-- Mathlib/CategoryTheory/Enriched/FunctorCategory.lean
@[reassoc]
lemma enriched_assoc [HasEnrichedHom V F₁ F₂] [HasEnrichedHom V F₁ F₃] [HasEnrichedHom V F₁ F₄]
    [HasEnrichedHom V F₂ F₃] [HasEnrichedHom V F₂ F₄] [HasEnrichedHom V F₃ F₄] :
    (α_ (enrichedHom V F₁ F₂) (enrichedHom V F₂ F₃) (enrichedHom V F₃ F₄)).inv ≫
      enrichedComp V F₁ F₂ F₃ ▷ enrichedHom V F₃ F₄ ≫ enrichedComp V F₁ F₃ F₄ =
      enrichedHom V F₁ F₂ ◁ enrichedComp V F₂ F₃ F₄ ≫ enrichedComp V F₁ F₂ F₄

Proof state:
case h
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
j : J
⊢ (end_.π (diagram V F₁ F₂) j ⊗ end_.π (diagram V F₂ F₃) j ⊗ end_.π (diagram V F₃ F₄) j) ≫
      (α_ (((diagram V F₁ F₂).obj (Opposite.op j)).obj j) (((diagram V F₂ F₃).obj (Opposite.op j)).obj j)
            (((diagram V F₃ F₄).obj (Opposite.op j)).obj j)).inv ≫
        eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₂.obj j) (F₃.obj j) ▷
            ((diagram V F₃ F₄).obj (Opposite.op j)).obj j ≫
          eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₃.obj j) (F₄.obj j) =
    (end_.π (diagram V F₁ F₂) j ⊗ end_.π (diagram V F₂ F₃) j ⊗ end_.π (diagram V F₃ F₄) j) ≫
      ((diagram V F₁ F₂).obj (Opposite.op j)).obj j ◁
          eComp V (Opposite.unop (F₂.op.obj (Opposite.op j))) (F₃.obj j) (F₄.obj j) ≫
        eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₂.obj j) (F₄.obj j)

Tactic:
dsimp

Result:
case h
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
j : J
⊢ (end_.π (diagram V F₁ F₂) j ⊗ end_.π (diagram V F₂ F₃) j ⊗ end_.π (diagram V F₃ F₄) j) ≫
      (α_ (EnrichedCategory.Hom (F₁.obj j) (F₂.obj j)) (EnrichedCategory.Hom (F₂.obj j) (F₃.obj j))
            (EnrichedCategory.Hom (F₃.obj j) (F₄.obj j))).inv ≫
        eComp V (F₁.obj j) (F₂.obj j) (F₃.obj j) ▷ EnrichedCategory.Hom (F₃.obj j) (F₄.obj j) ≫
          eComp V (F₁.obj j) (F₃.obj j) (F₄.obj j) =
    (end_.π (diagram V F₁ F₂) j ⊗ end_.π (diagram V F₂ F₃) j ⊗ end_.π (diagram V F₃ F₄) j) ≫
      EnrichedCategory.Hom (F₁.obj j) (F₂.obj j) ◁ eComp V (F₂.obj j) (F₃.obj j) (F₄.obj j) ≫
        eComp V (F₁.obj j) (F₂.obj j) (F₄.obj j)

Proof state:
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
⊢ (α_ (enrichedHom V F₁ F₂) (enrichedHom V F₂ F₃) (enrichedHom V F₃ F₄)).inv ≫
      enrichedComp V F₁ F₂ F₃ ▷ enrichedHom V F₃ F₄ ≫ enrichedComp V F₁ F₃ F₄ =
    enrichedHom V F₁ F₂ ◁ enrichedComp V F₂ F₃ F₄ ≫ enrichedComp V F₁ F₂ F₄

Tactic:
ext j

Result:
case h
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
j : J
⊢ ((α_ (enrichedHom V F₁ F₂) (enrichedHom V F₂ F₃) (enrichedHom V F₃ F₄)).inv ≫
        enrichedComp V F₁ F₂ F₃ ▷ enrichedHom V F₃ F₄ ≫ enrichedComp V F₁ F₃ F₄) ≫
      end_.π (diagram V F₁ F₄) j =
    (enrichedHom V F₁ F₂ ◁ enrichedComp V F₂ F₃ F₄ ≫ enrichedComp V F₁ F₂ F₄) ≫ end_.π (diagram V F₁ F₄) j

Proof state:
case h
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
j : J
⊢ ((α_ (enrichedHom V F₁ F₂) (enrichedHom V F₂ F₃) (enrichedHom V F₃ F₄)).inv ≫
        enrichedComp V F₁ F₂ F₃ ▷ enrichedHom V F₃ F₄ ≫ enrichedComp V F₁ F₃ F₄) ≫
      end_.π (diagram V F₁ F₄) j =
    (enrichedHom V F₁ F₂ ◁ enrichedComp V F₂ F₃ F₄ ≫ enrichedComp V F₁ F₂ F₄) ≫ end_.π (diagram V F₁ F₄) j

Tactic:
conv_lhs =>
  rw [assoc, assoc, enrichedComp_π,
    tensorHom_def_assoc, ← comp_whiskerRight_assoc, enrichedComp_π,
    comp_whiskerRight_assoc, ← whisker_exchange_assoc,
    ← whisker_exchange_assoc, ← tensorHom_def'_assoc, ← associator_inv_naturality_assoc]

Result:
case h
V : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} V
inst✝⁹ : MonoidalCategory V
C : Type u₂
inst✝⁸ : Category.{v₂, u₂} C
J : Type u₃
inst✝⁷ : Category.{v₃, u₃} J
inst✝⁶ : EnrichedOrdinaryCategory V C
F₁ F₂ F₃ F₄ : J ⥤ C
inst✝⁵ : HasEnrichedHom V F₁ F₂
inst✝⁴ : HasEnrichedHom V F₁ F₃
inst✝³ : HasEnrichedHom V F₁ F₄
inst✝² : HasEnrichedHom V F₂ F₃
inst✝¹ : HasEnrichedHom V F₂ F₄
inst✝ : HasEnrichedHom V F₃ F₄
j : J
⊢ (end_.π (diagram V F₁ F₂) j ⊗ end_.π (diagram V F₂ F₃) j ⊗ end_.π (diagram V F₃ F₄) j) ≫
      (α_ (((diagram V F₁ F₂).obj (Opposite.op j)).obj j) (((diagram V F₂ F₃).obj (Opposite.op j)).obj j)
            (((diagram V F₃ F₄).obj (Opposite.op j)).obj j)).inv ≫
        eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₂.obj j) (F₃.obj j) ▷
            ((diagram V F₃ F₄).obj (Opposite.op j)).obj j ≫
          eComp V (Opposite.unop (F₁.op.obj (Opposite.op j))) (F₃.obj j) (F₄.obj j) =
    (enrichedHom V F₁ F₂ ◁ enrichedComp V F₂ F₃ F₄ ≫ enrichedComp V F₁ F₂ F₄) ≫ end_.π (diagram V F₁ F₄) j