Theorem: quasiIsoAt_iff_exactAt
File: Mathlib/Algebra/Homology/QuasiIso.lean

tactic: simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff, exactAt_iff,
  ShortComplex.exact_iff_isZero_homology] at hK ⊢
state_before: ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : K.ExactAt i
⊢ QuasiIsoAt f i ↔ L.ExactAt i
state_after: ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) ↔ IsZero (L.sc i).homology

tactic: constructor
state_before: ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) ↔ IsZero (L.sc i).homology
state_after: case mp
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) → IsZero (L.sc i).homology

case mpr
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsZero (L.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))

tactic: intro h
state_before: case mp
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) → IsZero (L.sc i).homology
state_after: case mp
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
h : IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
⊢ IsZero (L.sc i).homology

tactic: exact IsZero.of_iso hK (@asIso _ _ _ _ _ h).symm
state_before: case mp
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
h : IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
⊢ IsZero (L.sc i).homology
state_after: no goals

tactic: intro hL
state_before: case mpr
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
⊢ IsZero (L.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
state_after: case mpr
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
hL : IsZero (L.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))

tactic: exact ⟨⟨0, IsZero.eq_of_src hK _ _, IsZero.eq_of_tgt hL _ _⟩⟩
state_before: case mpr
ι✝ : Type u_1
ι : Type u_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hK : IsZero (K.sc i).homology
hL : IsZero (L.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
state_after: no goals