Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: case pos
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j

pp: case neg
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : ¬i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j

Tactic:
· rw [h, Matrix.diagonal_apply_eq, Finsupp.single_eq_same]

Goal state after:
case neg
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : ¬i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j