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chore(StdBasis): Fintype -> Finite (#10914)
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urkud committed Feb 24, 2024
1 parent d6542aa commit a25ebf0
Showing 1 changed file with 3 additions and 2 deletions.
5 changes: 3 additions & 2 deletions Mathlib/LinearAlgebra/StdBasis.lean
Expand Up @@ -299,13 +299,14 @@ end Pi

namespace Module

variable (ι R M N : Type*) [Fintype ι] [CommSemiring R]
variable (ι R M N : Type*) [Finite ι] [CommSemiring R]
[AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]

/-- The natural linear equivalence: `Mⁱ ≃ Hom(Rⁱ, M)` for an `R`-module `M`. -/
noncomputable def piEquiv : (ι → M) ≃ₗ[R] ((ι → R) →ₗ[R] M) := Basis.constr (Pi.basisFun R ι) R

lemma piEquiv_apply_apply (v : ι → M) (w : ι → R) :
lemma piEquiv_apply_apply (ι R M : Type*) [Fintype ι] [CommSemiring R]
[AddCommMonoid M] [Module R M] (v : ι → M) (w : ι → R) :
piEquiv ι R M v w = ∑ i, w i • v i := by
simp only [piEquiv, Basis.constr_apply_fintype, Basis.equivFun_apply]
congr
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