Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 126 | 129 | theorem congrArg_cast_hom_right {X Y Z : C} (p : X ⟶ Y) (q : Z = Y) :
cast (congrArg (fun W : C => X ⟶ W) q.symm) p = p ≫ eqToHom q.symm := by |
cases q
simp
| 490 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 138 | 141 | theorem congrArg_mpr_hom_right {X Y Z : C} (p : X ⟶ Y) (q : Z = Y) :
(congrArg (fun W : C => X ⟶ W) q).mpr p = p ≫ eqToHom q.symm := by |
cases q
simp
| 490 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 | theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p ≪≫ eqToIso q = eqToIso (p.trans q) := by | ext; simp
| 490 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 174 | 176 | theorem eqToHom_op {X Y : C} (h : X = Y) : (eqToHom h).op = eqToHom (congr_arg op h.symm) := by |
cases h
rfl
| 490 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
open Bicategory
universe w v u
variable (B : Type u) [Bicategory.{w, v} B]... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 78 | 81 | theorem whiskerLeft_eqToHom {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g = h) :
f ◁ eqToHom η = eqToHom (congr_arg₂ (· ≫ ·) rfl η) := by |
cases η
simp only [whiskerLeft_id, eqToHom_refl]
| 491 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
open Bicategory
universe w v u
variable (B : Type u) [Bicategory.{w, v} B]... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 85 | 88 | theorem eqToHom_whiskerRight {a b c : B} {f g : a ⟶ b} (η : f = g) (h : b ⟶ c) :
eqToHom η ▷ h = eqToHom (congr_arg₂ (· ≫ ·) η rfl) := by |
cases η
simp only [id_whiskerRight, eqToHom_refl]
| 491 |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe v₁ u₁
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 62 | 63 | theorem hom_inl_inr_false {X : C} {Y : D} (f : Sum.inl X ⟶ Sum.inr Y) : False := by |
cases f
| 492 |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe v₁ u₁
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 66 | 67 | theorem hom_inr_inl_false {X : C} {Y : D} (f : Sum.inr X ⟶ Sum.inl Y) : False := by |
cases f
| 492 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Prod.Basic
#align_import category_theory.products.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
-- declare ... | Mathlib/CategoryTheory/Products/Basic.lean | 64 | 75 | theorem isIso_prod_iff {P Q : C} {S T : D} {f : (P, S) ⟶ (Q, T)} :
IsIso f ↔ IsIso f.1 ∧ IsIso f.2 := by |
constructor
· rintro ⟨g, hfg, hgf⟩
simp? at hfg hgf says simp only [prod_Hom, prod_comp, prod_id, Prod.mk.injEq] at hfg hgf
rcases hfg with ⟨hfg₁, hfg₂⟩
rcases hgf with ⟨hgf₁, hgf₂⟩
exact ⟨⟨⟨g.1, hfg₁, hgf₁⟩⟩, ⟨⟨g.2, hfg₂, hgf₂⟩⟩⟩
· rintro ⟨⟨g₁, hfg₁, hgf₁⟩, ⟨g₂, hfg₂, hgf₂⟩⟩
dsimp at hfg₁ hg... | 493 |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 31 | 34 | theorem map_id_comp (F : C × D ⥤ E) (W : C) {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((𝟙 W, f ≫ g) : (W, X) ⟶ (W, Z)) =
F.map ((𝟙 W, f) : (W, X) ⟶ (W, Y)) ≫ F.map ((𝟙 W, g) : (W, Y) ⟶ (W, Z)) := by |
rw [← Functor.map_comp, prod_comp, Category.comp_id]
| 494 |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 38 | 41 | theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by |
rw [← Functor.map_comp, prod_comp, Category.comp_id]
| 494 |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 45 | 48 | theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
| 494 |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 52 | 55 | theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
| 494 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 59 | 62 | theorem obj_map_id {X Y : C} (f : op X ⟶ op Y) :
(yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) := by |
dsimp
simp
| 495 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 221 | 224 | theorem reprW_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.reprX) :
(F.reprW.app X).hom f = F.map f.op F.reprx := by |
simp only [yoneda_obj_obj, Iso.app_hom, op_unop, reprx, ← FunctorToTypes.naturality,
yoneda_obj_map, unop_op, Quiver.Hom.unop_op, Category.comp_id]
| 495 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 255 | 258 | theorem coreprW_app_hom (X : C) (f : F.coreprX ⟶ X) :
(F.coreprW.app X).hom f = F.map f F.coreprx := by |
simp only [coyoneda_obj_obj, unop_op, Iso.app_hom, coreprx, ← FunctorToTypes.naturality,
coyoneda_obj_map, Category.id_comp]
| 495 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 113 | 116 | theorem LaxMonoidalFunctor.μ_natural (F : LaxMonoidalFunctor C D) {X Y X' Y' : C}
(f : X ⟶ Y) (g : X' ⟶ Y') :
(F.map f ⊗ F.map g) ≫ F.μ Y Y' = F.μ X X' ≫ F.map (f ⊗ g) := by |
simp [tensorHom_def]
| 496 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 164 | 167 | theorem LaxMonoidalFunctor.left_unitality_inv (F : LaxMonoidalFunctor C D) (X : C) :
(λ_ (F.obj X)).inv ≫ F.ε ▷ F.obj X ≫ F.μ (𝟙_ C) X = F.map (λ_ X).inv := by |
rw [Iso.inv_comp_eq, F.left_unitality, Category.assoc, Category.assoc, ← F.toFunctor.map_comp,
Iso.hom_inv_id, F.toFunctor.map_id, comp_id]
| 496 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 52 | 53 | theorem tensor_zero {W X Y Z : C} (f : W ⟶ X) : f ⊗ (0 : Y ⟶ Z) = 0 := by |
simp [tensorHom_def]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 57 | 58 | theorem zero_tensor {W X Y Z : C} (f : Y ⟶ Z) : (0 : W ⟶ X) ⊗ f = 0 := by |
simp [tensorHom_def]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 60 | 61 | theorem tensor_add {W X Y Z : C} (f : W ⟶ X) (g h : Y ⟶ Z) : f ⊗ (g + h) = f ⊗ g + f ⊗ h := by |
simp [tensorHom_def]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 63 | 64 | theorem add_tensor {W X Y Z : C} (f g : W ⟶ X) (h : Y ⟶ Z) : (f + g) ⊗ h = f ⊗ h + g ⊗ h := by |
simp [tensorHom_def]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 82 | 102 | theorem monoidalPreadditive_of_faithful {D} [Category D] [Preadditive D] [MonoidalCategory D]
(F : MonoidalFunctor D C) [F.Faithful] [F.Additive] :
MonoidalPreadditive D :=
{ whiskerLeft_zero := by |
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerLeft]
zero_whiskerRight := by
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerRight]
whiskerLeft_add := by
intros
apply F.toFunctor.map_injective
simp only [F.map_whiskerLeft, Functor.map_a... | 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 113 | 115 | theorem tensor_sum {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(f ⊗ ∑ j ∈ s, g j) = ∑ j ∈ s, f ⊗ g j := by |
simp only [tensorHom_def, whiskerLeft_sum, Preadditive.comp_sum]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 118 | 120 | theorem sum_tensor {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(∑ j ∈ s, g j) ⊗ f = ∑ j ∈ s, g j ⊗ f := by |
simp only [tensorHom_def, sum_whiskerRight, Preadditive.sum_comp]
| 497 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 151 | 158 | theorem leftDistributor_hom {J : Type} [Fintype J] (X : C) (f : J → C) :
(leftDistributor X f).hom =
∑ j : J, (X ◁ biproduct.π f j) ≫ biproduct.ι (fun j => X ⊗ f j) j := by |
ext
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
erw [biproduct.lift_π]
simp only [Preadditive.sum_comp, Category.assoc, biproduct.ι_π, comp_dite, comp_zero,
Finset.sum_dite_eq', Finset.mem_univ, ite_true, eqToHom_refl, Category.comp_id]
| 497 |
import Mathlib.CategoryTheory.Linear.LinearFunctor
import Mathlib.CategoryTheory.Monoidal.Preadditive
#align_import category_theory.monoidal.linear from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
namespace CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.MonoidalCateg... | Mathlib/CategoryTheory/Monoidal/Linear.lean | 58 | 70 | theorem monoidalLinearOfFaithful {D : Type*} [Category D] [Preadditive D] [Linear R D]
[MonoidalCategory D] [MonoidalPreadditive D] (F : MonoidalFunctor D C) [F.Faithful]
[F.toFunctor.Additive] [F.toFunctor.Linear R] : MonoidalLinear R D :=
{ whiskerLeft_smul := by |
intros X Y Z r f
apply F.toFunctor.map_injective
rw [F.map_whiskerLeft]
simp
smul_whiskerRight := by
intros r X Y f Z
apply F.toFunctor.map_injective
rw [F.map_whiskerRight]
simp }
| 498 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 75 | 78 | theorem tensor_id (M N : ModuleCat R) : tensorHom (𝟙 M) (𝟙 N) = 𝟙 (ModuleCat.of R (M ⊗ N)) := by |
-- Porting note: even with high priority ext fails to find this
apply TensorProduct.ext
rfl
| 499 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 81 | 85 | theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : ModuleCat R} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
(g₂ : Y₂ ⟶ Z₂) : tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂ := by |
-- Porting note: even with high priority ext fails to find this
apply TensorProduct.ext
rfl
| 499 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 151 | 155 | theorem associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : ModuleCat R} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂)
(f₃ : X₃ ⟶ Y₃) :
tensorHom (tensorHom f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom =
(associator X₁ X₂ X₃).hom ≫ tensorHom f₁ (tensorHom f₂ f₃) := by |
convert associator_naturality_aux f₁ f₂ f₃ using 1
| 499 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 158 | 162 | theorem pentagon (W X Y Z : ModuleCat R) :
whiskerRight (associator W X Y).hom Z ≫
(associator W (tensorObj X Y) Z).hom ≫ whiskerLeft W (associator X Y Z).hom =
(associator (tensorObj W X) Y Z).hom ≫ (associator W X (tensorObj Y Z)).hom := by |
convert pentagon_aux R W X Y Z using 1
| 499 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
| 500 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 186 | 187 | theorem functor_map {I : Type u₁} (F : I → C) {i : Discrete I} (f : i ⟶ i) :
(Discrete.functor F).map f = 𝟙 (F i.as) := by | aesop_cat
| 500 |
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.monoidal.free.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786cadc4c35ff"
universe u
namespace CategoryTheory
open Mo... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 91 | 95 | theorem inclusion_map {X Y : N C} (f : X ⟶ Y) :
inclusion.map f = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom f))) := by |
rcases f with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
apply inclusion.map_id
| 501 |
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.monoidal.free.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786cadc4c35ff"
universe u
namespace CategoryTheory
open Mo... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 184 | 191 | theorem tensorFunc_obj_map (Z : F C) {n n' : N C} (f : n ⟶ n') :
((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z := by |
cases n
cases n'
rcases f with ⟨⟨h⟩⟩
dsimp at h
subst h
simp
| 501 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 93 | 99 | theorem braiding_tensor_left (X Y Z : C) :
(β_ (X ⊗ Y) Z).hom =
(α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
(β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by |
apply (cancel_epi (α_ X Y Z).inv).1
apply (cancel_mono (α_ Z X Y).inv).1
simp [hexagon_reverse]
| 502 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 102 | 108 | theorem braiding_tensor_right (X Y Z : C) :
(β_ X (Y ⊗ Z)).hom =
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by |
apply (cancel_epi (α_ X Y Z).hom).1
apply (cancel_mono (α_ Y Z X).hom).1
simp [hexagon_forward]
| 502 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 125 | 128 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
(f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by |
rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
| 502 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 146 | 153 | theorem yang_baxter (X Y Z : C) :
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
(α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom := by |
rw [← braiding_tensor_right_assoc X Y Z, ← cancel_mono (α_ Z Y X).inv]
repeat rw [assoc]
rw [Iso.hom_inv_id, comp_id, ← braiding_naturality_right, braiding_tensor_right]
| 502 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 267 | 271 | theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by |
coherence
| 502 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 34 | 38 | theorem braiding_naturality {X₁ X₂ Y₁ Y₂ : ModuleCat.{u} R} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
(f ⊗ g) ≫ (Y₁.braiding Y₂).hom = (X₁.braiding X₂).hom ≫ (g ⊗ f) := by |
apply TensorProduct.ext'
intro x y
rfl
| 503 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 43 | 46 | theorem braiding_naturality_left {X Y : ModuleCat R} (f : X ⟶ Y) (Z : ModuleCat R) :
f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
| 503 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 49 | 52 | theorem braiding_naturality_right (X : ModuleCat R) {Y Z : ModuleCat R} (f : Y ⟶ Z) :
X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
| 503 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 55 | 60 | theorem hexagon_forward (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
(braiding X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (braiding X Z).hom := by |
apply TensorProduct.ext_threefold
intro x y z
rfl
| 503 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 65 | 71 | theorem hexagon_reverse (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).inv ≫ (braiding _ Z).hom ≫ (α_ Z X Y).inv =
X ◁ (Y.braiding Z).hom ≫ (α_ X Z Y).inv ≫ (X.braiding Z).hom ▷ Y := by |
apply (cancel_epi (α_ X Y Z).hom).1
apply TensorProduct.ext_threefold
intro x y z
rfl
| 503 |
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.CategoryTheory.Linear.Yoneda
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric
#align_import algebra.category.Module.monoidal.closed from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe ... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean | 88 | 91 | theorem ihom_ev_app (M N : ModuleCat.{u} R) :
(ihom.ev M).app N = TensorProduct.uncurry _ _ _ _ LinearMap.id.flip := by |
apply TensorProduct.ext'
apply ModuleCat.monoidalClosed_uncurry
| 504 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 34 | 39 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
tensorHom ℬ f g ≫ (Limits.BinaryFan.braiding (ℬ Y Y').isLimit (ℬ Y' Y).isLimit).hom =
(Limits.BinaryFan.braiding (ℬ X X').isLimit (ℬ X' X).isLimit).hom ≫ tensorHom ℬ g f := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 505 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 42 | 54 | theorem hexagon_forward (X Y Z : C) :
(BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
(Limits.BinaryFan.braiding (ℬ X (tensorObj ℬ Y Z)).isLimit
(ℬ (tensorObj ℬ Y Z) X).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom =
tensorHom ℬ (Limits.BinaryFan.braiding (ℬ X Y)... |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext; rintro ⟨⟨⟩⟩
· dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
· apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 505 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 57 | 74 | theorem hexagon_reverse (X Y Z : C) :
(BinaryFan.associatorOfLimitCone ℬ X Y Z).inv ≫
(Limits.BinaryFan.braiding (ℬ (tensorObj ℬ X Y) Z).isLimit
(ℬ Z (tensorObj ℬ X Y)).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Z X Y).inv =
tensorHom ℬ (𝟙 X) (Limits.BinaryFan.braiding ... |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext; rintro ⟨⟨⟩⟩
· apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;>
· dsimp [BinaryFan.associatorOfLimitCone, BinaryFan.associator,
Limits.IsLimit.conePointUniqueUpToIso]
simp
· dsimp [BinaryFan.associatorOfLimitCon... | 505 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 77 | 83 | theorem symmetry (X Y : C) :
(Limits.BinaryFan.braiding (ℬ X Y).isLimit (ℬ Y X).isLimit).hom ≫
(Limits.BinaryFan.braiding (ℬ Y X).isLimit (ℬ X Y).isLimit).hom =
𝟙 (tensorObj ℬ X Y) := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext;
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 505 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 82 | 85 | theorem associator_hom_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom := by |
dsimp
simp
| 506 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 90 | 93 | theorem associator_inv_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by |
dsimp
simp
| 506 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 98 | 100 | theorem leftUnitor_hom_hom {X : Action V G} : Hom.hom (λ_ X).hom = (λ_ X.V).hom := by |
dsimp
simp
| 506 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 105 | 107 | theorem leftUnitor_inv_hom {X : Action V G} : Hom.hom (λ_ X).inv = (λ_ X.V).inv := by |
dsimp
simp
| 506 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 112 | 114 | theorem rightUnitor_hom_hom {X : Action V G} : Hom.hom (ρ_ X).hom = (ρ_ X.V).hom := by |
dsimp
simp
| 506 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 119 | 121 | theorem rightUnitor_inv_hom {X : Action V G} : Hom.hom (ρ_ X).inv = (ρ_ X.V).inv := by |
dsimp
simp
| 506 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 30 | 32 | theorem leftUnitor_tensor'' (X Y : C) :
(α_ (𝟙_ C) X Y).hom ≫ (λ_ (X ⊗ Y)).hom = (λ_ X).hom ⊗ 𝟙 Y := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 36 | 38 | theorem leftUnitor_tensor' (X Y : C) :
(λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y) := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 42 | 43 | theorem leftUnitor_tensor_inv' (X Y : C) :
(λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom := by | coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 47 | 48 | theorem id_tensor_rightUnitor_inv (X Y : C) : 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ _).inv ≫ (α_ _ _ _).hom := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 52 | 53 | theorem leftUnitor_inv_tensor_id (X Y : C) : (λ_ X).inv ⊗ 𝟙 Y = (λ_ _).inv ≫ (α_ _ _ _).inv := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 57 | 60 | theorem pentagon_inv_inv_hom (W X Y Z : C) :
(α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ X) Y Z).hom =
(𝟙 W ⊗ (α_ X Y Z).hom) ≫ (α_ W X (Y ⊗ Z)).inv := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 63 | 64 | theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 67 | 68 | theorem unitors_inv_equal : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 72 | 75 | theorem pentagon_hom_inv {W X Y Z : C} :
(α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) =
(α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom := by |
coherence
| 507 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 79 | 82 | theorem pentagon_inv_hom (W X Y Z : C) :
(α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) =
(α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv := by |
coherence
| 507 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 75 | 76 | theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f := by |
rw [tensorHom_def'_assoc, M.one_mul, leftUnitor_naturality]
| 508 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 80 | 81 | theorem mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f := by |
rw [tensorHom_def_assoc, M.mul_one, rightUnitor_naturality]
| 508 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 84 | 85 | theorem assoc_flip :
(M.X ◁ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ▷ M.X) ≫ M.mul := by | simp
| 508 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 372 | 382 | theorem one_associator {M N P : Mon_ C} :
((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom =
(λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) := by |
simp only [Category.assoc, Iso.cancel_iso_inv_left]
slice_lhs 1 3 => rw [← Category.id_comp P.one, tensor_comp]
slice_lhs 2 3 => rw [associator_naturality]
slice_rhs 1 2 => rw [← Category.id_comp M.one, tensor_comp]
slice_lhs 1 2 => rw [tensorHom_id, ← leftUnitor_tensor_inv]
rw [← cancel_epi (λ_ (𝟙_ C)).i... | 508 |
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory.{v₁} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 37 | 38 | theorem assoc_flip :
(A.X ◁ M.act) ≫ M.act = (α_ A.X A.X M.X).inv ≫ (A.mul ▷ M.X) ≫ M.act := by | simp
| 509 |
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory.{v₁} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 81 | 82 | theorem id_hom' (M : Mod_ A) : (𝟙 M : M ⟶ M).hom = 𝟙 M.X := by |
rfl
| 509 |
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
universe v₁ v₂ u₁ u₂ u
open CategoryTheory MonoidalCategor... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 73 | 74 | theorem counit_comul_hom {Z : C} (f : M.X ⟶ Z) : M.comul ≫ (M.counit ⊗ f) = f ≫ (λ_ Z).inv := by |
rw [leftUnitor_inv_naturality, tensorHom_def, counit_comul_assoc]
| 510 |
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Monoidal.Braided.Opposite
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
universe v₁ v₂ u₁ u₂ u
open CategoryTheory MonoidalCategor... | Mathlib/CategoryTheory/Monoidal/Comon_.lean | 77 | 78 | theorem comul_counit_hom {Z : C} (f : M.X ⟶ Z) : M.comul ≫ (f ⊗ M.counit) = f ≫ (ρ_ Z).inv := by |
rw [rightUnitor_inv_naturality, tensorHom_def', comul_counit_assoc]
| 510 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 26 | 27 | theorem numElems_step_1 (start stop) : numElems ⟨start, stop, 1⟩ = stop - start := by |
simp [numElems]
| 511 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 40 | 47 | theorem mem_range'_elems (r : Range) (h : x ∈ List.range' r.start r.numElems r.step) : x ∈ r := by |
obtain ⟨i, h', rfl⟩ := List.mem_range'.1 h
refine ⟨Nat.le_add_right .., ?_⟩
unfold numElems at h'; split at h'
· split at h' <;> [cases h'; simp_all]
· next step0 =>
refine Nat.not_le.1 fun h =>
Nat.not_le.2 h' <| (numElems_le_iff (Nat.pos_of_ne_zero step0)).2 h
| 511 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 49 | 92 | theorem forIn'_eq_forIn_range' [Monad m] (r : Std.Range)
(init : β) (f : (a : Nat) → a ∈ r → β → m (ForInStep β)) :
forIn' r init f =
forIn
((List.range' r.start r.numElems r.step).pmap Subtype.mk fun _ => mem_range'_elems r)
init (fun ⟨a, h⟩ => f a h) := by |
let ⟨start, stop, step⟩ := r
let L := List.range' start (numElems ⟨start, stop, step⟩) step
let f' : { a // start ≤ a ∧ a < stop } → _ := fun ⟨a, h⟩ => f a h
suffices ∀ H, forIn' [start:stop:step] init f = forIn (L.pmap Subtype.mk H) init f' from this _
intro H; dsimp only [forIn', Range.forIn']
if h : sta... | 511 |
import Batteries.Tactic.SeqFocus
import Batteries.Data.List.Lemmas
import Batteries.Data.List.Init.Attach
namespace Std.Range
def numElems (r : Range) : Nat :=
if r.step = 0 then
-- This is a very weird choice, but it is chosen to coincide with the `forIn` impl
if r.stop ≤ r.start then 0 else r.stop
els... | .lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | 94 | 107 | theorem forIn_eq_forIn_range' [Monad m] (r : Std.Range)
(init : β) (f : Nat → β → m (ForInStep β)) :
forIn r init f = forIn (List.range' r.start r.numElems r.step) init f := by |
refine Eq.trans ?_ <| (forIn'_eq_forIn_range' r init (fun x _ => f x)).trans ?_
· simp [forIn, forIn', Range.forIn, Range.forIn']
suffices ∀ fuel i hl b, forIn'.loop r.start r.stop r.step (fun x _ => f x) fuel i hl b =
forIn.loop f fuel i r.stop r.step b from (this _ ..).symm
intro fuel; induction ... | 511 |
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.BilinearMap
#align_import topology.algebra.module.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Filter
open Topology
variable {α 𝕜 𝕝 R E F M : Type*}
section WeakTo... | Mathlib/Topology/Algebra/Module/WeakDual.lean | 133 | 137 | theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by |
rw [← tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]
rfl
| 512 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 83 | 86 | theorem map_mem (x : (inducedMap A).1) (n : ℕ) : (inducedMap A).2 x ∈ A (res x n) := by |
have := x.property.some_mem
rw [mem_iInter] at this
exact this n
| 513 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 99 | 115 | theorem Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
refine Subtype.coe_injective (res_injective ?_)
dsimp
ext n : 1
induction' n with n ih; · simp
simp only [res_succ, cons.injEq]
refine ⟨?_, ih⟩
contrapose hA
simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
refine ⟨res x n, _, _, hA, ?_⟩
r... | 513 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 131 | 144 | theorem VanishingDiam.dist_lt (hA : VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
∃ n : ℕ, ∀ (y) (_ : y ∈ A (res x n)) (z) (_ : z ∈ A (res x n)), dist y z < ε := by |
specialize hA x
rw [ENNReal.tendsto_atTop_zero] at hA
cases' hA (ENNReal.ofReal (ε / 2)) (by
simp only [gt_iff_lt, ENNReal.ofReal_pos]
linarith) with n hn
use n
intro y hy z hz
rw [← ENNReal.ofReal_lt_ofReal_iff ε_pos, ← edist_dist]
apply lt_of_le_of_lt (EMetric.edist_le_diam_of_mem hy hz)
appl... | 513 |
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"356447fe00e75e54777321045cdff7c9ea212e60"
namespace LieAlgebra
variable (R L : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
variable {R L} in
theorem Has... | Mathlib/Algebra/Lie/Semisimple/Basic.lean | 71 | 77 | theorem hasTrivialRadical_iff_no_abelian_ideals :
HasTrivialRadical R L ↔ ∀ I : LieIdeal R L, IsLieAbelian I → I = ⊥ := by |
rw [hasTrivialRadical_iff_no_solvable_ideals]
constructor <;> intro h₁ I h₂
· exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable R I
· rw [← abelian_of_solvable_ideal_eq_bot_iff]
exact h₁ _ <| abelian_derivedAbelianOfIdeal I
| 514 |
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 129 | 131 | theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(MonoidalFunctor.εIso F).inv.app X ≫ (𝟙_ (C ⥤ C)).map f = F.εIso.inv.app X ≫ f := by |
aesop_cat
| 515 |
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
import Mathlib.Algebra.Order.Group.Units
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.Basic
imp... | Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 128 | 129 | theorem le_of_le_mul_right (h : c ≠ 0) (hab : a * c ≤ b * c) : a ≤ b := by |
simpa only [mul_inv_cancel_right₀ h] using mul_le_mul_right' hab c⁻¹
| 516 |
import Mathlib.Init.Algebra.Classes
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
#align_import order.rel_iso.basic from "leanprover-community/mathlib"@"f29120f82f6e24a6f6579896dfa2de6769fec962"
set_option autoImplicit true
open Function
universe u v w
v... | Mathlib/Order/RelIso/Basic.lean | 168 | 180 | theorem injective_of_increasing (r : α → α → Prop) (s : β → β → Prop) [IsTrichotomous α r]
[IsIrrefl β s] (f : α → β) (hf : ∀ {x y}, r x y → s (f x) (f y)) : Injective f := by |
intro x y hxy
rcases trichotomous_of r x y with (h | h | h)
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
· exact h
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
| 517 |
import Mathlib.Order.RelIso.Basic
import Mathlib.Logic.Embedding.Set
import Mathlib.Logic.Equiv.Set
#align_import order.rel_iso.set from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
open Function
universe u v w
variable {α β γ δ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ ... | Mathlib/Order/RelIso/Set.lean | 111 | 112 | theorem RelIso.preimage_eq_image_symm (e : r ≃r s) (t : Set β) : e ⁻¹' t = e.symm '' t := by |
rw [e.symm.image_eq_preimage_symm]; rfl
| 518 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 | theorem mod_add_div' (m k : R) : m % k + m / k * k = m := by |
rw [mul_comm]
exact mod_add_div _ _
| 519 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 136 | 138 | theorem div_add_mod' (m k : R) : m / k * k + m % k = m := by |
rw [mul_comm]
exact div_add_mod _ _
| 519 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 141 | 144 | theorem mod_eq_sub_mul_div {R : Type*} [EuclideanDomain R] (a b : R) : a % b = a - b * (a / b) :=
calc
a % b = b * (a / b) + a % b - b * (a / b) := (add_sub_cancel_left _ _).symm
_ = a - b * (a / b) := by | rw [div_add_mod]
| 519 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 151 | 153 | theorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by |
rw [mul_comm]
exact mul_left_not_lt b h
| 519 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 157 | 157 | theorem mod_zero (a : R) : a % 0 = a := by | simpa only [zero_mul, zero_add] using div_add_mod a 0
| 519 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
| 519 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 63 | 64 | theorem dvd_mod_iff {a b c : R} (h : c ∣ b) : c ∣ a % b ↔ c ∣ a := by |
rw [← dvd_add_right (h.mul_right _), div_add_mod]
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 84 | 85 | theorem div_self {a : R} (a0 : a ≠ 0) : a / a = 1 := by |
simpa only [one_mul] using mul_div_cancel_right₀ 1 a0
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 88 | 89 | theorem eq_div_of_mul_eq_left {a b c : R} (hb : b ≠ 0) (h : a * b = c) : a = c / b := by |
rw [← h, mul_div_cancel_right₀ _ hb]
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 92 | 93 | theorem eq_div_of_mul_eq_right {a b c : R} (ha : a ≠ 0) (h : a * b = c) : b = c / a := by |
rw [← h, mul_div_cancel_left₀ _ ha]
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 96 | 101 | theorem mul_div_assoc (x : R) {y z : R} (h : z ∣ y) : x * y / z = x * (y / z) := by |
by_cases hz : z = 0
· subst hz
rw [div_zero, div_zero, mul_zero]
rcases h with ⟨p, rfl⟩
rw [mul_div_cancel_left₀ _ hz, mul_left_comm, mul_div_cancel_left₀ _ hz]
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 114 | 120 | theorem div_dvd_of_dvd {p q : R} (hpq : q ∣ p) : p / q ∣ p := by |
by_cases hq : q = 0
· rw [hq, zero_dvd_iff] at hpq
rw [hpq]
exact dvd_zero _
use q
rw [mul_comm, ← EuclideanDomain.mul_div_assoc _ hpq, mul_comm, mul_div_cancel_right₀ _ hq]
| 520 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 123 | 128 | theorem dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [div_zero, dvd_zero]
rcases h with ⟨d, rfl⟩
refine ⟨d, ?_⟩
rw [mul_assoc, mul_div_cancel_left₀ _ ha]
| 520 |
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