Theorem: Basis.localizationLocalization_repr_algebraMap
File: Mathlib/RingTheory/Localization/Module.lean

tactic: simp_rw [IsScalarTower.algebraMap_smul, Algebra.smul_def,
  IsScalarTower.algebraMap_apply R A Aₛ, ← _root_.map_mul, ← map_finsupp_sum, ←
  Algebra.smul_def, ← Finsupp.total_apply, Basis.total_repr]
state_before: R : Type u_2
Rₛ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₛ
inst✝⁸ : Algebra R Rₛ
S : Submonoid R
hT : IsLocalization S Rₛ
A : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
Aₛ : Type u_5
inst✝⁵ : CommRing Aₛ
inst✝⁴ : Algebra A Aₛ
inst✝³ : Algebra Rₛ Aₛ
inst✝² : Algebra R Aₛ
inst✝¹ : IsScalarTower R Rₛ Aₛ
inst✝ : IsScalarTower R A Aₛ
hA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ
ι : Type u_1
b : Basis ι R A
x : A
i : ι
⊢ ↑(↑(localizationLocalization Rₛ S Aₛ b).repr (↑(algebraMap A Aₛ) x)) i =
    ↑(↑(localizationLocalization Rₛ S Aₛ b).repr
          (Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(algebraMap A Aₛ) (↑b j)))
      i

tactic: simp_rw [← b.localizationLocalization_apply Rₛ S Aₛ, map_finsupp_sum, LinearEquiv.map_smul,
  Basis.repr_self, Finsupp.sum_apply, Finsupp.smul_apply]
state_before: R : Type u_2
Rₛ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₛ
inst✝⁸ : Algebra R Rₛ
S : Submonoid R
hT : IsLocalization S Rₛ
A : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
Aₛ : Type u_5
inst✝⁵ : CommRing Aₛ
inst✝⁴ : Algebra A Aₛ
inst✝³ : Algebra Rₛ Aₛ
inst✝² : Algebra R Aₛ
inst✝¹ : IsScalarTower R Rₛ Aₛ
inst✝ : IsScalarTower R A Aₛ
hA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ
ι : Type u_1
b : Basis ι R A
x : A
i : ι
⊢ ↑(↑(localizationLocalization Rₛ S Aₛ b).repr
          (Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(algebraMap A Aₛ) (↑b j)))
      i =
    Finsupp.sum (↑b.repr x) fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i

tactic: simp [hj]
state_before: R : Type u_2
Rₛ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₛ
inst✝⁸ : Algebra R Rₛ
S : Submonoid R
hT : IsLocalization S Rₛ
A : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
Aₛ : Type u_5
inst✝⁵ : CommRing Aₛ
inst✝⁴ : Algebra A Aₛ
inst✝³ : Algebra Rₛ Aₛ
inst✝² : Algebra R Aₛ
inst✝¹ : IsScalarTower R Rₛ Aₛ
inst✝ : IsScalarTower R A Aₛ
hA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ
ι : Type u_1
b : Basis ι R A
x : A
i j : ι
x✝ : j ∈ (↑b.repr x).support
hj : j ≠ i
⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) j (↑(↑b.repr x) j) = 0

tactic: simp [Finsupp.not_mem_support_iff.mp hi]
state_before: R : Type u_2
Rₛ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₛ
inst✝⁸ : Algebra R Rₛ
S : Submonoid R
hT : IsLocalization S Rₛ
A : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
Aₛ : Type u_5
inst✝⁵ : CommRing Aₛ
inst✝⁴ : Algebra A Aₛ
inst✝³ : Algebra Rₛ Aₛ
inst✝² : Algebra R Aₛ
inst✝¹ : IsScalarTower R Rₛ Aₛ
inst✝ : IsScalarTower R A Aₛ
hA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ
ι : Type u_1
b : Basis ι R A
x : A
i : ι
hi : ¬i ∈ (↑b.repr x).support
⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) i (↑(↑b.repr x) i) = 0

tactic: simp [Algebra.smul_def]
state_before: R : Type u_2
Rₛ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing Rₛ
inst✝⁸ : Algebra R Rₛ
S : Submonoid R
hT : IsLocalization S Rₛ
A : Type u_3
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
Aₛ : Type u_5
inst✝⁵ : CommRing Aₛ
inst✝⁴ : Algebra A Aₛ
inst✝³ : Algebra Rₛ Aₛ
inst✝² : Algebra R Aₛ
inst✝¹ : IsScalarTower R Rₛ Aₛ
inst✝ : IsScalarTower R A Aₛ
hA : IsLocalization (Algebra.algebraMapSubmonoid A S) Aₛ
ι : Type u_1
b : Basis ι R A
x : A
i : ι
⊢ (fun j c => ↑(algebraMap R Rₛ) c • ↑(Finsupp.single j 1) i) i (↑(↑b.repr x) i) = ↑(algebraMap R Rₛ) (↑(↑b.repr x) i)