-- Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean
lemma RightHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.RightHomologyData}
    {h₂ : S₂.RightHomologyData} (γ : RightHomologyMapData φ h₁ h₂) :
    QuasiIso φ ↔ IsIso γ.φH

Proof state:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) → IsIso γ.φH

Tactic:
intro h

Result:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH

Proof state:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
this : IsIso (γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH

Tactic:
exact IsIso.of_isIso_comp_right _ (RightHomologyData.homologyIso h₂).inv

Result:
no goals

Proof state:
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH

Tactic:
constructor

Result:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) → IsIso γ.φH

case mpr
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH → IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)

Proof state:
case mpr
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso γ.φH
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)

Tactic:
infer_instance

Result:
no goals

Proof state:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH

Tactic:
haveI : IsIso (γ.φH ≫ (RightHomologyData.homologyIso h₂).inv) :=
  IsIso.of_isIso_comp_left (RightHomologyData.homologyIso h₁).hom _

Result:
case mp
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
this : IsIso (γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH

Proof state:
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ QuasiIso φ ↔ IsIso γ.φH

Tactic:
rw [ShortComplex.quasiIso_iff, γ.homologyMap_eq]

Result:
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH

Proof state:
case mpr
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH → IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)

Tactic:
intro h

Result:
case mpr
C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso γ.φH
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)