Theorem: SymplecticGroup.transpose_mem
File: Mathlib/LinearAlgebra/SymplecticGroup.lean

tactic: rw [mem_iff] at hA ⊢
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A ∈ symplecticGroup l R
⊢ Aᵀ ∈ symplecticGroup l R
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
⊢ Aᵀ * J l R * Aᵀᵀ = J l R

tactic: rw [transpose_transpose]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
⊢ Aᵀ * J l R * Aᵀᵀ = J l R
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
⊢ Aᵀ * J l R * A = J l R

tactic: have huA := symplectic_det hA
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
⊢ Aᵀ * J l R * A = J l R
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
⊢ Aᵀ * J l R * A = J l R

tactic: have huAT : IsUnit Aᵀ.det := by
  rw [Matrix.det_transpose]
  exact huA
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
⊢ Aᵀ * J l R * A = J l R
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ Aᵀ * J l R * A = J l R

tactic: calc
  Aᵀ * J l R * A = (-Aᵀ) * (J l R)⁻¹ * A := by
    rw [J_inv]
    simp
  _ = (-Aᵀ) * (A * J l R * Aᵀ)⁻¹ * A := by rw [hA]
  _ = -(Aᵀ * (Aᵀ⁻¹ * (J l R)⁻¹)) * A⁻¹ * A := by
    simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]
  _ = -(J l R)⁻¹ := by
    rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]
  _ = J l R := by simp [J_inv]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ Aᵀ * J l R * A = J l R
state_after: no goals

tactic: rw [Matrix.det_transpose]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
⊢ IsUnit Aᵀ.det
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
⊢ IsUnit A.det

tactic: exact huA
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
⊢ IsUnit A.det
state_after: no goals

tactic: rw [J_inv]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ Aᵀ * J l R * A = -Aᵀ * (J l R)⁻¹ * A
state_after: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ Aᵀ * J l R * A = -Aᵀ * -J l R * A

tactic: simp
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ Aᵀ * J l R * A = -Aᵀ * -J l R * A
state_after: no goals

tactic: rw [hA]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ -Aᵀ * (J l R)⁻¹ * A = -Aᵀ * (A * J l R * Aᵀ)⁻¹ * A
state_after: no goals

tactic: simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ -Aᵀ * (A * J l R * Aᵀ)⁻¹ * A = -(Aᵀ * (Aᵀ⁻¹ * (J l R)⁻¹)) * A⁻¹ * A
state_after: no goals

tactic: rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ -(Aᵀ * (Aᵀ⁻¹ * (J l R)⁻¹)) * A⁻¹ * A = -(J l R)⁻¹
state_after: no goals

tactic: simp [J_inv]
state_before: l : Type u_1
R : Type u_2
inst✝² : DecidableEq l
inst✝¹ : Fintype l
inst✝ : CommRing R
A : Matrix (l ⊕ l) (l ⊕ l) R
hA : A * J l R * Aᵀ = J l R
huA : IsUnit A.det
huAT : IsUnit Aᵀ.det
⊢ -(J l R)⁻¹ = J l R
state_after: no goals