-- Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
theorem rightUnitor_monoidal (X₁ X₂ : C) :
    (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
      tensorμ X₁ (𝟙_ C) X₂ (𝟙_ C) ≫ ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom = tensorμ X₁ (𝟙_ C) X₂ (𝟙_ C) ≫ (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
dsimp only [tensorμ]

Result:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom

Tactic:
monoidal

Result:
no goals

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
this :
  (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
clear this

Result:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
have :
  (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
      (X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv) ≫ (X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom) ≫ (X₁ ◁ (λ_ X₂).hom) := by
  monoidal

Result:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
this :
  (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
this :
  (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom
⊢ (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
rw [this]

Result:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
this :
  (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
    (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
      X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ ((β_ (𝟙_ C) X₂).hom ≫ (ρ_ X₂).hom) =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
monoidal

Result:
no goals

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom

Tactic:
rw [← braiding_rightUnitor]

Result:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
      X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ ((β_ (𝟙_ C) X₂).hom ≫ (ρ_ X₂).hom) =
    ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
        X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫
          X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫
      (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom