Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: case comm₁₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            f :=
              (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            f :=
              (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            zero := ⋯ }.g)
x✝ : Ext S.X₁ Y n₀
⊢ ((ConcreteCategory.hom
            {
                X₁ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
                X₂ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
                X₃ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
                f :=
                  (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
                g :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
                zero := ⋯ }.f).comp
        ↑homAddEquiv)
      x✝ =
    ((↑homAddEquiv).comp
        (ConcreteCategory.hom
          { X₁ := AddCommGrpCat.of (Ext S.X₁ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₃ Y n₁),
              X₃ := AddCommGrpCat.of (Ext S.X₂ Y n₁), f := AddCommGrpCat.ofHom (hS.extClass.precomp Y h),
              g := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), zero := ⋯ }.f))
      x✝

Tactic:
;

Goal state after:
case comm₁₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
  Function.Exact
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            f :=
              (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            zero := ⋯ }.f)
    ⇑(ConcreteCategory.hom
        {
            X₁ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
            X₂ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
            X₃ :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
            f :=
              (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
            g :=
              ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
            zero := ⋯ }.g)
x✝ : Ext S.X₁ Y n₀
⊢ ((ConcreteCategory.hom
            {
                X₁ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₀).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₃,
                X₂ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₁,
                X₃ :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).obj
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).obj₂,
                f :=
                  (preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).homologySequenceδ
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)) ↑n₀ ↑n₁ ⋯,
                g :=
                  ((preadditiveYoneda.obj ((singleFunctor C 0).obj Y)).shift ↑n₁).map
                    ((triangleOpEquivalence (DerivedCategory C)).functor.obj (op hS.singleTriangle)).mor₁,
                zero := ⋯ }.f).comp
        ↑homAddEquiv)
      x✝ =
    ((↑homAddEquiv).comp
        (ConcreteCategory.hom
          { X₁ := AddCommGrpCat.of (Ext S.X₁ Y n₀), X₂ := AddCommGrpCat.of (Ext S.X₃ Y n₁),
              X₃ := AddCommGrpCat.of (Ext S.X₂ Y n₁), f := AddCommGrpCat.ofHom (hS.extClass.precomp Y h),
              g := AddCommGrpCat.ofHom ((mk₀ S.g).precomp Y ⋯), zero := ⋯ }.f))
      x✝