-- Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
theorem mono_of_epi_of_epi_mono' (hR₁ : R₁.map' 0 2 = 0) (hR₁' : Epi (R₁.map' 1 2))
    (hR₂ : R₂.Exact) (h₀ : Epi (app' φ 0)) (h₁ : Mono (app' φ 1)) :
    Mono (app' φ 2)

Proof state:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
⊢ Mono (app' φ 2 ⋯)

Tactic:
let ψ : mk₃ (R₁.map' 0 1) (R₁.map' 1 2) (0 : _ ⟶ R₁.obj' 0) ⟶
  mk₃ (R₂.map' 0 1) (R₂.map' 1 2) (0 : _ ⟶ R₁.obj' 0) := homMk₃ (app' φ 0) (app' φ 1)
    (app' φ 2) (𝟙 _) (naturality' φ 0 1) (naturality' φ 1 2) (by simp)

Result:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ Mono (app' φ 2 ⋯)

Proof state:
case refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ (mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 0 2 ⋯ ⋯ = 0

Tactic:
dsimp

Result:
case refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ R₁.map (homOfLE ⋯) ≫ R₁.map (homOfLE ⋯) = 0

Proof state:
case refine_1
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ R₁.map (homOfLE ⋯ ≫ homOfLE ⋯) = 0

Tactic:
exact hR₁

Result:
no goals

Proof state:
case refine_2
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ Epi
    (ShortComplex.mk
        ((mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯)
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map'
          0 1 ⋯ ⋯)
        ((mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯)
              ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map'
          1 2 ⋯ ⋯)
        ⋯).f

Tactic:
exact hR₁'

Result:
no goals

Proof state:
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 ⋯ ⋯ = 0
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 ⋯)
h₁ : Mono (app' φ 1 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 :=
  homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯)
            ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map'
        0 1 ⋯ ⋯ ≫
      (mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯)
            ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map'
        1 2 ⋯ ⋯ =
    0

Tactic:
simp

Result:
no goals