Theorem: HomotopyEquiv.toQuasiIso'_inv
File: Mathlib/Algebra/Homology/QuasiIso.lean

tactic: symm
state_before: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : HasZeroMorphisms V
inst✝¹¹ : HasZeroObject V
inst✝¹⁰ : HasEqualizers V
inst✝⁹ : HasImages V
inst✝⁸ : HasImageMaps V
inst✝⁷ : HasCokernels V
c : ComplexShape ι
C✝ D✝ E : HomologicalComplex V c
W : Type u_2
inst✝⁶ : Category.{u_3, u_2} W
inst✝⁵ : Preadditive W
inst✝⁴ : HasCokernels W
inst✝³ : HasImages W
inst✝² : HasEqualizers W
inst✝¹ : HasZeroObject W
inst✝ : HasImageMaps W
C D : HomologicalComplex W c
e : HomotopyEquiv C D
i : ι
⊢ (asIso ((homology'Functor W c i).map e.hom)).inv = (homology'Functor W c i).map e.inv
state_after: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : HasZeroMorphisms V
inst✝¹¹ : HasZeroObject V
inst✝¹⁰ : HasEqualizers V
inst✝⁹ : HasImages V
inst✝⁸ : HasImageMaps V
inst✝⁷ : HasCokernels V
c : ComplexShape ι
C✝ D✝ E : HomologicalComplex V c
W : Type u_2
inst✝⁶ : Category.{u_3, u_2} W
inst✝⁵ : Preadditive W
inst✝⁴ : HasCokernels W
inst✝³ : HasImages W
inst✝² : HasEqualizers W
inst✝¹ : HasZeroObject W
inst✝ : HasImageMaps W
C D : HomologicalComplex W c
e : HomotopyEquiv C D
i : ι
⊢ (homology'Functor W c i).map e.inv = (asIso ((homology'Functor W c i).map e.hom)).inv

tactic: haveI := e.toQuasiIso'.1 i
state_before: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : HasZeroMorphisms V
inst✝¹¹ : HasZeroObject V
inst✝¹⁰ : HasEqualizers V
inst✝⁹ : HasImages V
inst✝⁸ : HasImageMaps V
inst✝⁷ : HasCokernels V
c : ComplexShape ι
C✝ D✝ E : HomologicalComplex V c
W : Type u_2
inst✝⁶ : Category.{u_3, u_2} W
inst✝⁵ : Preadditive W
inst✝⁴ : HasCokernels W
inst✝³ : HasImages W
inst✝² : HasEqualizers W
inst✝¹ : HasZeroObject W
inst✝ : HasImageMaps W
C D : HomologicalComplex W c
e : HomotopyEquiv C D
i : ι
⊢ (homology'Functor W c i).map e.inv = (asIso ((homology'Functor W c i).map e.hom)).inv
state_after: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : HasZeroMorphisms V
inst✝¹¹ : HasZeroObject V
inst✝¹⁰ : HasEqualizers V
inst✝⁹ : HasImages V
inst✝⁸ : HasImageMaps V
inst✝⁷ : HasCokernels V
c : ComplexShape ι
C✝ D✝ E : HomologicalComplex V c
W : Type u_2
inst✝⁶ : Category.{u_3, u_2} W
inst✝⁵ : Preadditive W
inst✝⁴ : HasCokernels W
inst✝³ : HasImages W
inst✝² : HasEqualizers W
inst✝¹ : HasZeroObject W
inst✝ : HasImageMaps W
C D : HomologicalComplex W c
e : HomotopyEquiv C D
i : ι
this : IsIso ((homology'Functor W c i).map e.hom)
⊢ (homology'Functor W c i).map e.inv = (asIso ((homology'Functor W c i).map e.hom)).inv

tactic: simp only [← Iso.hom_comp_eq_id, asIso_hom, ← Functor.map_comp,
  ← (homology'Functor W c i).map_id, homology'_map_eq_of_homotopy e.homotopyHomInvId _]
state_before: ι : Type u_1
V : Type u
inst✝¹³ : Category.{v, u} V
inst✝¹² : HasZeroMorphisms V
inst✝¹¹ : HasZeroObject V
inst✝¹⁰ : HasEqualizers V
inst✝⁹ : HasImages V
inst✝⁸ : HasImageMaps V
inst✝⁷ : HasCokernels V
c : ComplexShape ι
C✝ D✝ E : HomologicalComplex V c
W : Type u_2
inst✝⁶ : Category.{u_3, u_2} W
inst✝⁵ : Preadditive W
inst✝⁴ : HasCokernels W
inst✝³ : HasImages W
inst✝² : HasEqualizers W
inst✝¹ : HasZeroObject W
inst✝ : HasImageMaps W
C D : HomologicalComplex W c
e : HomotopyEquiv C D
i : ι
this : IsIso ((homology'Functor W c i).map e.hom)
⊢ (homology'Functor W c i).map e.inv = (asIso ((homology'Functor W c i).map e.hom)).inv
state_after: no goals