refactor(Analysis/NormedSpace/Exponential): remove the 𝕂 argument from exp
#8370
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…ditive monoids While many of the lemmas cease to apply, the definitions of sequences and their sums are still perfectly well-behaved in additive monoids; no negation is needed. We could use this to generalize `NormedSpace.exp` such that it works with `NNReal`, but unless leanprover-community/mathlib3#16554 or similar lands and provides a `DivisionSemiring.nnqsmul` field, that would conflict with #8370.
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| analyticAt_exp_of_mem_ball x ((expSeries_radius_eq_top 𝕂 𝔸).symm ▸ edist_lt_top _ _) | ||
| #align exp_analytic NormedSpace.exp_analytic | ||
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| variable (𝕂) |
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Why do we need this? If it's so that we pull in a NormedAlgebra assumption, couldn't we always do that over the reals (since complex normed algebras are real normed algebras)?
| [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i) (i : ι) : | ||
| exp 𝕂 x i = exp 𝕂 (x i) := | ||
| map_exp _ (Pi.evalRingHom 𝔸 i) (continuous_apply _) x | ||
| [∀ i, Algebra ℚ (𝔸 i)] [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i) |
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Could we switched the normed algebra to be always over the reals, as complex normed algebras are real normed algebras (resp. below)?
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Turns out this could be generalized anyway, #10865
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This argument is still needed for almost all the lemmas, which means it can longer be found by unification.
We keep around
expSeries 𝕂 A, as it's needed for talking about the radius of convergence over different base fields.This is a prerequisite for #8372, as we can't merge the functions until they have the same interface.
This is started from the mathport output on leanprover-community/mathlib3#19244