( A -cn-> D ) ) $= - ( vy cfv cmpt ccncf co wcel wf cv syl cc wss wa cncff fvmptelrn ffvelcdmd - adantr fmpttd cncfrss2 ccom eqidd feqmptd fveq2 fmptco ssid cncfss sseldd - wb sylancl cncfco eqeltrrd cncffvrn syl2anc mpbird ) ABCDGKZLZCFMNOZCFVDP - ZABCVCFABQCOZUAEFDGAEFGPZVGAGEFMNZOZVHIEFGUBRZUEABCDEABCDLZCEMNOCEVLPHCEV - LUBRUCZUDUFAFSTZVDCSMNZOVEVFUPAVJVNIEFGUGRZAGVLUHVDVOABJCEDJQZGKVCVLGVMAV - LUIAJEFGVKUJVQDGUKULACESVLGHAVIESMNZGAVNSSTVIVRTVPSUMEFSUNUQIUOURUSCSFVDU + ( vy cfv cmpt ccncf co wcel wf cv syl cc wss wa cncff adantr ffvelcdmd wb + fvmptelcdm fmpttd cncfrss2 ccom eqidd feqmptd fveq2 fmptco cncfss sylancl + ssid sseldd cncfco eqeltrrd cncfcdm syl2anc mpbird ) ABCDGKZLZCFMNOZCFVDP + ZABCVCFABQCOZUAEFDGAEFGPZVGAGEFMNZOZVHIEFGUBRZUCABCDEABCDLZCEMNOCEVLPHCEV + LUBRUFZUDUGAFSTZVDCSMNZOVEVFUEAVJVNIEFGUHRZAGVLUIVDVOABJCEDJQZGKVCVLGVMAV + LUJAJEFGVKUKVQDGULUMACESVLGHAVIESMNZGAVNSSTVIVRTVPSUPEFSUNUOIUQURUSCSFVDU TVAVB $. $} @@ -713563,23 +713558,23 @@ distinct definitions for the 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C |-> D ) ) ) $= - ( vm cmpt csumge0 cfv c0 csn cdif cvv wcel wfo fornex sylc difssd cv cpnf - wa cc0 cicc co simpl sselda syl2anc cin dfin4 eqcomi inss2 eqsstri sselid - wceq id elsni syl adantl sge0ss eqcomd cres difexd wf1o wne crab crn eqid - fof ffvelcdmda fveq2 neeq1d cbvrabv cbvmptv rneqi difeq1i disjf1o feqmptd - wf wal ccnv cima eldifi adantr fvex elsn sylibr jca adantll ffnd elpreima - wb wfn ad2antrr mpbird eleqtrrdi wn eldifn ad2antlr pm2.65da neqned elrab - wral wss simplbi eleq2i biimpi mpbid simprd adantlr simprbi neneqd eldifd - ex ralrimi dfss3 sseld eqtr2d wi nfv nfan nfim anbi2d imbi12d vtoclgf + ( vm cmpt csumge0 cfv c0 csn cdif cvv wcel wfo focdmex sylc difssd cv cc0 + wa cpnf cicc co simpl sselda syl2anc wceq cin dfin4 eqcomi eqsstri sselid + inss2 id elsni syl adantl sge0ss eqcomd cres difexd wf1o wne crab eqid wf + crn ffvelcdmda fveq2 neeq1d cbvrabv cbvmptv rneqi difeq1i disjf1o feqmptd + fof wal ccnv cima eldifi adantr fvex elsn sylibr jca adantll wfn elpreima 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wbr crab nfan csalg adantr adantlr csmblfn cfv wceq + simpr oveq2 breq1d cbvrabv mpteq2i smfaddlem2 issmfdmpt ) ABCEUDZDFUEUFZG + UAIAUAUGJAWCCGUHZABWCCIBQZWCRZWFCRZAWFCEUIUJZUKACBCDUMZULZWEAWKCABWJCDSIW + JUNLUOUPAWKGWJJNWKUNUQURUSABWCWDSABWCWDSBWCWDUMZIAWGUTDFAWGWHDSRZWILVAAWG + WFERZFSRZWGWNAWFCEVBUJMVAVCWLUNVDVEAUAQZSRZUTBCDEFWPGPTPQZUBQZUEUFZWPVFVG + ZUBTVHZUMHUCPAWQBIWQBUGVIAGVJRWQJVKACHRWQKVKAWHWMWQLVLAWNWOWQMVLAWJGVMVNZ + RWQNVKABEFUMXCRWQOVKAWQVPPTXBWRUCQZUEUFZWPVFVGZUCTVHXAXFUBUCTWSXDVOWTXEWP + VFWSXDWRUEVQVRVSVTWAWB $. $} ${ @@ -752794,10 +752788,10 @@ convergence can be decidedly irregular (Remark 121G of [Fremlin1] smfpimioompt $p |- ( ph -> { x e. A | B e. ( L (,) R ) } e. ( S |`t A ) ) $= ( co crab cr eqid cioo wcel clt wbr cin crest cv wa cmpt cdm smff dmmptdf - wf feq2d mpbid fvmptelrn pimiooltgt subsalsal smfpimltxrmpt smfpimgtxrmpt - rexrd salincld eqeltrd ) ADGEUAQUBBCRDEUCUDBCRZGDUCUDBCRZUEFCUFQZABCDEGJO - PABUGCUBUHDABCDSABCDUIZUJZSVGUMCSVGUMAVHFVGKNVHTUKAVHCSVGABVGCDIJVGTMULUN - UOUPVAUQAVFVDVEACFVFHKLVFTURABCDEFIJKMNPUSABCDFGIJKMNOUTVBVC $. + wf feq2d mpbid fvmptelcdm pimiooltgt smfpimltxrmpt smfpimgtxrmpt salincld + rexrd subsalsal eqeltrd ) ADGEUAQUBBCRDEUCUDBCRZGDUCUDBCRZUEFCUFQZABCDEGJ + OPABUGCUBUHDABCDSABCDUIZUJZSVGUMCSVGUMAVHFVGKNVHTUKAVHCSVGABVGCDIJVGTMULU + NUOUPVAUQAVFVDVEACFVFHKLVFTVBABCDEFIJKMNPURABCDFGIJKMNOUSUTVC $. $} ${ @@ -753953,27 +753947,27 @@ that every function in the sequence can have a different (partial) smfinfmpt $p |- ( ph -> G e. ( SMblFn ` S ) ) $= ( cv cmpt cfv cle wbr wral cr wrex cdm ciin crab crn clt cinf csmblfn a1i wceq wcel wa eqidd fvmpt2d dmeqd nfcv nfel nfan eqid csalg adantr smffmpt - wal 3expa fvmptelrn dmmptdf 3eqtrrd iineq2d nfmpt1 nfmpt nffv nfdm rabeqf - nfiin syl nfv nfii1 wb simpll simpr eliinid adantll eqcomd adantlr fveq1d - eleqtrd 3adant3 simp3 fvmpt2 syl2anc eqtr2d breq2d syl3anc ralbida rexbid - w3a rabbida eqtrd alrimi nfra1 nfrex nfrabw nfcxfr eleq2i biimpi rabidim1 - wi idi mpteq2da rneqd infeq1d ex ralrimi mpteq12f fmptdf smfinf eqeltrd ) - AIBCUCZBUCZHUCZHLBDEUDZUDZUEZUEZUFUGZHLUHZCUIUJZBHLYLUKZULZUMZHLYMUDZUNZU - IUOUPZUDZGUQUEZAIBFHLEUDZUNZUIUOUPZUDZUUCIUUHUSAUBURAFYSUSZBVLUUGUUBUSZBF - UHUUHUUCUSAUUIBNAFYGEUFUGZHLUHZCUIUJZBHLDULZUMZYSFUUOUSAUAURAUUOUUMBYRUMZ - YSAUUNYRUSUUOUUPUSAHLDYQMAYILUTZVAZYQYJUKDDUURYLYJAHLYJYKUUDAYKVBTVCZVDUU - RBYJDEUIAUUQBNBYILBYIVEZBLVEZVFVGZYJVHZUURBDEUIUURBDEGKUVBAGVIUTUUQRVJAUU - QYHDUTZEKUTZSVMTVKVNVOUURDVBVPZVQUUMBUUNYRBUUNVEHBLYQUVABYLBYIYKBHLYJUVAB - DEVRVSZUUTVTWAWCWBWDAUUMYPBYRNAYHYRUTZVAZUULYOCUIAUVHCOUVHCWEVGUVIUUKYNHL - AUVHHMHYHYRHYHVEZHLYQWFVFVGUVIUUQVAZAUUQUVDUUKYNWGAUVHUUQWHUVIUUQWIUVKYHY - QDUVHUUQYHYQUTAHYHLYQWJWKAUUQYQDUSUVHUURDYQUVFWLWMWOAUUQUVDXEZEYMYGUFUVLY - MYHYJUEZEAUUQYMUVMUSUVDUURYHYLYJUUSWNWPUVLUVDUVEUVMEUSAUUQUVDWQSBDEKYJUVC - WRWSWTZXAXBXCXDXFXGXGXHAUUJBFNAYHFUTZUUJAUVOVAZUIUUFUUAUOUVPUUEYTUVPHLEYM - AUVOHMHYHFUVJHFUUOUAUUMHBUUNUULHCUIHUIVEUUKHLXIXJHLDWFXKXLVFVGUVPUUQVAAUU - QUVDEYMUSZAUVOUUQWHUVPUUQWIUVOUUQUVDAUVOUUQVAYHUUNUTZUUQUVDUVOUVRUUQUVOYH - UUOUTZUVRUVOUVSFUUOYHUAXMXNUUMBUUNXOWDVJUVOUUQWIHYHLDWJWSWKUVLUVQXPUVNXQX - BXRXSXTYAYBBFUUGYSUUBYCWSXGABCYSGHYKUUCJLHLYJVRUVGPQRAHLYJUUDYKMTYKVHYDYS - VHUUCVHYEYF $. + wal 3expa fvmptelcdm dmmptdf 3eqtrrd iineq2d nfmpt1 nfmpt nffv nfdm nfiin + rabeqf syl nfv nfii1 wb simpll eliinid adantll eqcomd adantlr eleqtrd w3a + simpr fveq1d 3adant3 fvmpt2 syl2anc eqtr2d breq2d syl3anc ralbida rabbida + simp3 rexbid eqtrd alrimi nfra1 nfrabw nfcxfr eleq2i rabidim1 wi mpteq2da + nfrex biimpi idi rneqd infeq1d ex ralrimi mpteq12f fmptdf smfinf eqeltrd + ) AIBCUCZBUCZHUCZHLBDEUDZUDZUEZUEZUFUGZHLUHZCUIUJZBHLYLUKZULZUMZHLYMUDZUN + ZUIUOUPZUDZGUQUEZAIBFHLEUDZUNZUIUOUPZUDZUUCIUUHUSAUBURAFYSUSZBVLUUGUUBUSZ + BFUHUUHUUCUSAUUIBNAFYGEUFUGZHLUHZCUIUJZBHLDULZUMZYSFUUOUSAUAURAUUOUUMBYRU + MZYSAUUNYRUSUUOUUPUSAHLDYQMAYILUTZVAZYQYJUKDDUURYLYJAHLYJYKUUDAYKVBTVCZVD + UURBYJDEUIAUUQBNBYILBYIVEZBLVEZVFVGZYJVHZUURBDEUIUURBDEGKUVBAGVIUTUUQRVJA + UUQYHDUTZEKUTZSVMTVKVNVOUURDVBVPZVQUUMBUUNYRBUUNVEHBLYQUVABYLBYIYKBHLYJUV + ABDEVRVSZUUTVTWAWBWCWDAUUMYPBYRNAYHYRUTZVAZUULYOCUIAUVHCOUVHCWEVGUVIUUKYN + HLAUVHHMHYHYRHYHVEZHLYQWFVFVGUVIUUQVAZAUUQUVDUUKYNWGAUVHUUQWHUVIUUQWOUVKY + HYQDUVHUUQYHYQUTAHYHLYQWIWJAUUQYQDUSUVHUURDYQUVFWKWLWMAUUQUVDWNZEYMYGUFUV + LYMYHYJUEZEAUUQYMUVMUSUVDUURYHYLYJUUSWPWQUVLUVDUVEUVMEUSAUUQUVDXESBDEKYJU + VCWRWSWTZXAXBXCXFXDXGXGXHAUUJBFNAYHFUTZUUJAUVOVAZUIUUFUUAUOUVPUUEYTUVPHLE + YMAUVOHMHYHFUVJHFUUOUAUUMHBUUNUULHCUIHUIVEUUKHLXIXPHLDWFXJXKVFVGUVPUUQVAA + UUQUVDEYMUSZAUVOUUQWHUVPUUQWOUVOUUQUVDAUVOUUQVAYHUUNUTZUUQUVDUVOUVRUUQUVO + YHUUOUTZUVRUVOUVSFUUOYHUAXLXQUUMBUUNXMWDVJUVOUUQWOHYHLDWIWSWJUVLUVQXNUVNX + RXBXOXSXTYAYBBFUUGYSUUBYCWSXGABCYSGHYKUUCJLHLYJVRUVGPQRAHLYJUUDYKMTYKVHYD + YSVHUUCVHYEYF $. $} ${ @@ -758807,7 +758801,7 @@ Introduce the Axiom of Power Sets (extension) $( A function's value belongs to its codomain, analogous to ~ ffvelcdm . (Contributed by Alexander van der Vekens, 25-May-2017.) $) - fafvelrn $p |- ( ( F : A --> B /\ C e. A ) -> ( F ''' C ) e. B ) $= + fafvelcdm $p |- ( ( F : A --> B /\ C e. A ) -> ( F ''' C ) e. B ) $= ( wf wcel wa cafv crn wfn ffn fnafvelrn sylan wi frn sseld adantr mpd ) ABD EZCAFZGCDHZDIZFZUABFZSDAJTUCABDKACDLMSUCUDNTSUBBUAABDOPQR $. @@ -758817,11 +758811,11 @@ Introduce the Axiom of Power Sets (extension) ~ ffnfv . (Contributed by Alexander van der Vekens, 25-May-2017.) $) ffnafv $p |- ( F : A --> B <-> ( F Fn A /\ A. x e. A ( F ''' x ) e. B ) ) $= - ( vy wf wfn cv cafv wcel wral wa ffn fafvelrn ralrimiva jca crn wss simpl - wceq wrex afvelrnb0 nfra1 nfv rsp eleq1 biimpcd syl6 rexlimd sylan9 ssrdv - wi df-f sylanbrc impbii ) BCDFZDBGZAHZDIZCJZABKZLZUPUQVABCDMUPUTABBCURDNO - PVBUQDQZCRUPUQVASVBEVCCUQEHZVCJUSVDTZABUAVAVDCJZABVDDUBVAVEVFABUTABUCVFAU - DVAURBJUTVEVFULUTABUEVEUTVFUSVDCUFUGUHUIUJUKBCDUMUNUO $. + ( vy wf wfn cv cafv wcel wral wa ffn fafvelcdm ralrimiva jca crn wss wceq + simpl wrex afvelrnb0 nfra1 nfv wi eleq1 biimpcd syl6 rexlimd sylan9 ssrdv + rsp df-f sylanbrc impbii ) BCDFZDBGZAHZDIZCJZABKZLZUPUQVABCDMUPUTABBCURDN + OPVBUQDQZCRUPUQVATVBEVCCUQEHZVCJUSVDSZABUAVAVDCJZABVDDUBVAVEVFABUTABUCVFA + UDVAURBJUTVEVFUEUTABULVEUTVFUSVDCUFUGUHUIUJUKBCDUMUNUO $. $} $( The value of a restricted function, analogous to ~ fvres . (Contributed @@ -759575,7 +759569,7 @@ known as function application (and called "alternate function value" in $( An alternate function value belongs to the codomain of the function, analogous to ~ ffvelcdm . (Contributed by AV, 2-Sep-2022.) $) - fafv2elrn $p |- ( ( F : A --> B /\ C e. A ) -> ( F '''' C ) e. B ) $= + fafv2elcdm $p |- ( ( F : A --> B /\ C e. A ) -> ( F '''' C ) e. B ) $= ( wf wcel wa cafv2 crn wfn ffn fnafv2elrn sylan wi frn sseld adantr mpd ) A BDEZCAFZGCDHZDIZFZUABFZSDAJTUCABDKACDLMSUCUDNTSUBBUAABDOPQR $. @@ -759590,7 +759584,7 @@ known as function application (and called "alternate function value" in $( If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.) $) - frnvafv2v $p |- ( ( F : A --> B /\ B e. V ) -> ( F '''' C ) e. _V ) $= + fcdmvafv2v $p |- ( ( F : A --> B /\ B e. V ) -> ( F '''' C ) e. _V ) $= ( wf wcel wa crn cvv cafv2 wfn wss wi df-f ssexg ex simplbiim imp afv2ex syl ) ABDFZBEGZHDIZJGZCDKJGUBUCUEUBDALUDBMZUCUENABDOUFUCUEUDBEPQRSCDJTUA $. @@ -760244,14 +760238,14 @@ Negated membership (alternative) f1oresf1o2 $p |- ( ph -> ( F |` D ) : D -1-1-onto-> { y e. B | ch } ) $= ( cv wceq wrex wcel wa syl adantr wi ex cfv w3a wf1o f1of sselda ffvelcdm wf jca 3adant3 wb eleq1 3ad2ant3 mpbid eqcom biimpd com23 3imp rexlimdv3a - syl5bi wfo f1ofo foelrni sylan nfv nfim rspe expcom eqcoms adantl sylbird - nfre1 rexlimd syld impd impbid f1oresf1o ) ABCDEFGHIJACLZHUAZDLZMZCGNZVSF - OZBPZAVTWCCGAVQGOZVTUBZWBBWEVRFOZWBWEEFHUGZVQEOZPZWFAWDWIVTAWDPWGWHAWGWDA - EFHUCZWGIEFHUDQRAGEVQJUEUHUIEFVQHUFQVTAWFWBUJWDVRVSFUKULUMAWDVTBAVTWDBVTV - SVRMZAWDBSZVRVSUNZAWKWLAWKPZWDBKUOTUSUPUQUHURAWBBWAAWBVTCENZBWASZAWBWOAEF - HUTZWBWOAWJWQIEFHVAQCEFHVSVBVCTAVTWPCEACVDBWACBCVDVTCGVKVEAWHVTWPSVTWKAWH - PWPWMAWKWPSWHAWKWPWNBWDWAKWKWDWASZAWRVRVSWDVTWAVTCGVFVGVHVIVJTRUSTVLVMVNV - OVP $. + syl5bi wfo f1ofo foelcdmi sylan nfv nfre1 nfim rspe expcom eqcoms sylbird + adantl rexlimd syld impd impbid f1oresf1o ) ABCDEFGHIJACLZHUAZDLZMZCGNZVS + FOZBPZAVTWCCGAVQGOZVTUBZWBBWEVRFOZWBWEEFHUGZVQEOZPZWFAWDWIVTAWDPWGWHAWGWD + AEFHUCZWGIEFHUDQRAGEVQJUEUHUIEFVQHUFQVTAWFWBUJWDVRVSFUKULUMAWDVTBAVTWDBVT + VSVRMZAWDBSZVRVSUNZAWKWLAWKPZWDBKUOTUSUPUQUHURAWBBWAAWBVTCENZBWASZAWBWOAE + FHUTZWBWOAWJWQIEFHVAQCEFHVSVBVCTAVTWPCEACVDBWACBCVDVTCGVEVFAWHVTWPSVTWKAW + HPWPWMAWKWPSWHAWKWPWNBWDWAKWKWDWASZAWRVRVSWDVTWAVTCGVGVHVIVKVJTRUSTVLVMVN + VOVP $. $} @@ -780999,7 +780993,7 @@ this concept is not fully developed yet (it is within a mathbox), and it is 2arympt $p |- ( ( X e. V /\ O : ( X X. X ) --> X ) -> F e. ( 2 -aryF X ) ) $= ( wcel cxp wf wa c2 cnaryf co cc0 c1 cpr cfv a1i ffvelcdmd adantl cmap cv - simplr elmapi c0ex prid1 1ex prid2 fovrnd fmptd 2aryfvalel adantr mpbird + simplr elmapi c0ex prid1 1ex prid2 fovcdmd fmptd 2aryfvalel adantr mpbird wb ) EDGZEEHECIZJZBKELMGZENOPZUAMZEBIZUQAUTNAUBZQZOVBQZCMEBUQVBUTGZJVCVDE EECUOUPVEUCVEVCEGUQVEUSENVBVBEUSUDZNUSGVENOUEUFRSTVEVDEGUQVEUSEOVBVFOUSGV ENOUGUHRSTUIFUJUOURVAUNUPBDEUKULUM $.