Theorem: CategoryTheory.IsFiltered.coeq₃_condition₂
File: Mathlib/CategoryTheory/Filtered/Basic.lean

Proof steps:
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫ coeq₃Hom f g h = h ≫ coeq₃Hom f g h
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
tactic: dsimp [coeq₃Hom]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
tactic: slice_lhs 2 4 => rw [← Category.assoc, coeq_condition _ _]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      coeqHom f g ≫
        leftToMax (coeq f g) (coeq g h) ≫
          coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
tactic: slice_rhs 2 4 => rw [← Category.assoc, coeq_condition _ _]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ g ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
state_after: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ ((h ≫ coeqHom g h) ≫ rightToMax (coeq f g) (coeq g h)) ≫
      coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
tactic: slice_lhs 1 3 => rw [← Category.assoc, coeq_condition _ _]
state_before: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : IsFilteredOrEmpty C
j₁ j₂ : C
f g h : j₁ ⟶ j₂
⊢ ((h ≫ coeqHom g h) ≫ rightToMax (coeq f g) (coeq g h)) ≫
      coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) =
    h ≫
      (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h)) ≫
        coeqHom (coeqHom f g ≫ leftToMax (coeq f g) (coeq g h)) (coeqHom g h ≫ rightToMax (coeq f g) (coeq g h))
state_after: no goals
tactic: simp only [Category.assoc]