-- Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Preadditive.{v, u} C
K L : CochainComplex.{v, u, 0} C ℤ
n : ℤ
x y : Cocycle K L n
h :
  Eq (α :=
    Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
        (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    ((toHom :
          (a : CohomologyClass K L n) →
            (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
        (mk x) :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    ((toHom :
          (a : CohomologyClass K L n) →
            (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
        (mk y) :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
refine_1 : mk x - mk y = 0
refine_1₁ : mk (x - y) = 0
refine_1₂ : x - y ∈ coboundaries K L n
refine_1₃ :
  Eq (α :=
    Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
        (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    ((toHom :
          (a : CohomologyClass K L n) →
            (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
        (mk (x - y)) :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    (0 :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
refine_1₄ :
  Eq (α :=
    Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
        (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    ((toHom :
          (a : CohomologyClass K L n) →
            (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
        (mk x - mk y) :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    (0 :
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
refine_1₅ :
  HSub.hSub (α :=
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      (β :=
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      ((toHom :
            (a : CohomologyClass K L n) →
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
                (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                  ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
          (mk x) :
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
          Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
            (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      ((toHom :
            (a : CohomologyClass K L n) →
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
                (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                  ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
          (mk y) :
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
          Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
            (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L)) =
    0
refine_1₆ :
  HSub.hSub (α :=
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      (β :=
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
          (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      ((toHom :
            (a : CohomologyClass K L n) →
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
                (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                  ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
          (mk y) :
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
          Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
            (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
      ((toHom :
            (a : CohomologyClass K L n) →
              (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶
                (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj
                  ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) n).obj L))
          (mk y) :
        Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
          Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
            (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L)) =
    0
refine_1₇ refine_1₈ :
  Eq.{v + 1} (α :=
    Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ)) K ⟶
      Functor.obj (HomotopyCategory.quotient C (ComplexShape.up ℤ))
        (Functor.obj (CategoryTheory.shiftFunctor (HomologicalComplex.{v, u, 0} C (ComplexShape.up ℤ)) n) L))
    0 0
⊢ mk x = mk y

Tactic:
rw [← sub_eq_zero, ← mk_sub, mk_eq_zero_iff, ← toHom_mk_eq_zero_iff,
      mk_sub, map_sub, h, sub_self]

Premises used: map_sub, sub_self, CochainComplex.HomComplex.CohomologyClass.mk_eq_zero_iff, sub_eq_zero, CochainComplex.HomComplex.CohomologyClass.toHom_mk_eq_zero_iff, CochainComplex.HomComplex.CohomologyClass.mk_sub