state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
| Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X))) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
| Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫
(tensor_μ C (𝟙_ C, 𝟙_ C) (M.X, N.X) ≫ ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X)) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← tensor_comp, one_mul M, one_mul N] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
| Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, one_mul M, one_mul N] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, one_mul M, one_mul N] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.one ⊗ 𝟙 M.X) ⊗ N.one ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, one_mul M, one_mul N] | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (𝟙_ C, 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom) =
(λ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | symm | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (λ_ (M.X ⊗ N.X)).hom =
((λ_ (𝟙_ C)).inv ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (𝟙_ C, 𝟙_ C) (M.X, N.X) ≫ ((λ_ M.X).hom ⊗ (λ_ N.X).hom) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | exact tensor_left_unitality C M.X N.X | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.407_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_one_mul (M N : Mon_ C) :
((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(λ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
| Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ ((𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv) ≫ (𝟙 (M.X ⊗ N.X) ⊗ M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
| Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ M.one ⊗ N.one) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mul
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ M.one ⊗ N.one) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mul
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ M.one ⊗ N.one) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mul
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv) ≫
(tensor_μ C (M.X, N.X) (𝟙_ C, 𝟙_ C) ≫ ((𝟙 M.X ⊗ M.one) ⊗ 𝟙 N.X ⊗ N.one)) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← tensor_comp, mul_one M, mul_one N] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
| Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((𝟙 M.X ⊗ M.one) ⊗ 𝟙 N.X ⊗ N.one) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| 𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_one M, mul_one N] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((𝟙 M.X ⊗ M.one) ⊗ 𝟙 N.X ⊗ N.one) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| 𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_one M, mul_one N] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((𝟙 M.X ⊗ M.one) ⊗ 𝟙 N.X ⊗ N.one) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| 𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_one M, mul_one N] | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv) ≫ tensor_μ C (M.X, N.X) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ M.X).hom ⊗ (ρ_ N.X).hom) =
(ρ_ (M.X ⊗ N.X)).hom | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | symm | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (ρ_ (M.X ⊗ N.X)).hom =
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv) ≫ tensor_μ C (M.X, N.X) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ M.X).hom ⊗ (ρ_ N.X).hom) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | exact tensor_right_unitality C M.X N.X | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom := by
rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]
slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]
slice_lhs ... | Mathlib.CategoryTheory.Monoidal.Mon_.418_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_one (M N : Mon_ C) :
(𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(ρ_ (M.X ⊗ N.X)).hom | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tens... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X))) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_id, tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
(tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).1, ((M.X, N.X) ⊗ (M.X, N.X)).2) (M.X, N.X) ≫
((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X)) ≫
(M.mul ⊗ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X, N.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X, N.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| ((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X) ≫ (M.mul ⊗ N.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X, N.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).1, ((M.X, N.X) ⊗ (M.X, N.X)).2) (M.X, N.X) ≫
((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom) ≫ ((𝟙 M.X ⊗ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 3 => dsimp; rw [tensor_associativity] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).1, ((M.X, N.X) ⊗ (M.X, N.X)).2) (M.X, N.X) ≫
((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom)
case a.a... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp; rw [tensor_associativity] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).1, ((M.X, N.X) ⊗ (M.X, N.X)).2) (M.X, N.X) ≫
((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom)
case a.a... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp; rw [tensor_associativity] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).1, ((M.X, N.X) ⊗ (M.X, N.X)).2) (M.X, N.X) ≫
((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom)
case a.a... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X ⊗ M.X, N.X ⊗ N.X) (M.X, N.X) ≫ ((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [tensor_associativity] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (((α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X)) ≫
((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)) ≫
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [← tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X) ≫ ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X) ≫ ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X) ≫ ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_μ_natural] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫
(((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ (M.mul ⊗ N.mul) =
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_comp, tensor_id] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ ((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, tensor_id] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ ((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, tensor_id] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
| (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ ((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, tensor_id] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
((𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X)) ≫
(M.mul ⊗ N.mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc] | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib.CategoryTheory.Monoidal.Mon_.429_0.NTUMzhXPwXsmsYt | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫
... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [tensor_obj, prodMonoidal_tensorObj, Category.assoc] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← Category.id_comp P.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp P.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [associator_naturality] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| ((M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_naturality] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫
(α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =
((α_ M.X N.X P.X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 3 4 => rw [← Category.id_comp M.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (α_ M.X N.X P.X).hom ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (α_ M.X N.X P.X).hom ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul)
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| (α_ M.X N.X P.X).hom ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.mul, tensor_comp] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫
(α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =
((α_ M.X N.X P.X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 3 => rw [associator_monoidal] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ins... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_monoidal] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ins... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_monoidal] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
| tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ins... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [associator_monoidal] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N P : Mon_ C
⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X))) ≫
(M.mul ⊗ N.mul ⊗ P.mul) =
((α_ M.X N.X ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc] | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib.CategoryTheory.Monoidal.Mon_.445_0.NTUMzhXPwXsmsYt | theorem mul_associator {M N P : Mon_ C} :
(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
(tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
(α_ M.X N.X P.X).hom =
((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
(M... | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ (tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
| Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ (tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (M.X ⊗ M.X)) ≫ (𝟙 (𝟙_ C) ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [leftUnitor_naturality] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
| Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.mul) ≫ (λ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.mul) ≫ (λ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (𝟙 (𝟙_ C) ⊗ M.mul) ≫ (λ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [leftUnitor_naturality] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (M.X ⊗ M.X)) ≫ (λ_ (M.X ⊗ M.X)).hom ≫ M.mul =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 3 => rw [← leftUnitor_monoidal] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => rw [leftUnitor_naturality]
| Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (M.X ⊗ M.X)) ≫ (λ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_monoidal] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3... | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (M.X ⊗ M.X)) ≫ (λ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_monoidal] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3... | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (M.X ⊗ M.X)) ≫ (λ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← leftUnitor_monoidal] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3... | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ ((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul = ((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc, Category.id_comp] | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
slice_lhs 3 4 => rw [leftUnitor_naturality]
slice_lhs 1 3... | Mathlib.CategoryTheory.Monoidal.Mon_.460_0.NTUMzhXPwXsmsYt | theorem mul_leftUnitor {M : Mon_ C} :
(tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ (tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
| Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ (tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (𝟙 (M.X ⊗ M.X) ⊗ (λ_ (𝟙_ C)).hom) ≫ (M.mul ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 3 4 => rw [rightUnitor_naturality] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
| Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (M.mul ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (M.mul ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| (M.mul ⊗ 𝟙 (𝟙_ C)) ≫ (ρ_ M.X).hom
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C)
case a.a
C : Ty... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [rightUnitor_naturality] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (𝟙 (M.X ⊗ M.X) ⊗ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (M.X ⊗ M.X)).hom ≫ M.mul =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 1 3 => rw [← rightUnitor_monoidal] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => rw [rightUnitor_naturality]
| Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (𝟙 (M.X ⊗ M.X) ⊗ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← rightUnitor_monoidal] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => rw [rightUnitor_naturality]
slice_lhs 1... | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (𝟙 (M.X ⊗ M.X) ⊗ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← rightUnitor_monoidal] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => rw [rightUnitor_naturality]
slice_lhs 1... | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
| tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (𝟙 (M.X ⊗ M.X) ⊗ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (M.X ⊗ M.X)).hom
case a C : Type u₁ inst✝² : Category.{v₁, u₁} C inst✝¹ : MonoidalCategory C inst✝ : BraidedCategory C M : M... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← rightUnitor_monoidal] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => rw [rightUnitor_naturality]
slice_lhs 1... | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M : Mon_ C
⊢ ((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul = ((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ 𝟙 (M.X ⊗ M.X) ≫ M.mul | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc, Category.id_comp] | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
slice_lhs 3 4 => rw [rightUnitor_naturality]
slice_lhs 1... | Mathlib.CategoryTheory.Monoidal.Mon_.469_0.NTUMzhXPwXsmsYt | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ (tensorObj X₁ X₂).on... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ ((λ_ (𝟙_ C)).inv ≫ ... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_comp, Hom.one_hom f, Hom.one_hom g] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.one ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.one_hom f, Hom.one_hom g] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.one ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.one_hom f, Hom.one_hom g] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.one ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.one_hom f, Hom.one_hom g] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ (tensorObj X₁ X₂).mu... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | dsimp | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ (tensor_μ C (X₁.X, X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_rhs 1 2 => rw [tensor_μ_natural] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| ((f.hom ⊗ g.h... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [tensor_μ_natural] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| ((f.hom ⊗ g.h... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [tensor_μ_natural] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| ((f.hom ⊗ g.h... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [tensor_μ_natural] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ (tensor_μ C (X₁.X, X... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | slice_lhs 2 3 => rw [← tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.mul ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.mul ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
| (X₁.mul ⊗ X₂.... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | rw [← tensor_comp, Hom.mul_hom f, Hom.mul_hom g, tensor_comp] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
tensorObj : Mon_ C → Mon_ C → Mon_ C :=
fun M N => mk (M.X ⊗ N.X) ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))
X₁ Y₁ X₂ Y₂ : Mon_ C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
⊢ tensor_μ C (X₁.X, X₂... | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | simp only [Category.assoc] | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
let tensorObj (M N : Mon_ C) : Mon_ C :=
{ X := M.X ⊗ N.X
one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
one_mul := Mon_tensor_one_mul M N
mul_one := Mon_tensor_mul_one M N
... | Mathlib.CategoryTheory.Monoidal.Mon_.478_0.NTUMzhXPwXsmsYt | instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
⊢ ∀ (X₁ X₂ : Mon_ C), 𝟙 X₁ ⊗ 𝟙 X₂ = 𝟙 (X₁ ⊗ X₂) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | intros | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁✝ X₂✝ : Mon_ C
⊢ 𝟙 X₁✝ ⊗ 𝟙 X₂✝ = 𝟙 (X₁✝ ⊗ X₂✝) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.C... | ext | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
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