Theorem: Matrix.inv_fromBlocks_zero₁₂_of_isUnit_iff
File: Mathlib/LinearAlgebra/Matrix/SchurComplement.lean

tactic: by_cases hA : IsUnit A
state_before: l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case pos
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: have hD := hAD.mp hA
state_before: case pos
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case pos
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: cases hA.nonempty_invertible
state_before: case pos
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case pos.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝ : Invertible A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: cases hD.nonempty_invertible
state_before: case pos.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : IsUnit A
hD : IsUnit D
val✝ : Invertible A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case pos.intro.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m 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 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: letI := fromBlocksZero₁₂Invertible A C D
state_before: case pos.intro.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m 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 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case pos.intro.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m 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 0 C D) := A.fromBlocksZero₁₂Invertible C D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: simp_rw [← invOf_eq_nonsing_inv, invOf_fromBlocks_zero₁₂_eq]
state_before: case pos.intro.intro
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m 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 0 C D) := A.fromBlocksZero₁₂Invertible C D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: no goals

tactic: have hD := hAD.not.mp hA
state_before: case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

tactic: have : ¬IsUnit (fromBlocks A 0 C D) :=
  isUnit_fromBlocks_zero₁₂.not.mpr (not_and'.mpr fun _ => hA)
state_before: case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
this : ¬IsUnit (fromBlocks A 0 C D)
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) 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]
state_before: case neg
l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁶ : Fintype l
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : DecidableEq l
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
hAD : IsUnit A ↔ IsUnit D
hA : ¬IsUnit A
hD : ¬IsUnit D
this : ¬IsUnit (fromBlocks A 0 C D)
⊢ (fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹
state_after: no goals