-- Mathlib/LinearAlgebra/Matrix/Adjugate.lean
theorem adjugate_mul_distrib (A B : Matrix n n α) : adjugate (A * B) = adjugate B * adjugate A

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Tactic:
have hu : ∀ M : Matrix n n α, IsRegular (g M).det := by
  intro M
  refine Polynomial.Monic.isRegular ?_
  simp only [g, Polynomial.Monic.def, ← Polynomial.leadingCoeff_det_X_one_add_C M, add_comm]

Result:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
hu : ∀ (M : Matrix n n α), IsRegular (g M).det
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Proof state:
case a
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
M✝ : Matrix n n α
i✝ j✝ : n
⊢ f' (g M✝) i✝ j✝ = M✝ i✝ j✝

Tactic:
simp [f', g]

Result:
no goals

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
M : Matrix n n α
⊢ IsRegular (g M).det

Tactic:
refine Polynomial.Monic.isRegular ?_

Result:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
M : Matrix n n α
⊢ (g M).det.Monic

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
M✝ : Matrix n n α
⊢ f' (g M✝) = M✝

Tactic:
ext

Result:
case a
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
M✝ : Matrix n n α
i✝ j✝ : n
⊢ f' (g M✝) i✝ j✝ = M✝ i✝ j✝

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
⊢ ∀ (M : Matrix n n α), IsRegular (g M).det

Tactic:
intro M

Result:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
M : Matrix n n α
⊢ IsRegular (g M).det

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Tactic:
have f'_g_mul : ∀ M N : Matrix n n α, f' (g M * g N) = M * N := by
  intros M N
  rw [RingHom.map_mul, f'_inv, f'_inv]

Result:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
⊢ ∀ (M N : Matrix n n α), f' (g M * g N) = M * N

Tactic:
intros M N

Result:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
M N : Matrix n n α
⊢ f' (g M * g N) = M * N

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
M : Matrix n n α
⊢ (g M).det.Monic

Tactic:
simp only [g, Polynomial.Monic.def, ← Polynomial.leadingCoeff_det_X_one_add_C M, add_comm]

Result:
no goals

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
hu : ∀ (M : Matrix n n α), IsRegular (g M).det
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Tactic:
rw [← f'_adj, ← f'_adj, ← f'_adj, ← f'.map_mul, ←
  adjugate_mul_distrib_aux _ _ (hu A).left (hu B).left, RingHom.map_adjugate,
  RingHom.map_adjugate, f'_inv, f'_g_mul]

Result:
no goals

Proof state:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M => M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
M✝ : Matrix n n α
⊢ f' (g M✝).adjugate = M✝.adjugate

Tactic:
rw [RingHom.map_adjugate, f'_inv]

Result:
no goals