-- Mathlib/Algebra/Homology/TotalComplexSymmetry.lean
lemma flip_totalFlipIso : K.flip.totalFlipIso c = (K.totalFlipIso c).symm

Proof state:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
⊢ K.flip.totalFlipIso c = (K.totalFlipIso c).symm

Tactic:
ext j i₁ i₂ h

Result:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.flip.totalFlipIso c).hom.f j =
    K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).symm.hom.f j

Proof state:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j

Tactic:
rw [ιTotal_totalFlipIso_f_inv, ComplexShape.σ_symm]

Result:
no goals

Proof state:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j

Tactic:
dsimp only [flip_flip]

Result:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j

Proof state:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.flip.totalFlipIso c).hom.f j =
    K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).symm.hom.f j

Tactic:
rw [Iso.symm_hom, ιTotal_totalFlipIso_f_hom]

Result:
case w.h.h
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{u_5, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j