Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided

Goal state before:
pp: C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
this :
  ((unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z)) ≫ (H ⊛ F).map (β_ z x).inv =
    ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x) ≫ (H ⊛ F).map (β_ z x).hom) ≫ (H ⊛ F).map (β_ z x).inv
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)

Tactic:
simp only [Category.assoc, Iso.map_hom_inv_id, Category.comp_id] at this

Goal state after:
C : Type u₁
inst✝¹⁸ : Category.{v₁, u₁} C
V : Type u₂
inst✝¹⁷ : Category.{v₂, u₂} V
inst✝¹⁶ : MonoidalCategory C
inst✝¹⁵ : BraidedCategory C
inst✝¹⁴ : MonoidalCategory V
inst✝¹³ : BraidedCategory V
F G : C ⥤ V
inst✝¹² : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorLeft v)
inst✝¹¹ : ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow (tensor C) d) (tensorRight v)
H : C ⥤ V
inst✝¹⁰ : DayConvolution F G
inst✝⁹ : DayConvolution G H
inst✝⁸ : DayConvolution F (G ⊛ H)
inst✝⁷ : DayConvolution (F ⊛ G) H
inst✝⁶ : DayConvolution H F
inst✝⁵ : DayConvolution G (H ⊛ F)
inst✝⁴ : DayConvolution (G ⊛ H) F
inst✝³ : DayConvolution G F
inst✝² : DayConvolution (G ⊛ F) H
inst✝¹ : DayConvolution F H
inst✝ : DayConvolution G (F ⊛ H)
z x y : C
this :
  (unit F H).app (x, z) ≫ (braiding F H).hom.app (x ⊗ z) ≫ (H ⊛ F).map (β_ z x).inv =
    (β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)
⊢ (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ ((β_ (F.obj x) (H.obj z)).hom ≫ (unit H F).app (z, x)) ≫
          (unit G (H ⊛ F)).app (y, z ⊗ x) ≫ (G ⊛ H ⊛ F).map ((α_ y z x).inv ≫ (β_ (y ⊗ z) x).hom ≫ (α_ x y z).inv) =
    (β_ (F.obj x) (G.obj y)).hom ▷ H.obj z ≫
      (α_ (G.obj y) (F.obj x) (H.obj z)).hom ≫
        G.obj y ◁ (unit F H).app (x, z) ≫
          G.obj y ◁ (braiding F H).hom.app (x ⊗ z) ≫
            (unit G (H ⊛ F)).app (y, x ⊗ z) ≫ (G ⊛ H ⊛ F).map ((α_ y x z).inv ≫ (β_ y x).hom ▷ z)