Theorem: CategoryTheory.Functor.mapHomologicalComplex_upToQuasiIso_Q_inverts_quasiIso
File: Mathlib/Algebra/Homology/Localization.lean

tactic: apply Localization.inverts _ (HomologicalComplex.quasiIso D c)
state_before: C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_6, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : CategoryWithHomology C
inst✝⁴ : CategoryWithHomology D
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
K L : HomologicalComplex C c
f : K ⟶ L
hf : HomologicalComplex.quasiIso C c f
⊢ IsIso ((F.mapHomologicalComplex c ⋙ HomologicalComplexUpToQuasiIso.Q).map f)
state_after: case x
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_6, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : CategoryWithHomology C
inst✝⁴ : CategoryWithHomology D
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
K L : HomologicalComplex C c
f : K ⟶ L
hf : HomologicalComplex.quasiIso C c f
⊢ HomologicalComplex.quasiIso D c ((F.mapHomologicalComplex c).map f)

tactic: have : QuasiIso f := hf
state_before: case x
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_6, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : CategoryWithHomology C
inst✝⁴ : CategoryWithHomology D
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
K L : HomologicalComplex C c
f : K ⟶ L
hf : HomologicalComplex.quasiIso C c f
⊢ HomologicalComplex.quasiIso D c ((F.mapHomologicalComplex c).map f)
state_after: case x
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_6, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : CategoryWithHomology C
inst✝⁴ : CategoryWithHomology D
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
K L : HomologicalComplex C c
f : K ⟶ L
hf : HomologicalComplex.quasiIso C c f
this : QuasiIso f
⊢ HomologicalComplex.quasiIso D c ((F.mapHomologicalComplex c).map f)

tactic: apply HomologicalComplex.quasiIso_map_of_preservesHomology
state_before: case x
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_6, u_2} D
F : C ⥤ D
ι : Type u_3
c : ComplexShape ι
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : CategoryWithHomology C
inst✝⁴ : CategoryWithHomology D
inst✝³ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝² : (HomologicalComplex.quasiIso D c).HasLocalization
inst✝¹ : F.Additive
inst✝ : F.PreservesHomology
K L : HomologicalComplex C c
f : K ⟶ L
hf : HomologicalComplex.quasiIso C c f
this : QuasiIso f
⊢ HomologicalComplex.quasiIso D c ((F.mapHomologicalComplex c).map f)
state_after: no goals