Theorem: Bimod.LeftUnitorBimod.hom_right_act_hom'
File: Mathlib/CategoryTheory/Monoidal/Bimod.lean

Proof steps:
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((regular R).tensorBimod P).actRight ≫ hom P = hom P ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ TensorBimod.actRight (regular R) P ≫ hom P = hom P ▷ S.X ≫ P.actRight
tactic: dsimp
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ TensorBimod.actRight (regular R) P ≫ hom P = hom P ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
        colimMap
          (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
            ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
            ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
              (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
            ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
      coequalizer.desc P.actLeft ⋯ =
    coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
tactic: dsimp [hom, TensorBimod.actRight, regular]
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
        colimMap
          (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
            ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
            ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
              (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
            ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
      coequalizer.desc P.actLeft ⋯ =
    coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
tactic: refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    (tensorRight S.X).map
        (coequalizer.π
          ({ X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷
            P.X)
          ((α_
                { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                    right_assoc := ⋯, middle_assoc := ⋯ }.X
                R.X P.X).hom ≫
            { X := R.X, actLeft := R.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := R.mul, actRight_one := ⋯,
                  right_assoc := ⋯, middle_assoc := ⋯ }.X ◁
              P.actLeft)) ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
tactic: dsimp
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      ((PreservesCoequalizer.iso (tensorRight S.X) (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)).inv ≫
          colimMap
            (parallelPairHom (R.mul ▷ P.X ▷ S.X) (((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X) (R.mul ▷ P.X)
              ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft)
              ((α_ (R.X ⊗ R.X) P.X S.X).hom ≫
                (α_ R.X R.X (P.X ⊗ S.X)).hom ≫ R.X ◁ R.X ◁ P.actRight ≫ (α_ R.X R.X P.X).inv)
              ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ⋯ ⋯)) ≫
        coequalizer.desc P.actLeft ⋯ =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
tactic: slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft =
    coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ▷ S.X ≫
      coequalizer.desc P.actLeft ⋯ ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight
tactic: slice_rhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft = P.actLeft ▷ S.X ≫ P.actRight
state_after: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft = (α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight ≫ P.actLeft
tactic: slice_rhs 1 2 => rw [middle_assoc]
state_before: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight) ≫ P.actLeft = (α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight ≫ P.actLeft
state_after: no goals
tactic: simp only [Category.assoc]