Theorem: CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesRightHomology
File: Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ
state_after: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ
tactic: have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default
state_before: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ QuasiIso (F.mapShortComplex.map φ) ↔ QuasiIso φ
state_after: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH
tactic: rw [γ.quasiIso_iff, (γ.map F).quasiIso_iff, RightHomologyMapData.map_φH]
state_before: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) ↔ IsIso γ.φH
state_after: case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso (F.map γ.φH) → IsIso γ.φH

case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_3, u_1} C
inst✝¹⁰ : Category.{u_4, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : F.ReflectsIsomorphisms
γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData
⊢ IsIso γ.φH → IsIso (F.map γ.φH)
tactic: constructor