Theorem: CochainComplex.mappingCone.quasiIso_descShortComplex
File: Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean

tactic: rw [quasiIsoAt_iff_isIso_homologyMap]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ QuasiIsoAt (descShortComplex S) n
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)

tactic: let φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅
    (triangleh S.f) n _ rfl).δlast ⟶ (composableArrows₅ hS n _ rfl).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.app _)
    ((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)
    (by
      erw [(homologyFunctorFactors C (up ℤ) n).hom.naturality_assoc]
      dsimp
      rw [← HomologicalComplex.homologyMap_comp, inr_descShortComplex])
    (by
      dsimp
      erw [homologySequenceδ_triangleh hS]
      simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj, assoc,
        Iso.inv_hom_id_app, comp_id])
    ((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)

tactic: have : IsIso ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
    HomologicalComplex.homologyMap (descShortComplex S) n) := by
  apply Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
    ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅_exact _
      (mappingCone_triangleh_distinguished S.f) n _ rfl).δlast
    (composableArrows₅_exact hS n _ rfl).δlast φ
  all_goals dsimp [φ]; infer_instance
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
this :
  IsIso
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)

tactic: apply IsIso.of_isIso_comp_left ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f))
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
this :
  IsIso
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)
state_after: no goals

tactic: erw [(homologyFunctorFactors C (up ℤ) n).hom.naturality_assoc]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast.map' 1 2 ⋯ ⋯ ≫
      (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂ ≫
      (composableArrows₅ hS n (n + 1) ⋯).δlast.map' 1 2 ⋯ ⋯
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂ ≫
      (HomologicalComplex.homologyFunctor C (up ℤ) n).map (triangle S.f).mor₂ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂ ≫
      (composableArrows₅ hS n (n + 1) ⋯).δlast.map' 1 2 ⋯ ⋯

tactic: dsimp
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂ ≫
      (HomologicalComplex.homologyFunctor C (up ℤ) n).map (triangle S.f).mor₂ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂ ≫
      (composableArrows₅ hS n (n + 1) ⋯).δlast.map' 1 2 ⋯ ⋯
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫
      HomologicalComplex.homologyMap (inr S.f) n ≫ HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫ HomologicalComplex.homologyMap S.g n

tactic: rw [← HomologicalComplex.homologyMap_comp, inr_descShortComplex]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫
      HomologicalComplex.homologyMap (inr S.f) n ≫ HomologicalComplex.homologyMap (descShortComplex S) n =
    (homologyFunctorFactors C (up ℤ) n).hom.app S.X₂ ≫ HomologicalComplex.homologyMap S.g n
state_after: no goals

tactic: dsimp
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast.map' 2 3 ⋯ ⋯ ≫
      (homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁ =
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
        HomologicalComplex.homologyMap (descShortComplex S) n) ≫
      (composableArrows₅ hS n (n + 1) ⋯).δlast.map' 2 3 ⋯ ⋯
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctor C (up ℤ) 0).homologySequenceδ
        (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
          ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁))
        n (n + 1) ⋯ ≫
      (homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₁ =
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n) ≫
      hS.δ n (n + 1) ⋯

tactic: erw [homologySequenceδ_triangleh hS]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ (homologyFunctor C (up ℤ) 0).homologySequenceδ
        (Triangle.mk ((quotient C (up ℤ)).map S.f) ((quotient C (up ℤ)).map (inr S.f))
          ((quotient C (up ℤ)).map (triangle S.f).mor₃ ≫ ((quotient C (up ℤ)).commShiftIso 1).hom.app S.X₁))
        n (n + 1) ⋯ ≫
      (homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₁ =
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n) ≫
      hS.δ n (n + 1) ⋯
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n ≫
          hS.δ n (n + 1) ⋯ ≫ (homologyFunctorFactors C (up ℤ) (n + 1)).inv.app S.X₁) ≫
      (homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₁ =
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n) ≫
      hS.δ n (n + 1) ⋯

tactic: simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj, assoc,
  Iso.inv_hom_id_app, comp_id]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
⊢ ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n ≫
          hS.δ n (n + 1) ⋯ ≫ (homologyFunctorFactors C (up ℤ) (n + 1)).inv.app S.X₁) ≫
      (homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₁ =
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
        HomologicalComplex.homologyMap (descShortComplex S) n) ≫
      hS.δ n (n + 1) ⋯
state_after: no goals

tactic: apply Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
  ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅_exact _
    (mappingCone_triangleh_distinguished S.f) n _ rfl).δlast
  (composableArrows₅_exact hS n _ rfl).δlast φ
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
state_after: case h₀
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Epi (app' φ 0 ⋯)

case h₁
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 1 ⋯)

case h₂
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 3 ⋯)

case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono (app' φ 4 ⋯)

tactic: all_goals dsimp [φ]; infer_instance
state_before: case h₀
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Epi (app' φ 0 ⋯)

case h₁
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 1 ⋯)

case h₂
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ IsIso (app' φ 3 ⋯)

case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono (app' φ 4 ⋯)
state_after: no goals

tactic: dsimp [φ]
state_before: case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono (app' φ 4 ⋯)
state_after: case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₂)

tactic: infer_instance
state_before: case h₃
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
⊢ Mono ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app S.X₂)
state_after: no goals