-- Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
theorem inv_fromBlocks_zero₂₁_of_isUnit_iff (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α)
    (hAD : IsUnit A ↔ IsUnit D) :
    (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case pos.intro.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝¹ : Invertible A
val✝ : Invertible D
this : Invertible (fromBlocks A B 0 D) := A.fromBlocksZero₂₁Invertible B D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
simp_rw [← invOf_eq_nonsing_inv, invOf_fromBlocks_zero₂₁_eq]

Result:
no goals

Proof state:
case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
have hD := hAD.not.mp hA

Result:
case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
have : ¬IsUnit (fromBlocks A B 0 D) :=
  isUnit_fromBlocks_zero₂₁.not.mpr (not_and'.mpr fun _ => hA)

Result:
case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
this : ¬IsUnit (fromBlocks A B 0 D)
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case pos.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝ : Invertible A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
cases hD.nonempty_invertible

Result:
case pos.intro.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝¹ : Invertible A
val✝ : Invertible D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case pos
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
have hD := hAD.mp hA

Result:
case pos
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case pos.intro.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝¹ : Invertible A
val✝ : Invertible D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
letI := fromBlocksZero₂₁Invertible A B D

Result:
case pos.intro.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝¹ : Invertible A
val✝ : Invertible D
this : Invertible (fromBlocks A B 0 D) := A.fromBlocksZero₂₁Invertible B D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
by_cases hA : IsUnit A

Result:
case pos
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case pos
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
cases hA.nonempty_invertible

Result:
case pos.intro
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝ : Invertible A
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Proof state:
case neg
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
this : ¬IsUnit (fromBlocks A B 0 D)
⊢ (fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

Tactic:
simp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ hA, Ring.inverse_non_unit _ hD,
  Ring.inverse_non_unit _ this, Matrix.zero_mul, neg_zero, fromBlocks_zero]

Result:
no goals