-- Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
R₁ R₂ : ComposableArrows.{v_1, u_1} C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₀ : Epi (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
    R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 0 1 ⋯ ⋯ : R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
    (R₁.map' 1 2 ⋯ ⋯ :
      R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) ⟶
  mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj ⟨0, ⋯⟩) (X₂ := R₂.obj ⟨1, ⋯⟩) (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj ⟨0, ⋯⟩)
    (R₂.map' 0 1 ⋯ ⋯ : R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨1, ⋯⟩)
    (R₂.map' 1 2 ⋯ ⋯ : R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj ⟨2, ⋯⟩) :=
  homMk₃
    (𝟙
      ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
            R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (0 : R₁.obj' 0 ⋯ ⟶ R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 0 1 ⋯ ⋯ :
              R₁.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
            (R₁.map' 1 2 ⋯ ⋯ :
              R₁.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₁.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).obj'
        0 _proof_297✝))
    (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯
refine_1 :
  Mono
    {
        X₁ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨0, ⋯⟩,
        X₂ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨1, ⋯⟩,
        X₃ :=
          (mk₂
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                  mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                  mono_of_epi_of_mono_of_mono'._proof_4)).obj
            ⟨2, ⋯⟩,
        f :=
          (mk₂
                ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 0 1 ⋯ ⋯ :
                        R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 1 2 ⋯ ⋯ :
                        R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                          R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                  0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 0 1 ⋯ ⋯ :
                        R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 1 2 ⋯ ⋯ :
                        R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                          R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                  1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).map'
            0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
        g :=
          (mk₂
                ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 0 1 ⋯ ⋯ :
                        R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 1 2 ⋯ ⋯ :
                        R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                          R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                  0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
                ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 0 1 ⋯ ⋯ :
                        R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                      (R₂.map' 1 2 ⋯ ⋯ :
                        R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                          R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                  1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).map'
            1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
        zero := ⋯ : ShortComplex.{v_1, u_1} C }.g
⊢ {
      X₁ :=
        (mk₂
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                mono_of_epi_of_mono_of_mono'._proof_11)
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)).obj
          ⟨0, ⋯⟩,
      X₂ :=
        (mk₂
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                mono_of_epi_of_mono_of_mono'._proof_11)
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)).obj
          ⟨1, ⋯⟩,
      X₃ :=
        (mk₂
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9
                mono_of_epi_of_mono_of_mono'._proof_11)
              ((mk₃ 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6
                mono_of_epi_of_mono_of_mono'._proof_4)).obj
          ⟨2, ⋯⟩,
      f :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                    R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 0 1 ⋯ ⋯ :
                      R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 1 2 ⋯ ⋯ :
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                        R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
              ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                    R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 0 1 ⋯ ⋯ :
                      R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 1 2 ⋯ ⋯ :
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                        R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).map'
          0 1 isComplex₂_iff._proof_2 isComplex₂_iff._proof_4,
      g :=
        (mk₂
              ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                    R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 0 1 ⋯ ⋯ :
                      R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 1 2 ⋯ ⋯ :
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                        R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
              ((mk₃ (X₀ := R₁.obj' 0 ⋯) (X₁ := R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1))) (X₂ :=
                    R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (0 : R₁.obj' 0 ⋯ ⟶ R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 0 1 ⋯ ⋯ :
                      R₂.obj (⟨0, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶ R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))
                    (R₂.map' 1 2 ⋯ ⋯ :
                      R₂.obj (⟨1, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)) ⟶
                        R₂.obj (⟨2, ⋯⟩ : Fin (HAdd.hAdd (α := ℕ) 2 1)))).map'
                1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).map'
          1 2 isComplex₂_iff._proof_4 isComplex₂_iff._proof_5,
      zero := ⋯ : ShortComplex.{v_1, u_1} C }.Exact

Tactic:
rw [ShortComplex.exact_iff_mono _ (by simp)]

Premises used: CategoryTheory.ShortComplex.exact_iff_mono