-- Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
theorem inv_submatrix_equiv (A : Matrix m m α) (e₁ e₂ : n ≃ m) :
    (A.submatrix e₁ e₂)⁻¹ = A⁻¹.submatrix e₂ e₁

Proof state:
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
by_cases h : IsUnit A

Result:
case pos
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

case neg
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : ¬IsUnit A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Proof state:
case pos.intro
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
val✝ : Invertible A
this : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
rw [← invOf_eq_nonsing_inv, ← invOf_eq_nonsing_inv, invOf_submatrix_equiv_eq A]

Result:
no goals

Proof state:
case neg
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : ¬IsUnit A
this : ¬IsUnit (A.submatrix ⇑e₁ ⇑e₂)
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
simp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ h, Ring.inverse_non_unit _ this,
  submatrix_zero, Pi.zero_apply]

Result:
no goals

Proof state:
case pos
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
cases h.nonempty_invertible

Result:
case pos.intro
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
val✝ : Invertible A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Proof state:
case pos.intro
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
val✝ : Invertible A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
letI := submatrixEquivInvertible A e₁ e₂

Result:
case pos.intro
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : IsUnit A
val✝ : Invertible A
this : Invertible (A.submatrix ⇑e₁ ⇑e₂) := A.submatrixEquivInvertible e₁ e₂
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Proof state:
case neg
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : ¬IsUnit A
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁

Tactic:
have := (isUnit_submatrix_equiv e₁ e₂).not.mpr h

Result:
case neg
m : Type u
n : Type u'
α : Type v
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m α
e₁ e₂ : n ≃ m
h : ¬IsUnit A
this : ¬IsUnit (A.submatrix ⇑e₁ ⇑e₂)
⊢ (A.submatrix ⇑e₁ ⇑e₂)⁻¹ = A⁻¹.submatrix ⇑e₂ ⇑e₁