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 |
|---|---|---|---|---|---|---|
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁✝ X₂✝ : Mon_ C
⊢ (𝟙 X₁✝ ⊗ 𝟙 X₂✝).hom = (𝟙 (X₁✝ ⊗ X₂✝)).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... | apply tensor_id | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; | 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 Y₁ Z₁ X₂ Y₂ Z₂ : Mon_ C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂),
f₁ ≫ g₁ ⊗ f₂ ≫ g₂ = (f₁ ⊗ f₂) ≫ (g₁ ⊗ g₂) | /-
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 intros; ext; apply tensor_id)
(tensor_comp := 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✝ Y₁✝ Z₁✝ X₂✝ Y₂✝ Z₂✝ : Mon_ C
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
g₁✝ : Y₁✝ ⟶ Z₁✝
g₂✝ : Y₂✝ ⟶ Z₂✝
⊢ f₁✝ ≫ g₁✝ ⊗ f₂✝ ≫ g₂✝ = (f₁✝ ⊗ f₂✝) ≫ (g₁✝ ⊗ g₂✝) | /-
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; ext; apply tensor_id)
(tensor_comp := by intros; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁✝ Y₁✝ Z₁✝ X₂✝ Y₂✝ Z₂✝ : Mon_ C
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
g₁✝ : Y₁✝ ⟶ Z₁✝
g₂✝ : Y₂✝ ⟶ Z₂✝
⊢ (f₁✝ ≫ g₁✝ ⊗ f₂✝ ≫ g₂✝).hom = ((f₁✝ ⊗ f₂✝) ≫ (g₁✝ ⊗ g₂✝)).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... | apply tensor_comp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; | 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 X₃ Y₁ Y₂ Y₃ : Mon_ C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃),
((f₁ ⊗ f₂) ⊗ f₃) ≫ (α_ Y₁ Y₂ Y₃).hom = (α_ X₁ X₂ X₃).hom ≫ (f₁ ⊗ f₂ ⊗ f₃) | /-
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 intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := 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✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : Mon_ C
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
f₃✝ : X₃✝ ⟶ Y₃✝
⊢ ((f₁✝ ⊗ f₂✝) ⊗ f₃✝) ≫ (α_ Y₁✝ Y₂✝ Y₃✝).hom = (α_ X₁✝ X₂✝ X₃✝).hom ≫ (f₁✝ ⊗ f₂✝ ⊗ f₃✝) | /-
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; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : Mon_ C
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
f₃✝ : X₃✝ ⟶ Y₃✝
⊢ (((f₁✝ ⊗ f₂✝) ⊗ f₃✝) ≫ (α_ Y₁✝ Y₂✝ Y₃✝).hom).hom = ((α_ X₁✝ X₂✝ X₃✝).hom ≫ (f₁✝ ⊗ f₂✝ ⊗ f₃✝)).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... | dsimp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : Mon_ C
f₁✝ : X₁✝ ⟶ Y₁✝
f₂✝ : X₂✝ ⟶ Y₂✝
f₃✝ : X₃✝ ⟶ Y₃✝
⊢ ((f₁✝ ⊗ f₂✝) ⊗ f₃✝).hom ≫ (α_ Y₁✝ Y₂✝ Y₃✝).hom.hom = (α_ X₁✝ X₂✝ X₃✝).hom.hom ≫ (f₁✝ ⊗ f₂✝ ⊗ f₃✝).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... | apply associator_naturality | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; | 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
⊢ ∀ {X Y : Mon_ C} (f : X ⟶ Y), (𝟙 (𝟙_ (Mon_ C)) ⊗ f) ≫ (λ_ Y).hom = (λ_ X).hom ≫ f | /-
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 intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := 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
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ (𝟙 (𝟙_ (Mon_ C)) ⊗ f✝) ≫ (λ_ Y✝).hom = (λ_ X✝).hom ≫ f✝ | /-
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; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ ((𝟙 (𝟙_ (Mon_ C)) ⊗ f✝) ≫ (λ_ Y✝).hom).hom = ((λ_ X✝).hom ≫ f✝).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... | dsimp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ (𝟙 (𝟙_ (Mon_ C)) ⊗ f✝).hom ≫ (λ_ Y✝).hom.hom = (λ_ X✝).hom.hom ≫ f✝.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... | apply leftUnitor_naturality | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; | 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
⊢ ∀ {X Y : Mon_ C} (f : X ⟶ Y), (f ⊗ 𝟙 (𝟙_ (Mon_ C))) ≫ (ρ_ Y).hom = (ρ_ X).hom ≫ f | /-
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 intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | 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
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ (f✝ ⊗ 𝟙 (𝟙_ (Mon_ C))) ≫ (ρ_ Y✝).hom = (ρ_ X✝).hom ≫ f✝ | /-
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; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ ((f✝ ⊗ 𝟙 (𝟙_ (Mon_ C))) ≫ (ρ_ Y✝).hom).hom = ((ρ_ X✝).hom ≫ f✝).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... | dsimp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
f✝ : X✝ ⟶ Y✝
⊢ (f✝ ⊗ 𝟙 (𝟙_ (Mon_ C))).hom ≫ (ρ_ Y✝).hom.hom = (ρ_ X✝).hom.hom ≫ f✝.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... | apply rightUnitor_naturality | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | 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
⊢ ∀ (W X Y Z : Mon_ C),
((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom ≫ (𝟙 W ⊗ (α_ X Y Z).hom) = (α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).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... | intros | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | 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
W✝ X✝ Y✝ Z✝ : Mon_ C
⊢ ((α_ W✝ X✝ Y✝).hom ⊗ 𝟙 Z✝) ≫ (α_ W✝ (X✝ ⊗ Y✝) Z✝).hom ≫ (𝟙 W✝ ⊗ (α_ X✝ Y✝ Z✝).hom) =
(α_ (W✝ ⊗ X✝) Y✝ Z✝).hom ≫ (α_ W✝ X✝ (Y✝ ⊗ Z✝)).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... | ext | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
W✝ X✝ Y✝ Z✝ : Mon_ C
⊢ (((α_ W✝ X✝ Y✝).hom ⊗ 𝟙 Z✝) ≫ (α_ W✝ (X✝ ⊗ Y✝) Z✝).hom ≫ (𝟙 W✝ ⊗ (α_ X✝ Y✝ Z✝).hom)).hom =
((α_ (W✝ ⊗ X✝) Y✝ Z✝).hom ≫ (α_ W✝ X✝ (Y✝ ⊗ Z✝)).hom).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... | dsimp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
W✝ X✝ Y✝ Z✝ : Mon_ C
⊢ ((α_ W✝ X✝ Y✝).hom ⊗ 𝟙 Z✝).hom ≫ (α_ W✝ (X✝ ⊗ Y✝) Z✝).hom.hom ≫ (𝟙 W✝ ⊗ (α_ X✝ Y✝ Z✝).hom).hom =
(α_ (W✝ ⊗ X✝) Y✝ Z✝).hom.hom ≫ (α_ W✝ X✝ (Y✝ ⊗ Z✝)).hom.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... | apply pentagon | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | 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
⊢ ∀ (X Y : Mon_ C), (α_ X (𝟙_ (Mon_ C)) Y).hom ≫ (𝟙 X ⊗ (λ_ Y).hom) = (ρ_ X).hom ⊗ 𝟙 Y | /-
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 intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | 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
X✝ Y✝ : Mon_ C
⊢ (α_ X✝ (𝟙_ (Mon_ C)) Y✝).hom ≫ (𝟙 X✝ ⊗ (λ_ Y✝).hom) = (ρ_ X✝).hom ⊗ 𝟙 Y✝ | /-
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; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
⊢ ((α_ X✝ (𝟙_ (Mon_ C)) Y✝).hom ≫ (𝟙 X✝ ⊗ (λ_ Y✝).hom)).hom = ((ρ_ X✝).hom ⊗ 𝟙 Y✝).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... | dsimp | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
case w
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X✝ Y✝ : Mon_ C
⊢ (α_ X✝ (𝟙_ (Mon_ C)) Y✝).hom.hom ≫ (𝟙 X✝ ⊗ (λ_ Y✝).hom).hom = ((ρ_ X✝).hom ⊗ 𝟙 Y✝).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... | apply triangle | instance monMonoidal : MonoidalCategory (Mon_ C) := .ofTensorHom
(tensor_id := by intros; ext; apply tensor_id)
(tensor_comp := by intros; ext; apply tensor_comp)
(associator_naturality := by intros; ext; dsimp; apply associator_naturality)
(leftUnitor_naturality := by intros; ext; dsimp; apply leftUnitor_natur... | Mathlib.CategoryTheory.Monoidal.Mon_.506_0.NTUMzhXPwXsmsYt | instance monMonoidal : MonoidalCategory (Mon_ C) | Mathlib_CategoryTheory_Monoidal_Mon_ |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f g : CentroidHom α
h : (fun f => f.toFun) f = (fun f => f.toFun) g
⊢ f = g | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | cases f | instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
| Mathlib.Algebra.Ring.CentroidHom.89_0.FQQ3LT1tg3cKlkH | instance : CentroidHomClass (CentroidHom α) α where
coe f | Mathlib_Algebra_Ring_CentroidHom |
case mk
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
g : CentroidHom α
toAddMonoidHom✝ : α →+ α
map_mul_left'✝ : ∀ (a b : α), ZeroHom.toFun (↑toAddMonoidHom✝) (a * b) = a * ZeroHom.toFun (↑toAddMonoidHom✝) b
map_mul_right'✝ : ∀ (a b : α), ZeroHom.toFun (↑toAddMono... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | cases g | instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
| Mathlib.Algebra.Ring.CentroidHom.89_0.FQQ3LT1tg3cKlkH | instance : CentroidHomClass (CentroidHom α) α where
coe f | Mathlib_Algebra_Ring_CentroidHom |
case mk.mk
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
toAddMonoidHom✝¹ : α →+ α
map_mul_left'✝¹ : ∀ (a b : α), ZeroHom.toFun (↑toAddMonoidHom✝¹) (a * b) = a * ZeroHom.toFun (↑toAddMonoidHom✝¹) b
map_mul_right'✝¹ : ∀ (a b : α), ZeroHom.toFun (↑toAddMonoidHom✝¹) (... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | congr with x | instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
| Mathlib.Algebra.Ring.CentroidHom.89_0.FQQ3LT1tg3cKlkH | instance : CentroidHomClass (CentroidHom α) α where
coe f | Mathlib_Algebra_Ring_CentroidHom |
case mk.mk.e_toAddMonoidHom.h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
toAddMonoidHom✝¹ : α →+ α
map_mul_left'✝¹ : ∀ (a b : α), ZeroHom.toFun (↑toAddMonoidHom✝¹) (a * b) = a * ZeroHom.toFun (↑toAddMonoidHom✝¹) b
map_mul_right'✝¹ : ∀ (a b : α), ZeroHom.toFun (↑... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact congrFun h x | instance : CentroidHomClass (CentroidHom α) α where
coe f := f.toFun
coe_injective' f g h := by
cases f
cases g
congr with x
| Mathlib.Algebra.Ring.CentroidHom.89_0.FQQ3LT1tg3cKlkH | instance : CentroidHomClass (CentroidHom α) α where
coe f | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f : CentroidHom α
f' : α → α
h : f' = ⇑f
src✝ : α →+ α := AddMonoidHom.copy f.toAddMonoidHom f' h
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (↑src✝) 0 = 0) },
... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp_rw [h, map_mul_left] | /-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by | Mathlib.Algebra.Ring.CentroidHom.144_0.FQQ3LT1tg3cKlkH | /-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f : CentroidHom α
f' : α → α
h : f' = ⇑f
src✝ : α →+ α := AddMonoidHom.copy f.toAddMonoidHom f' h
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := { toFun := f', map_zero' := (_ : ZeroHom.toFun (↑src✝) 0 = 0) },
... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp_rw [h, map_mul_right] | /-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α :=
{ f.toAddMonoidHom.copy f' <| h with
toFun := f'
map_mul_left' := fun a b ↦ by simp_rw [h, map_mul_left]
map_mu... | Mathlib.Algebra.Ring.CentroidHom.144_0.FQQ3LT1tg3cKlkH | /-- Copy of a `CentroidHom` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : CentroidHom α) (f' : α → α) (h : f' = f) : CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
g f₁ f₂ : CentroidHom α
hg : Injective ⇑g
h : comp g f₁ = comp g f₂
a : α
⊢ g (f₁ a) = g (f₂ a) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [← comp_apply, h, comp_apply] | @[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h ↦ ext fun a ↦ hg <| by | Mathlib.Algebra.Ring.CentroidHom.235_0.FQQ3LT1tg3cKlkH | @[simp]
theorem cancel_left {g f₁ f₂ : CentroidHom α} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f g : CentroidHom α
src✝ : α →+ α := ↑f + ↑g
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.to... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | show f (a * b) + g (a * b) = a * (f b + g b) | instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.249_0.FQQ3LT1tg3cKlkH | instance : Add (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f g : CentroidHom α
src✝ : α →+ α := ↑f + ↑g
a b : α
⊢ f (a * b) + g (a * b) = a * (f b + g b) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_left, mul_add] | instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
| Mathlib.Algebra.Ring.CentroidHom.249_0.FQQ3LT1tg3cKlkH | instance : Add (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f g : CentroidHom α
src✝ : α →+ α := ↑f + ↑g
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.to... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | show f (a * b) + g (a * b) = (f a + g a) * b | instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.249_0.FQQ3LT1tg3cKlkH | instance : Add (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocSemiring α
f g : CentroidHom α
src✝ : α →+ α := ↑f + ↑g
a b : α
⊢ f (a * b) + g (a * b) = (f a + g a) * b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_right, add_mul] | instance : Add (CentroidHom α) :=
⟨fun f g ↦
{ (f + g : α →+ α) with
map_mul_left' := fun a b ↦ by
show f (a * b) + g (a * b) = a * (f b + g b)
simp [map_mul_left, mul_add]
map_mul_right' := fun a b ↦ by
show f (a * b) + g (a * b) = (f a + g a) * b
| Mathlib.Algebra.Ring.CentroidHom.249_0.FQQ3LT1tg3cKlkH | instance : Add (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change n • f (a * b) = a * n • f b | instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.267_0.FQQ3LT1tg3cKlkH | instance instSMul : SMul M (CentroidHom α) where
smul n f | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [map_mul_left f, ← mul_smul_comm] | instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
| Mathlib.Algebra.Ring.CentroidHom.267_0.FQQ3LT1tg3cKlkH | instance instSMul : SMul M (CentroidHom α) where
smul n f | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change n • f (a * b) = n • f a * b | instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.267_0.FQQ3LT1tg3cKlkH | instance instSMul : SMul M (CentroidHom α) where
smul n f | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [map_mul_right f, ← smul_mul_assoc] | instance instSMul : SMul M (CentroidHom α) where
smul n f :=
{ (n • f : α →+ α) with
map_mul_left' := fun a b ↦ by
change n • f (a * b) = a * n • f b
rw [map_mul_left f, ← mul_smul_comm]
map_mul_right' := fun a b ↦ by
change n • f (a * b) = n • f a * b
| Mathlib.Algebra.Ring.CentroidHom.267_0.FQQ3LT1tg3cKlkH | instance instSMul : SMul M (CentroidHom α) where
smul n f | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | induction' n with n ih | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case zero
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact rfl | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [pow_succ] | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
| Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _) | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
| Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | induction' n with n ih | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := ... | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case zero
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact rfl | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := ... | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := ... | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [pow_succ] | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := ... | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
case succ
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
in... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact (congr_arg f.toEnd ih).trans (f.map_mul_right' _ _) | instance hasNPowNat : Pow (CentroidHom α) ℕ :=
⟨fun f n ↦
{ (f.toEnd ^ n : AddMonoid.End α) with
map_mul_left' := fun a b ↦ by
induction' n with n ih
· exact rfl
· simp
rw [pow_succ]
exact (congr_arg f.toEnd ih).trans (f.map_mul_left' _ _)
map_mul_right' := ... | Mathlib.Algebra.Ring.CentroidHom.291_0.FQQ3LT1tg3cKlkH | instance hasNPowNat : Pow (CentroidHom α) ℕ | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp only [Function.comp_apply] | theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
| Mathlib.Algebra.Ring.CentroidHom.424_0.FQQ3LT1tg3cKlkH | theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [map_mul_right, map_mul_left, ← map_mul_right, ← map_mul_left] | theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) := by
simp only [Function.comp_apply]
| Mathlib.Algebra.Ring.CentroidHom.424_0.FQQ3LT1tg3cKlkH | theorem comp_mul_comm (T S : CentroidHom α) (a b : α) : (T ∘ S) (a * b) = (S ∘ T) (a * b) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | ext T | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
| Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | refine ⟨?_, fun h ↦ ?_⟩ | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
| Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_1
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩) | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_1.intro.inl.intro
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_1.intro.inr.intro
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact AddMonoidHom.ext fun a ↦ (map_mul_right f a b).symm | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_2
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rw [Subsemiring.mem_centralizer_iff] at h | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (... | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_2
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | refine ⟨⟨T, fun a b ↦ ?_, fun a b ↦ ?_⟩, rfl⟩ | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (... | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_2.refine_1
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCom... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact congr($(h (L a) (.inl ⟨a, rfl⟩)) b).symm | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (... | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
case h.refine_2.refine_2
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCom... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact congr($(h (R b) (.inr ⟨b, rfl⟩)) a).symm | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) := by
ext T
refine ⟨?_, fun h ↦ ?_⟩
· rintro ⟨f, rfl⟩ S (⟨a, rfl⟩ | ⟨b, rfl⟩)
· exact AddMonoidHom.ext fun b ↦ (map_mul_left f a b).symm
· exact AddMonoidHom.ext fun a ↦ (... | Mathlib.Algebra.Ring.CentroidHom.438_0.FQQ3LT1tg3cKlkH | lemma centroid_eq_centralizer_mulLeftRight :
RingHom.rangeS (toEndRingHom α) = Subsemiring.centralizer (Set.range L ∪ Set.range R) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | ext a | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
case h
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact (((Set.mem_center_iff _).mp z₁.prop).left_assoc z₂ a).symm | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp only [ZeroMemClass.coe_zero, map_zero] | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact rfl | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add] | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact rfl | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z :=
{ L (z : α) with
map_mul_left' := ((Set.mem_center_iff _).mp z.prop).left_comm
map_mul_right' := ((Set.mem_center_iff _).mp z.prop).left_assoc }
... | Mathlib.Algebra.Ring.CentroidHom.450_0.FQQ3LT1tg3cKlkH | /-- The canonical homomorphism from the center into the centroid -/
def centerToCentroid : NonUnitalSubsemiring.center α →ₙ+* CentroidHom α where
toFun z | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | constructor | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
| Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mp
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩ | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
ins... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | rintro ⟨hc, ⟨T, hT⟩⟩ | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommCla... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | have e1 (d : α) : T d = a * d := congr($hT d) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
| Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommCla... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | have e2 (d : α) : T d = d * a := congr($(hT.trans hc) d) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommCla... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | constructor | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro.comm
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCo... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case comm => exact (congr($hc ·)) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case comm => exact (congr($hc ·)) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact (congr($hc ·)) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro.left_assoc
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case left_assoc => simpa [e1] using (map_mul_right T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case left_assoc => simpa [e1] using (map_mul_right T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simpa [e1] using (map_mul_right T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro.mid_assoc
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : S... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case mid_assoc => exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | exact fun b c ↦ by simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simpa [e1 c, e2 b] using
(map_mul_right T b c).symm.trans <| map_mul_left T b c | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
case mpr.intro.intro.right_assoc
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ :... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case right_assoc => simpa [e2] using (map_mul_left T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | case right_assoc => simpa [e2] using (map_mul_left T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝¹² : NonUnitalNonAssocSemiring α
inst✝¹¹ : Monoid M
inst✝¹⁰ : Monoid N
inst✝⁹ : Semiring R
inst✝⁸ : DistribMulAction M α
inst✝⁷ : SMulCommClass M α α
inst✝⁶ : IsScalarTower M α α
inst✝⁵ : DistribMulAction N α
inst✝⁴ : SMulCommClass N α α
inst✝³ : IsS... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simpa [e2] using (map_mul_left T · ·) | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd := by
constructor
· exact fun ha ↦ ⟨AddMonoidHom.ext <| IsMulCentral.comm ha, ⟨centerToCentroid ⟨a, ha⟩, rfl⟩⟩
· rintro ⟨hc, ⟨T, hT⟩⟩
have e1 (d : α) : T d = a * d := congr($hT d)
... | Mathlib.Algebra.Ring.CentroidHom.469_0.FQQ3LT1tg3cKlkH | lemma center_iff_op_centroid (a : α) :
a ∈ NonUnitalSubsemiring.center α ↔ L a = R a ∧ (L a) ∈ Set.range CentroidHom.toEnd | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ T 1 ∈ NonUnitalSubsemiring.center α | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | refine ⟨?_, ?_, ?_, ?_⟩ | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
case refine_1
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (a : α), T 1 * a = a * T 1
case refine_2
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ :... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | all_goals simp [← map_mul_left, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
case refine_1
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (a : α), T 1 * a = a * T 1 | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [← map_mul_left, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
case refine_2
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (b c : α), T 1 * (b * c) = T 1 * b * c | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [← map_mul_left, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
case refine_3
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (a c : α), a * T 1 * c = a * (T 1 * c) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [← map_mul_left, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
case refine_4
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (a b : α), a * b * T 1 = a * (b * T 1) | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [← map_mul_left, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
z : ↥(NonUnitalSubsemiring.center α)
⊢ ↑((fun T => { val := T 1, property := (_ : IsMulCentral (T 1)) }) (MulHom.toFun src✝.toMulHom z)) = ↑z | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [centerToCentroid_apply] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
lef... | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonAssocSemiring α
src✝ : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α := centerToCentroid
T : CentroidHom α
⊢ ∀ (a : α), (MulHom.toFun src✝.toMulHom ((fun T => { val := T 1, property := (_ : IsMulCentral (T 1)) }) T)) a = T a | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [centerToCentroid_apply, ← map_mul_right] | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α :=
{ centerToCentroid with
invFun := fun T ↦
⟨T 1, by refine ⟨?_, ?_, ?_, ?_⟩; all_goals simp [← map_mul_left, ← map_mul_right]⟩
lef... | Mathlib.Algebra.Ring.CentroidHom.489_0.FQQ3LT1tg3cKlkH | /-- The canonical isomorphism from the center of a (non-associative) semiring onto its centroid. -/
def centerIsoCentroid : NonUnitalSubsemiring.center α ≃+* CentroidHom α | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f : CentroidHom α
src✝ : α →+ α := -↑f
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.toFun (↑src✝... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change -f (a * b) = a * (-f b) | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.503_0.FQQ3LT1tg3cKlkH | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f : CentroidHom α
src✝ : α →+ α := -↑f
a b : α
⊢ -f (a * b) = a * -f b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_left] | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
| Mathlib.Algebra.Ring.CentroidHom.503_0.FQQ3LT1tg3cKlkH | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f : CentroidHom α
src✝ : α →+ α := -↑f
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.toFun (↑src✝... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change -f (a * b) = (-f a) * b | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.503_0.FQQ3LT1tg3cKlkH | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f : CentroidHom α
src✝ : α →+ α := -↑f
a b : α
⊢ -f (a * b) = -f a * b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_right] | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_left' := fun a b ↦ by
change -f (a * b) = a * (-f b)
simp [map_mul_left]
map_mul_right' := fun a b ↦ by
change -f (a * b) = (-f a) * b
| Mathlib.Algebra.Ring.CentroidHom.503_0.FQQ3LT1tg3cKlkH | /-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f g : CentroidHom α
src✝ : α →+ α := ↑f - ↑g
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.toFun ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b | instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.514_0.FQQ3LT1tg3cKlkH | instance : Sub (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f g : CentroidHom α
src✝ : α →+ α := ↑f - ↑g
a b : α
⊢ (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | simp [map_mul_left, mul_sub] | instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
| Mathlib.Algebra.Ring.CentroidHom.514_0.FQQ3LT1tg3cKlkH | instance : Sub (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
F : Type u_1
M : Type u_2
N : Type u_3
R : Type u_4
α : Type u_5
inst✝ : NonUnitalNonAssocRing α
f g : CentroidHom α
src✝ : α →+ α := ↑f - ↑g
a b : α
⊢ ZeroHom.toFun
(↑{ toZeroHom := ↑src✝,
map_add' :=
(_ : ∀ (x y : α), ZeroHom.toFun (↑src✝) (x + y) = ZeroHom.toFun (↑src✝) x + ZeroHom.toFun ... | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
import Mathlib.RingTheory.Subsemiring.Basic
#align_import alge... | change (⇑f - ⇑g) (a * b) = ((⇑f - ⇑g) a) * b | instance : Sub (CentroidHom α) :=
⟨fun f g ↦
{ (f - g : α →+ α) with
map_mul_left' := fun a b ↦ by
change (⇑f - ⇑g) (a * b) = a * (⇑f - ⇑g) b
simp [map_mul_left, mul_sub]
map_mul_right' := fun a b ↦ by
| Mathlib.Algebra.Ring.CentroidHom.514_0.FQQ3LT1tg3cKlkH | instance : Sub (CentroidHom α) | Mathlib_Algebra_Ring_CentroidHom |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.