state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
C : Type u_1
inst✝² : Category.{u_3, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b c : A
h : a + b = c
⊢ (shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X = ((shiftFunctorAdd' C a b c h).inv.app X.unop).op | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# The (naive) shift on the opposite category
If `C` is a category equipped wi... | subst h | lemma oppositeShiftFunctorAdd'_hom_app :
(shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
((shiftFunctorAdd' C a b c h).inv.app X.unop).op := by
| Mathlib.CategoryTheory.Shift.Opposite.107_0.DcKDXhgVzsvY9hl | lemma oppositeShiftFunctorAdd'_hom_app :
(shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
((shiftFunctorAdd' C a b c h).inv.app X.unop).op | Mathlib_CategoryTheory_Shift_Opposite |
C : Type u_1
inst✝² : Category.{u_3, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b : A
⊢ (shiftFunctorAdd' (OppositeShift C A) a b (a + b) (_ : a + b = a + b)).hom.app X =
((shiftFunctorAdd' C a b (a + b) (_ : a + b = a + b)).inv.app X.unop).op | /-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# The (naive) shift on the opposite category
If `C` is a category equipped wi... | simp only [shiftFunctorAdd'_eq_shiftFunctorAdd, oppositeShiftFunctorAdd_hom_app] | lemma oppositeShiftFunctorAdd'_hom_app :
(shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
((shiftFunctorAdd' C a b c h).inv.app X.unop).op := by
subst h
| Mathlib.CategoryTheory.Shift.Opposite.107_0.DcKDXhgVzsvY9hl | lemma oppositeShiftFunctorAdd'_hom_app :
(shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
((shiftFunctorAdd' C a b c h).inv.app X.unop).op | Mathlib_CategoryTheory_Shift_Opposite |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α)... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' J.plusFunctorWhiskerLeftIso _ ≪≫ _ ≪≫ Functor.associator _ _ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.65_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α)... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' isoWhiskerRight _ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.65_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α)... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' J.plusFunctorWhiskerLeftIso _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.65_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `F`. -/
noncomputable def sheafificationWhiskerLeftIso (P : Cᵒᵖ ⥤ D)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso] | @[simp]
theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafif... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.78_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafif... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [Category.comp_id] | @[simp]
theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafif... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.78_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerLeftIso_hom_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).hom.app F = (J.sheafif... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | dsimp [sheafificationWhiskerLeftIso, sheafifyCompIso] | @[simp]
theorem sheafificationWhiskerLeftIso_inv_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).inv.app F = (J.sheafif... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.88_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerLeftIso_inv_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).inv.app F = (J.sheafif... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹⁰ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁹ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁸ : Category.{max v u, w₂} E
F✝ : D ⥤ E
inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | erw [Category.id_comp] | @[simp]
theorem sheafificationWhiskerLeftIso_inv_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).inv.app F = (J.sheafif... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.88_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerLeftIso_inv_app (P : Cᵒᵖ ⥤ D) (F : D ⥤ E)
[∀ (F : D ⥤ E) (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F]
[∀ (F : D ⥤ E) (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D),
PreservesLimit (W.index P).multicospan F] :
(sheafificationWhiskerLeftIso J P).inv.app F = (J.sheafif... | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' Functor.associator _ _ _ ≪≫ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.98_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' isoWhiskerLeft (J.plusFunctor D) (J.plusFunctorWhiskerRightIso _) ≪≫ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E := by
refine' Functor.a... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.98_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' _ ≪≫ Functor.associator _ _ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E := by
refine' Functor.a... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.98_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | refine' (Functor.associator _ _ _).symm ≪≫ _ | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E := by
refine' Functor.a... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.98_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | exact isoWhiskerRight (J.plusFunctorWhiskerRightIso _) (J.plusFunctor E) | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E := by
refine' Functor.a... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.98_0.A60cLXHj1ecBhch | /-- The isomorphism between the sheafification of `P` composed with `F` and
the sheafification of `P ⋙ F`, functorially in `P`. -/
noncomputable def sheafificationWhiskerRightIso :
J.sheafification D ⋙ (whiskeringRight _ _ _).obj F ≅
(whiskeringRight _ _ _).obj F ⋙ J.sheafification E | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | dsimp [sheafificationWhiskerRightIso, sheafifyCompIso] | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.110_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | simp only [Category.id_comp, Category.comp_id] | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom := by
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.110_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | erw [Category.id_comp] | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom := by
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
simp only [Category.id_comp, Category.comp_id]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.110_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | dsimp [sheafificationWhiskerRightIso, sheafifyCompIso] | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.118_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | simp only [Category.id_comp, Category.comp_id] | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv := by
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.118_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | erw [Category.id_comp] | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv := by
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
simp only [Category.id_comp, Category.comp_id]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.118_0.A60cLXHj1ecBhch | @[simp]
theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | dsimp [sheafifyCompIso] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | erw [whiskerRight_comp, Category.assoc] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | slice_lhs 2 3 => rw [plusCompIso_whiskerRight] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd ... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [plusCompIso_whiskerRight] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd ... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [plusCompIso_whiskerRight] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
case a.a
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd ... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [plusCompIso_whiskerRight] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => | Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [Category.assoc, ← J.plusMap_comp, whiskerRight_toPlus_comp_plusCompIso_hom, ←
Category.assoc, whiskerRight_toPlus_comp_plusCompIso_hom] | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => rw [plusCompIso_whiskerRight]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rfl | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ := by
dsimp [sheafifyCompIso]
erw [whiskerRight_comp, Category.assoc]
slice_lhs 2 3 => rw [plusCompIso_whiskerRight]
rw [Category.assoc, ← J.plusMap_comp, whi... | Mathlib.CategoryTheory.Sites.CompatibleSheafification.126_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [Iso.comp_inv_eq] | @[simp, reassoc]
theorem toSheafify_comp_sheafifyCompIso_inv :
J.toSheafify _ ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify _) _ := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.137_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem toSheafify_comp_sheafifyCompIso_inv :
J.toSheafify _ ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify _) _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α),... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | simp | @[simp, reassoc]
theorem toSheafify_comp_sheafifyCompIso_inv :
J.toSheafify _ ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify _) _ := by
rw [Iso.comp_inv_eq]; | Mathlib.CategoryTheory.Sites.CompatibleSheafification.137_0.A60cLXHj1ecBhch | @[simp, reassoc]
theorem toSheafify_comp_sheafifyCompIso_inv :
J.toSheafify _ ≫ (J.sheafifyCompIso F P).inv = whiskerRight (J.toSheafify _) _ | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → ... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | apply J.sheafifyLift_unique | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) := by
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.149_0.A60cLXHj1ecBhch | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
case a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | rw [Iso.comp_inv_eq] | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) := by
apply J.sheafifyLift_unique
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.149_0.A60cLXHj1ecBhch | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
case a
C : Type u
inst✝¹² : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝¹¹ : Category.{max v u, w₁} D
E : Type w₂
inst✝¹⁰ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd... | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification f... | simp | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) := by
apply J.sheafifyLift_unique
rw [Iso.comp_inv_eq]
| Mathlib.CategoryTheory.Sites.CompatibleSheafification.149_0.A60cLXHj1ecBhch | @[simp]
theorem sheafifyCompIso_inv_eq_sheafifyLift :
(J.sheafifyCompIso F P).inv =
J.sheafifyLift (whiskerRight (J.toSheafify P) F)
(HasSheafCompose.isSheaf _ ((J.sheafify_isSheaf _))) | Mathlib_CategoryTheory_Sites_CompatibleSheafification |
α : Type u_1
n : ℕ
⊢ Even (card (Fin (bit0 n))) | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Parity
#align_import data.fintype.parity from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df08... | rw [Fintype.card_fin] | /-- The cardinality of `Fin (bit0 n)` is even, `Fact` version.
This `Fact` is needed as an instance by `Matrix.SpecialLinearGroup.has_neg`. -/
theorem Fintype.card_fin_even {n : ℕ} : Fact (Even (Fintype.card (Fin (bit0 n)))) :=
⟨by | Mathlib.Data.Fintype.Parity.28_0.BIALH4coWJ17WOi | /-- The cardinality of `Fin (bit0 n)` is even, `Fact` version.
This `Fact` is needed as an instance by `Matrix.SpecialLinearGroup.has_neg`. -/
theorem Fintype.card_fin_even {n : ℕ} : Fact (Even (Fintype.card (Fin (bit0 n)))) | Mathlib_Data_Fintype_Parity |
α : Type u_1
n : ℕ
⊢ Even (bit0 n) | /-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Parity
#align_import data.fintype.parity from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df08... | exact even_bit0 _ | /-- The cardinality of `Fin (bit0 n)` is even, `Fact` version.
This `Fact` is needed as an instance by `Matrix.SpecialLinearGroup.has_neg`. -/
theorem Fintype.card_fin_even {n : ℕ} : Fact (Even (Fintype.card (Fin (bit0 n)))) :=
⟨by rw [Fintype.card_fin]; | Mathlib.Data.Fintype.Parity.28_0.BIALH4coWJ17WOi | /-- The cardinality of `Fin (bit0 n)` is even, `Fact` version.
This `Fact` is needed as an instance by `Matrix.SpecialLinearGroup.has_neg`. -/
theorem Fintype.card_fin_even {n : ℕ} : Fact (Even (Fintype.card (Fin (bit0 n)))) | Mathlib_Data_Fintype_Parity |
a : ℤ
h₁ : a ≠ 0
h₂ : -a = 0
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | have : -a = -0 := by rwa [Int.neg_zero] | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 := fun h₁ h₂ => by
| Mathlib.Init.Data.Int.CompLemmas.25_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
h₁ : a ≠ 0
h₂ : -a = 0
⊢ -a = -0 | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rwa [Int.neg_zero] | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 := fun h₁ h₂ => by
have : -a = -0 := by | Mathlib.Init.Data.Int.CompLemmas.25_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
h₁ : a ≠ 0
h₂ : -a = 0
this : -a = -0
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | have : a = 0 := Int.neg_eq_neg this | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 := fun h₁ h₂ => by
have : -a = -0 := by rwa [Int.neg_zero]
| Mathlib.Init.Data.Int.CompLemmas.25_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
h₁ : a ≠ 0
h₂ : -a = 0
this✝ : -a = -0
this : a = 0
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | contradiction | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 := fun h₁ h₂ => by
have : -a = -0 := by rwa [Int.neg_zero]
have : a = 0 := Int.neg_eq_neg this
| Mathlib.Init.Data.Int.CompLemmas.25_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_zero_of_ne {a : ℤ} : a ≠ 0 → -a ≠ 0 | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
h₁ : 0 < a
h₂ : 0 < b
h : -a = b
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rw [← h] at h₂ | protected theorem neg_ne_of_pos {a b : ℤ} : 0 < a → 0 < b → -a ≠ b := fun h₁ h₂ h => by
| Mathlib.Init.Data.Int.CompLemmas.35_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_of_pos {a b : ℤ} : 0 < a → 0 < b → -a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
h₁ : 0 < a
h₂ : 0 < -a
h : -a = b
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | exact absurd (le_of_lt h₁) (not_le_of_gt (Int.neg_of_neg_pos h₂)) | protected theorem neg_ne_of_pos {a b : ℤ} : 0 < a → 0 < b → -a ≠ b := fun h₁ h₂ h => by
rw [← h] at h₂
| Mathlib.Init.Data.Int.CompLemmas.35_0.Hx4Nk3JpwZvlmHq | protected theorem neg_ne_of_pos {a b : ℤ} : 0 < a → 0 < b → -a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
n : ℕ
⊢ 0 + ↑n = ofNat n | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rw [Int.zero_add, Int.coe_nat_eq] | theorem zero_le_ofNat (n : ℕ) : 0 ≤ ofNat n :=
@le.intro _ _ n (by | Mathlib.Init.Data.Int.CompLemmas.89_0.Hx4Nk3JpwZvlmHq | theorem zero_le_ofNat (n : ℕ) : 0 ≤ ofNat n | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : natAbs a ≠ natAbs b
h : a = b
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | have : (natAbs a : ℤ) = natAbs b := by
rwa [natAbs_of_nonneg ha, natAbs_of_nonneg hb] | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b := fun h => by
| Mathlib.Init.Data.Int.CompLemmas.95_0.Hx4Nk3JpwZvlmHq | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : natAbs a ≠ natAbs b
h : a = b
⊢ ↑(natAbs a) = ↑(natAbs b) | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rwa [natAbs_of_nonneg ha, natAbs_of_nonneg hb] | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b := fun h => by
have : (natAbs a : ℤ) = natAbs b := by
| Mathlib.Init.Data.Int.CompLemmas.95_0.Hx4Nk3JpwZvlmHq | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : natAbs a ≠ natAbs b
h : a = b
this : ↑(natAbs a) = ↑(natAbs b)
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | injection this | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b := fun h => by
have : (natAbs a : ℤ) = natAbs b := by
rwa [natAbs_of_nonneg ha, natAbs_of_nonneg hb]
| Mathlib.Init.Data.Int.CompLemmas.95_0.Hx4Nk3JpwZvlmHq | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
ha : 0 ≤ a
hb : 0 ≤ b
h✝ : natAbs a ≠ natAbs b
h : a = b
a_eq✝ : natAbs a = natAbs b
⊢ False | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | contradiction | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b := fun h => by
have : (natAbs a : ℤ) = natAbs b := by
rwa [natAbs_of_nonneg ha, natAbs_of_nonneg hb]
injection this
| Mathlib.Init.Data.Int.CompLemmas.95_0.Hx4Nk3JpwZvlmHq | theorem ne_of_natAbs_ne_natAbs_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(h : natAbs a ≠ natAbs b) : a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
a b : ℤ
n m : ℕ
ha : 0 ≤ a
hb : 0 ≤ b
e1 : natAbs a = n
e2 : natAbs b = m
h : n ≠ m
⊢ natAbs a ≠ natAbs b | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rwa [e1, e2] | protected theorem ne_of_nat_ne_nonneg_case {a b : ℤ} {n m : Nat} (ha : 0 ≤ a) (hb : 0 ≤ b)
(e1 : natAbs a = n) (e2 : natAbs b = m) (h : n ≠ m) : a ≠ b :=
have : natAbs a ≠ natAbs b := by | Mathlib.Init.Data.Int.CompLemmas.103_0.Hx4Nk3JpwZvlmHq | protected theorem ne_of_nat_ne_nonneg_case {a b : ℤ} {n m : Nat} (ha : 0 ≤ a) (hb : 0 ≤ b)
(e1 : natAbs a = n) (e2 : natAbs b = m) (h : n ≠ m) : a ≠ b | Mathlib_Init_Data_Int_CompLemmas |
n m : ℕ
x✝¹ : 0 ≤ ofNat n
x✝ : 0 ≤ ofNat m
⊢ natAbs (ofNat n + ofNat m) = natAbs (ofNat n) + natAbs (ofNat m) | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | simp only [ofNat_eq_coe, natAbs_ofNat] | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
| Mathlib.Init.Data.Int.CompLemmas.122_0.Hx4Nk3JpwZvlmHq | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
simp only [ofNat_eq_coe, natAbs_ofNat]
simp only [Int.ofNat_add_ofNat]
simp only [← ofNat_eq_coe, natAbs_ofNat_core]
| _, negSucc m, _, h₂ => absurd (negSucc_lt_ze... | Mathlib_Init_Data_Int_CompLemmas |
n m : ℕ
x✝¹ : 0 ≤ ofNat n
x✝ : 0 ≤ ofNat m
⊢ natAbs (↑n + ↑m) = n + m | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | simp only [Int.ofNat_add_ofNat] | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
simp only [ofNat_eq_coe, natAbs_ofNat]
| Mathlib.Init.Data.Int.CompLemmas.122_0.Hx4Nk3JpwZvlmHq | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
simp only [ofNat_eq_coe, natAbs_ofNat]
simp only [Int.ofNat_add_ofNat]
simp only [← ofNat_eq_coe, natAbs_ofNat_core]
| _, negSucc m, _, h₂ => absurd (negSucc_lt_ze... | Mathlib_Init_Data_Int_CompLemmas |
n m : ℕ
x✝¹ : 0 ≤ ofNat n
x✝ : 0 ≤ ofNat m
⊢ natAbs ↑(n + m) = n + m | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | simp only [← ofNat_eq_coe, natAbs_ofNat_core] | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
simp only [ofNat_eq_coe, natAbs_ofNat]
simp only [Int.ofNat_add_ofNat]
| Mathlib.Init.Data.Int.CompLemmas.122_0.Hx4Nk3JpwZvlmHq | protected theorem natAbs_add_nonneg :
∀ {a b : Int}, 0 ≤ a → 0 ≤ b → natAbs (a + b) = natAbs a + natAbs b
| ofNat n, ofNat m, _, _ => by
simp only [ofNat_eq_coe, natAbs_ofNat]
simp only [Int.ofNat_add_ofNat]
simp only [← ofNat_eq_coe, natAbs_ofNat_core]
| _, negSucc m, _, h₂ => absurd (negSucc_lt_ze... | Mathlib_Init_Data_Int_CompLemmas |
n m : ℕ
x✝¹ : -[n+1] < 0
x✝ : -[m+1] < 0
⊢ natAbs (-[n+1] + -[m+1]) = natAbs -[n+1] + natAbs -[m+1] | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | simp [negSucc_add_negSucc, natAbs_of_negSucc, Nat.succ_add, Nat.add_succ] | protected theorem natAbs_add_neg :
∀ {a b : Int}, a < 0 → b < 0 → natAbs (a + b) = natAbs a + natAbs b
| negSucc n, negSucc m, _, _ => by
| Mathlib.Init.Data.Int.CompLemmas.132_0.Hx4Nk3JpwZvlmHq | protected theorem natAbs_add_neg :
∀ {a b : Int}, a < 0 → b < 0 → natAbs (a + b) = natAbs a + natAbs b
| negSucc n, negSucc m, _, _ => by
simp [negSucc_add_negSucc, natAbs_of_negSucc, Nat.succ_add, Nat.add_succ] | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
n : ℕ
h : natAbs a = n
⊢ natAbs (bit0 a) = bit0 n | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rw [← h] | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit0_step {a : Int} {n : Nat} (h : natAbs a = n) :
natAbs (bit0 a) = bit0 n := by | Mathlib.Init.Data.Int.CompLemmas.145_0.Hx4Nk3JpwZvlmHq | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit0_step {a : Int} {n : Nat} (h : natAbs a = n) :
natAbs (bit0 a) = bit0 n | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
n : ℕ
h : natAbs a = n
⊢ natAbs (bit0 a) = bit0 (natAbs a) | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | apply Int.natAbs_bit0 | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit0_step {a : Int} {n : Nat} (h : natAbs a = n) :
natAbs (bit0 a) = bit0 n := by rw [← h]; | Mathlib.Init.Data.Int.CompLemmas.145_0.Hx4Nk3JpwZvlmHq | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit0_step {a : Int} {n : Nat} (h : natAbs a = n) :
natAbs (bit0 a) = bit0 n | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
h : 0 ≤ a
⊢ natAbs (bit0 a + 1) = bit0 (natAbs a) + natAbs 1 | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rw [Int.natAbs_add_nonneg (Int.bit0_nonneg h) (le_of_lt Int.zero_lt_one), Int.natAbs_bit0] | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg {a : Int} (h : 0 ≤ a) : natAbs (bit1 a) = bit1 (natAbs a) :=
show natAbs (bit0 a + 1) = bit0 (natAbs a) + natAbs 1 by
| Mathlib.Init.Data.Int.CompLemmas.151_0.Hx4Nk3JpwZvlmHq | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg {a : Int} (h : 0 ≤ a) : natAbs (bit1 a) = bit1 (natAbs a) | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
n : ℕ
h₁ : 0 ≤ a
h₂ : natAbs a = n
⊢ natAbs (bit1 a) = bit1 n | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | rw [← h₂] | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg_step {a : Int} {n : Nat} (h₁ : 0 ≤ a) (h₂ : natAbs a = n) :
natAbs (bit1 a) = bit1 n := by | Mathlib.Init.Data.Int.CompLemmas.158_0.Hx4Nk3JpwZvlmHq | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg_step {a : Int} {n : Nat} (h₁ : 0 ≤ a) (h₂ : natAbs a = n) :
natAbs (bit1 a) = bit1 n | Mathlib_Init_Data_Int_CompLemmas |
a : ℤ
n : ℕ
h₁ : 0 ≤ a
h₂ : natAbs a = n
⊢ natAbs (bit1 a) = bit1 (natAbs a) | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Data.Int.Order
#align_import init.data.int.comp_lemmas from "leanprover-community/lean"@"4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db"
/-... | apply Int.natAbs_bit1_nonneg h₁ | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg_step {a : Int} {n : Nat} (h₁ : 0 ≤ a) (h₂ : natAbs a = n) :
natAbs (bit1 a) = bit1 n := by rw [← h₂]; | Mathlib.Init.Data.Int.CompLemmas.158_0.Hx4Nk3JpwZvlmHq | set_option linter.deprecated false in
@[deprecated]
protected theorem natAbs_bit1_nonneg_step {a : Int} {n : Nat} (h₁ : 0 ≤ a) (h₂ : natAbs a = n) :
natAbs (bit1 a) = bit1 n | Mathlib_Init_Data_Int_CompLemmas |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | intro f | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f = f ≫ biprod.inr | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.fst = (f ≫ biprod.inr) ≫ biprod.fst | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.lift 0 f ≫ biprod.snd = (f ≫ biprod.inr) ≫ biprod.snd | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := by
intro f
ext
· aesop_cat
· simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
⊢ ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | intro f | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 = f ≫ biprod.inl | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 ≫ biprod.fst = (f ≫ biprod.inl) ≫ biprod.fst | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
f : X ⟶ Y
⊢ biprod.lift f 0 ≫ biprod.snd = (f ≫ biprod.inl) ≫ biprod.snd | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [biprod.lift_snd, Category.assoc, biprod.inl_snd, comp_zero] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
⊢ EckmannHilton.IsUnital (fun x x_1 => leftAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | exact ⟨⟨fun f => by simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc]⟩,
⟨fun f => by simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc]⟩⟩ | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
f : X ⟶ Y
⊢ leftAdd X Y 0 f = f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [hr f, leftAdd, Category.assoc, Category.comp_id, biprod.inr_desc] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
hr : ∀ (f : X ⟶ Y), biprod.lift 0 f = f ≫ biprod.inr
hl : ∀ (f : X ⟶ Y), biprod.lift f 0 = f ≫ biprod.inl
f : X ⟶ Y
⊢ leftAdd X Y f 0 = f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [hl f, leftAdd, Category.assoc, Category.comp_id, biprod.inl_desc] | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.56_0.XxgpFky1BRo6Llh | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
⊢ ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | intro f | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.desc 0 f = biprod.snd ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.inl ≫ biprod.desc 0 f = biprod.inl ≫ biprod.snd ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f : X ⟶ Y
⊢ biprod.inr ≫ biprod.desc 0 f = biprod.inr ≫ biprod.snd ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
⊢ ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | intro f | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.desc f 0 = biprod.fst ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.inl ≫ biprod.desc f 0 = biprod.inl ≫ biprod.fst ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
f : X ⟶ Y
⊢ biprod.inr ≫ biprod.desc f 0 = biprod.inr ≫ biprod.fst ≫ f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp only [biprod.inr_desc, BinaryBicone.inr_fst_assoc, zero_comp] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
⊢ EckmannHilton.IsUnital (fun x x_1 => rightAdd X Y x x_1) 0 | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | exact ⟨⟨fun f => by simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp]⟩,
⟨fun f => by simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp]⟩⟩ | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
f : X ⟶ Y
⊢ rightAdd X Y 0 f = f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [h₂ f, rightAdd, biprod.lift_snd_assoc, Category.id_comp] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
h₂ : ∀ (f : X ⟶ Y), biprod.desc 0 f = biprod.snd ≫ f
h₁ : ∀ (f : X ⟶ Y), biprod.desc f 0 = biprod.fst ≫ f
f : X ⟶ Y
⊢ rightAdd X Y f 0 = f | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [h₁ f, rightAdd, biprod.lift_fst_assoc, Category.id_comp] | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
in... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.71_0.XxgpFky1BRo6Llh | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k) | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k) | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (leftAdd X Y g k) | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ biprod.inl ≫ diag = biprod.lift f h | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ (biprod.inl ≫ diag) ≫ biprod.fst = biprod.lift f h ≫ biprod.fst | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
⊢ (biprod.inl ≫ diag) ≫ biprod.snd = biprod.lift f h ≫ biprod.snd | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X Y h k) = rightAdd X Y (leftAdd X Y f h) (... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ biprod.inr ≫ diag = biprod.lift g k | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ (biprod.inr ≫ diag) ≫ biprod.fst = biprod.lift g k ≫ biprod.fst | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ (biprod.inr ≫ diag) ≫ biprod.snd = biprod.lift g k ≫ biprod.snd | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ leftAdd X Y (rightAdd X Y f g) (rightAdd X... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have h₁ : biprod.lift (f +ᵣ g) (h +ᵣ k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag := by
ext <;> aesop_cat | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
| Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (rightAdd X... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (ri... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
⊢ biprod.lift (rightAdd X Y f g) (ri... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | aesop_cat | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAd... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | have h₂ : diag ≫ biprod.desc (𝟙 Y) (𝟙 Y) = biprod.desc (f +ₗ h) (g +ₗ k) := by
ext <;> simp [reassoc_of% hd₁, reassoc_of% hd₂] | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAd... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | ext | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₀
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) ... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [reassoc_of% hd₁, reassoc_of% hd₂] | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
case h₁
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) ... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | simp [reassoc_of% hd₁, reassoc_of% hd₂] | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
hd₂ : biprod.inr ≫ diag = biprod.lift g k
h₁ : biprod.lift (rightAdd X Y f g) (rightAd... | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | rw [leftAdd, h₁, Category.assoc, h₂, rightAdd] | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp
have h₁ : biprod.lift ... | Mathlib.CategoryTheory.Preadditive.OfBiproducts.86_0.XxgpFky1BRo6Llh | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | Mathlib_CategoryTheory_Preadditive_OfBiproducts |
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