Module: Mathlib.LinearAlgebra.Matrix.Determinant.Basic

Goal state before:
pp: case refine_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ a ∈ univ, (prodCongrLeft fun k ↦ a k ⋯) ∈ preserving_snd

pp: case refine_2
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ a₁ ∈ univ, ∀ a₂ ∈ univ, ((prodCongrLeft fun k ↦ a₁ k ⋯) = prodCongrLeft fun k ↦ a₂ k ⋯) → a₁ = a₂

pp: case refine_3
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ b ∈ preserving_snd, ∃ a, ∃ (_ : a ∈ univ), (prodCongrLeft fun k ↦ a k ⋯) = b

pp: case refine_4
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ a ∈ univ,
    ∏ x, ↑↑(sign (a x ⋯)) * ∏ x_1, M x ((a x ⋯) x_1) x_1 =
      ↑↑(sign (prodCongrLeft fun k ↦ a k ⋯)) * ∏ i, blockDiagonal M ((prodCongrLeft fun k ↦ a k ⋯) i) i

Tactic:
· intro σ _
      rw [mem_preserving_snd]
      rintro ⟨-, x⟩
      simp only [prodCongrLeft_apply]

Goal state after:
case refine_2
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ a₁ ∈ univ, ∀ a₂ ∈ univ, ((prodCongrLeft fun k ↦ a₁ k ⋯) = prodCongrLeft fun k ↦ a₂ k ⋯) → a₁ = a₂

case refine_3
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ b ∈ preserving_snd, ∃ a, ∃ (_ : a ∈ univ), (prodCongrLeft fun k ↦ a k ⋯) = b

case refine_4
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
R : Type v
inst✝² : CommRing R
o : Type u_3
inst✝¹ : Fintype o
inst✝ : DecidableEq o
M : o → Matrix n n R
preserving_snd : Finset (Perm (n × o)) := {σ | ∀ (x : n × o), (σ x).2 = x.2}
mem_preserving_snd : ∀ {σ : Perm (n × o)}, σ ∈ preserving_snd ↔ ∀ (x : n × o), (σ x).2 = x.2
⊢ ∀ a ∈ univ,
    ∏ x, ↑↑(sign (a x ⋯)) * ∏ x_1, M x ((a x ⋯) x_1) x_1 =
      ↑↑(sign (prodCongrLeft fun k ↦ a k ⋯)) * ∏ i, blockDiagonal M ((prodCongrLeft fun k ↦ a k ⋯) i) i