state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case a.w.h
C : Type ?u.46399
D : Type ?u.46402
inst✝³ : Category.{?u.46406, ?u.46399} C
inst✝² : Category.{?u.46410, ?u.46402} D
F : C ⥤ D
A : Type ?u.46441
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp [eqToHom_map, eqToHom_app] | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A :=
hasShiftMk D A
{ F := s
zero := Induced.zero F s i hF
add := Induced.add F s i hF
zero_add_ho... | Mathlib.CategoryTheory.Shift.Induced.96_0.hEyfGnVpBGMswTi | /-- When `F : C ⥤ D` is a functor satisfying suitable technical assumptions,
this is the induced term of type `HasShift D A` deduced from `[HasShift C A]`. -/
noncomputable def induced : HasShift D A | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_1
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_2, u_1} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a : A
⊢ shiftF... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | rfl | @[simp]
lemma shiftFunctor_of_induced (a : A) :
letI := HasShift.induced F A s i hF
shiftFunctor D a = s a := by
| Mathlib.CategoryTheory.Shift.Induced.168_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctor_of_induced (a : A) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
⊢ (shift... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := HasShift.induced F A s i | @[simp]
lemma shiftFunctorZero_hom_app_obj_of_induced (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorZero D A).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
| Mathlib.CategoryTheory.Shift.Induced.176_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorZero_hom_app_obj_of_induced (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : N... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [ShiftMkCore.shiftFunctorZero_eq, HasShift.Induced.zero_hom_app_obj] | @[simp]
lemma shiftFunctorZero_hom_app_obj_of_induced (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorZero D A).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
letI := HasShift.induced F A s i
| Mathlib.CategoryTheory.Shift.Induced.176_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorZero_hom_app_obj_of_induced (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
⊢ (shift... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := HasShift.induced F A s i | @[simp]
lemma shiftFunctorZero_inv_app_obj_of_induced (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorZero D A).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
| Mathlib.CategoryTheory.Shift.Induced.184_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorZero_inv_app_obj_of_induced (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_3, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
X : C
this : N... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [ShiftMkCore.shiftFunctorZero_eq, HasShift.Induced.zero_inv_app_obj] | @[simp]
lemma shiftFunctorZero_inv_app_obj_of_induced (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorZero D A).inv.app (F.obj X) =
F.map ((shiftFunctorZero C A).inv.app X) ≫ (i 0).inv.app X := by
letI := HasShift.induced F A s i
| Mathlib.CategoryTheory.Shift.Induced.184_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorZero_inv_app_obj_of_induced (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := HasShift.induced F A s i | @[simp]
lemma shiftFunctorAdd_hom_app_obj_of_induced (a b : A) (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorAdd D a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫
F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫
(s b).map ((i a).i... | Mathlib.CategoryTheory.Shift.Induced.194_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorAdd_hom_app_obj_of_induced (a b : A) (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [ShiftMkCore.shiftFunctorAdd_eq, HasShift.Induced.add_hom_app_obj] | @[simp]
lemma shiftFunctorAdd_hom_app_obj_of_induced (a b : A) (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorAdd D a b).hom.app (F.obj X) =
(i (a + b)).hom.app X ≫
F.map ((shiftFunctorAdd C a b).hom.app X) ≫
(i b).inv.app ((shiftFunctor C a).obj X) ≫
(s b).map ((i a).i... | Mathlib.CategoryTheory.Shift.Induced.194_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorAdd_hom_app_obj_of_induced (a b : A) (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := HasShift.induced F A s i | @[simp]
lemma shiftFunctorAdd_inv_app_obj_of_induced (a b : A) (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorAdd D a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫
(i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫
(i (a + b)).inv.ap... | Mathlib.CategoryTheory.Shift.Induced.205_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorAdd_inv_app_obj_of_induced (a b : A) (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type u_4
D : Type u_2
inst✝³ : Category.{u_5, u_4} C
inst✝² : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_3
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((whiskeringLeft C D D).obj F)
a b : A
X : C
... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [ShiftMkCore.shiftFunctorAdd_eq, HasShift.Induced.add_inv_app_obj] | @[simp]
lemma shiftFunctorAdd_inv_app_obj_of_induced (a b : A) (X : C) :
letI := HasShift.induced F A s i hF
(shiftFunctorAdd D a b).inv.app (F.obj X) =
(s b).map ((i a).hom.app X) ≫
(i b).hom.app ((shiftFunctor C a).obj X) ≫
F.map ((shiftFunctorAdd C a b).inv.app X) ≫
(i (a + b)).inv.ap... | Mathlib.CategoryTheory.Shift.Induced.205_0.hEyfGnVpBGMswTi | @[simp]
lemma shiftFunctorAdd_inv_app_obj_of_induced (a b : A) (X : C) :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | letI := HasShift.induced F A s i hF | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
| Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | exact
{ iso := fun a => (i a).symm
zero := by
ext X
dsimp
simp only [isoZero_hom_app, shiftFunctorZero_inv_app_obj_of_induced,
← F.map_comp_assoc, Iso.hom_inv_id_app, F.map_id, Category.id_comp]
add := fun a b => by
ext X
dsimp
simp only [isoAdd_... | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
| Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | ext X | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
case w.w.h
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ F... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | dsimp | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
case w.w.h
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ F... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [isoZero_hom_app, shiftFunctorZero_inv_app_obj_of_induced,
← F.map_comp_assoc, Iso.hom_inv_id_app, F.map_id, Category.id_comp] | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ Faithful ((w... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | ext X | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
case w.w.h
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ F... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | dsimp | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
case w.w.h
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ F... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | simp only [isoAdd_hom_app, Iso.symm_hom, shiftFunctorAdd_inv_app_obj_of_induced,
shiftFunctor_of_induced] | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
case w.w.h
C : Type ?u.125746
D : Type ?u.125749
inst✝³ : Category.{?u.125753, ?u.125746} C
inst✝² : Category.{?u.125757, ?u.125749} D
F : C ⥤ D
A : Type ?u.125788
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
hF : Nonempty (Full ((whiskeringLeft C D D).obj F)) ∧ F... | /-
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.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | erw [← Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.map_id,
Category.id_comp, Iso.inv_hom_id_app_assoc, ← F.map_comp_assoc, Iso.hom_inv_id_app,
F.map_id, Category.id_comp] | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI := HasShift.induced F A s i hF
F.CommShift A := by
letI := HasShift.induced F A s i hF
exact
... | Mathlib.CategoryTheory.Shift.Induced.218_0.hEyfGnVpBGMswTi | /-- When the target category of a functor `F : C ⥤ D` is equipped with
the induced shift, this is the compatibility of `F` with the shifts on
the categories `C` and `D`. -/
def Functor.CommShift.ofInduced :
letI | Mathlib_CategoryTheory_Shift_Induced |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Semigroup (f i)
⊢ ∀ (a b c : (i : I) → f i), a * b * c = a * (b * c) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) :=
{ mul := (· * ·)
--pi_instance
mul_assoc := by | Mathlib.Algebra.Group.Pi.54_0.DFGfFEDon0PHcgt | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Semigroup (f i)
a✝ b✝ c✝ : (i : I) → f i
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) :=
{ mul := (· * ·)
--pi_instance
mul_assoc := by intros; | Mathlib.Algebra.Group.Pi.54_0.DFGfFEDon0PHcgt | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Semigroup (f i)
a✝ b✝ c✝ : (i : I) → f i
x✝ : I
⊢ (a✝ * b✝ * c✝) x✝ = (a✝ * (b✝ * c✝)) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_assoc _ _ _ | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) :=
{ mul := (· * ·)
--pi_instance
mul_assoc := by intros; ext; | Mathlib.Algebra.Group.Pi.54_0.DFGfFEDon0PHcgt | @[to_additive]
instance semigroup [∀ i, Semigroup <| f i] : Semigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → CommSemigroup (f i)
src✝ : Semigroup ((i : I) → f i) := semigroup
⊢ ∀ (a b : (i : I) → f i), a * b = b * a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) :=
{ semigroup with
--pi_instance
mul_comm := by | Mathlib.Algebra.Group.Pi.62_0.DFGfFEDon0PHcgt | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → CommSemigroup (f i)
src✝ : Semigroup ((i : I) → f i) := semigroup
a✝ b✝ : (i : I) → f i
⊢ a✝ * b✝ = b✝ * a✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) :=
{ semigroup with
--pi_instance
mul_comm := by intros; | Mathlib.Algebra.Group.Pi.62_0.DFGfFEDon0PHcgt | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → CommSemigroup (f i)
src✝ : Semigroup ((i : I) → f i) := semigroup
a✝ b✝ : (i : I) → f i
x✝ : I
⊢ (a✝ * b✝) x✝ = (b✝ * a✝) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_comm _ _ | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) :=
{ semigroup with
--pi_instance
mul_comm := by intros; ext; | Mathlib.Algebra.Group.Pi.62_0.DFGfFEDon0PHcgt | @[to_additive]
instance commSemigroup [∀ i, CommSemigroup <| f i] : CommSemigroup (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
⊢ ∀ (a : (i : I) → f i), 1 * a = a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
a✝ : (i : I) → f i
⊢ 1 * a✝ = a✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by intros; | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
a✝ : (i : I) → f i
x✝ : I
⊢ (1 * a✝) x✝ = a✝ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact one_mul _ | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by intros; ext; | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
⊢ ∀ (a : (i : I) → f i), a * 1 = a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by intros; ext; exact one_mul _
mul_one := by | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
a✝ : (i : I) → f i
⊢ a✝ * 1 = a✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by intros; ext; exact one_mul _
mul_one := by intros; | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulOneClass (f i)
a✝ : (i : I) → f i
x✝ : I
⊢ (a✝ * 1) x✝ = a✝ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_one _ | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
mul := (· * ·)
--pi_instance
one_mul := by intros; ext; exact one_mul _
mul_one := by intros; ext; | Mathlib.Algebra.Group.Pi.71_0.DFGfFEDon0PHcgt | @[to_additive]
instance mulOneClass [∀ i, MulOneClass <| f i] : MulOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvOneClass (f i)
⊢ 1⁻¹ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
inv := (· ⁻¹)
inv_one := by | Mathlib.Algebra.Group.Pi.82_0.DFGfFEDon0PHcgt | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvOneClass (f i)
⊢ 1⁻¹ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
inv := (· ⁻¹)
inv_one := by intros; | Mathlib.Algebra.Group.Pi.82_0.DFGfFEDon0PHcgt | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvOneClass (f i)
x✝ : I
⊢ 1⁻¹ x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact inv_one | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) :=
{ one := (1 : ∀ i, f i)
inv := (· ⁻¹)
inv_one := by intros; ext; | Mathlib.Algebra.Group.Pi.82_0.DFGfFEDon0PHcgt | @[to_additive]
instance invOneClass [∀ i, InvOneClass <| f i] : InvOneClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
⊢ ∀ (x : (i : I) → f i), (fun n x i => x i ^ n) 0 x = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
x✝ : (i : I) → f i
⊢ (fun n x i => x i ^ n) 0 x✝ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by intros; | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
x✝¹ : (i : I) → f i
x✝ : I
⊢ (fun n x i => x i ^ n) 0 x✝¹ x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact Monoid.npow_zero _ | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by intros; ext; | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
⊢ ∀ (n : ℕ) (x : (i : I) → f i), (fun n x i => x i ^ n) (n + 1) x = x * (fun n x i => x i ^ n) n x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by intros; ext; exact Monoid.npow_zero _
npow_succ := by | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
n✝ : ℕ
x✝ : (i : I) → f i
⊢ (fun n x i => x i ^ n) (n✝ + 1) x✝ = x✝ * (fun n x i => x i ^ n) n✝ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by intros; ext; exact Monoid.npow_zero _
npow_succ := by intros; | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Monoid (f i)
src✝¹ : Semigroup ((i : I) → f i) := semigroup
src✝ : MulOneClass ((i : I) → f i) := mulOneClass
n✝ : ℕ
x✝¹ : (i : I) → f i
x✝ : I
⊢ (fun n x i => x i ^ n) (n✝ + 1) x✝¹ x✝ = (x✝¹ * (fun n x i => x i ^ n... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact Monoid.npow_succ _ _ | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) :=
{ semigroup, mulOneClass with
npow := fun n x i => x i ^ n
--pi_instance
npow_zero := by intros; ext; exact Monoid.npow_zero _
npow_succ := by intros; ext; | Mathlib.Algebra.Group.Pi.88_0.DFGfFEDon0PHcgt | @[to_additive]
instance monoid [∀ i, Monoid <| f i] : Monoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
⊢ ∀ (a b : (i : I) → f i), a / b = a * b⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
a✝ b✝ : (i : I) → f i
⊢ a✝ / b✝ = a✝ * b✝⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
a✝ b✝ : (i : I) → f i
x✝ : I
⊢ (a✝ / b✝) x✝ = (a✝ * b✝⁻¹) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact div_eq_mul_inv _ _ | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
⊢ ∀ (a : (i : I) → f i), (fun z x i => x i ^ z) 0 a = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
a✝ : (i : I) → f i
⊢ (fun z x i => x i ^ z) 0 a✝ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
a✝ : (i : I) → f i
x✝ : I
⊢ (fun z x i => x i ^ z) 0 a✝ x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact DivInvMonoid.zpow_zero' _ | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
⊢ ∀ (n : ℕ) (a : (i : I) → f i),
(fun z x i => x i ^ z) (Int.ofNat (Nat.succ n)) a = a * (fun z x i => x i ^ z) (Int.ofNat n) a | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
n✝ : ℕ
a✝ : (i : I) → f i
⊢ (fun z x i => x i ^ z) (Int.ofNat (Nat.succ n✝)) a✝ = a✝ * (fun z x i => x i ^ z) (Int.ofNat n✝) a✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
n✝ : ℕ
a✝ : (i : I) → f i
x✝ : I
⊢ (fun z x i => x i ^ z) (Int.ofNat (Nat.succ n✝)) a✝ x✝ = (a✝ * (fun z x i => x i ^ z) (Int.ofNat n✝) a✝) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact DivInvMonoid.zpow_succ' _ _ | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
⊢ ∀ (n : ℕ) (a : (i : I) → f i), (fun z x i => x i ^ z) (Int.negSucc n) a = ((fun z x i => x i ^ z) (↑(Nat.succ n)) a)⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
n✝ : ℕ
a✝ : (i : I) → f i
⊢ (fun z x i => x i ^ z) (Int.negSucc n✝) a✝ = ((fun z x i => x i ^ z) (↑(Nat.succ n✝)) a✝)⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvMonoid (f i)
src✝ : Monoid ((i : I) → f i) := monoid
n✝ : ℕ
a✝ : (i : I) → f i
x✝ : I
⊢ (fun z x i => x i ^ z) (Int.negSucc n✝) a✝ x✝ = ((fun z x i => x i ^ z) (↑(Nat.succ n✝)) a✝)⁻¹ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact DivInvMonoid.zpow_neg' _ _ | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) :=
{ monoid with
inv := Inv.inv
div := Div.div
zpow := fun z x i => x i ^ z
--pi_instance
div_eq_mul_inv := by intros; ext; exact div_eq_mul_inv _ _
zpow_zero' := by intros; ext; exact... | Mathlib.Algebra.Group.Pi.112_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegMonoid]
instance divInvMonoid [∀ i, DivInvMonoid <| f i] : DivInvMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvOneMonoid (f i)
src✝ : DivInvMonoid ((i : I) → f i) := divInvMonoid
⊢ 1⁻¹ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subNegZeroMonoid]
instance divInvOneMonoid [∀ i, DivInvOneMonoid <| f i] : DivInvOneMonoid (∀ i : I, f i) :=
{ divInvMonoid with
inv_one := by | Mathlib.Algebra.Group.Pi.125_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegZeroMonoid]
instance divInvOneMonoid [∀ i, DivInvOneMonoid <| f i] : DivInvOneMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivInvOneMonoid (f i)
src✝ : DivInvMonoid ((i : I) → f i) := divInvMonoid
x✝ : I
⊢ 1⁻¹ x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact inv_one | @[to_additive Pi.subNegZeroMonoid]
instance divInvOneMonoid [∀ i, DivInvOneMonoid <| f i] : DivInvOneMonoid (∀ i : I, f i) :=
{ divInvMonoid with
inv_one := by ext; | Mathlib.Algebra.Group.Pi.125_0.DFGfFEDon0PHcgt | @[to_additive Pi.subNegZeroMonoid]
instance divInvOneMonoid [∀ i, DivInvOneMonoid <| f i] : DivInvOneMonoid (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvolutiveInv (f i)
⊢ ∀ (x : (i : I) → f i), x⁻¹⁻¹ = x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) :=
{ inv := Inv.inv
--pi_instance
inv_inv := by | Mathlib.Algebra.Group.Pi.130_0.DFGfFEDon0PHcgt | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvolutiveInv (f i)
x✝ : (i : I) → f i
⊢ x✝⁻¹⁻¹ = x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) :=
{ inv := Inv.inv
--pi_instance
inv_inv := by intros; | Mathlib.Algebra.Group.Pi.130_0.DFGfFEDon0PHcgt | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → InvolutiveInv (f i)
x✝¹ : (i : I) → f i
x✝ : I
⊢ x✝¹⁻¹⁻¹ x✝ = x✝¹ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact inv_inv _ | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) :=
{ inv := Inv.inv
--pi_instance
inv_inv := by intros; ext; | Mathlib.Algebra.Group.Pi.130_0.DFGfFEDon0PHcgt | @[to_additive]
instance involutiveInv [∀ i, InvolutiveInv <| f i] : InvolutiveInv (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
⊢ ∀ (a b : (i : I) → f i), (a * b)⁻¹ = b⁻¹ * a⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by | Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
a✝ b✝ : (i : I) → f i
⊢ (a✝ * b✝)⁻¹ = b✝⁻¹ * a✝⁻¹ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by intros; | Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
a✝ b✝ : (i : I) → f i
x✝ : I
⊢ (a✝ * b✝)⁻¹ x✝ = (b✝⁻¹ * a✝⁻¹) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_inv_rev _ _ | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by intros; ext; | Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
⊢ ∀ (a b : (i : I) → f i), a * b = 1 → a⁻¹ = b | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros _ _ h | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by intros; ext; exact mul_inv_rev _ _
inv_eq_of_mul := by
| Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
a✝ b✝ : (i : I) → f i
h : a✝ * b✝ = 1
⊢ a✝⁻¹ = b✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by intros; ext; exact mul_inv_rev _ _
inv_eq_of_mul := by
intros _ _ h; | Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → DivisionMonoid (f i)
src✝¹ : DivInvMonoid ((i : I) → f i) := divInvMonoid
src✝ : InvolutiveInv ((i : I) → f i) := involutiveInv
a✝ b✝ : (i : I) → f i
h : a✝ * b✝ = 1
x✝ : I
⊢ a✝⁻¹ x✝ = b✝ x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact DivisionMonoid.inv_eq_of_mul _ _ (congrFun h _) | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) :=
{ divInvMonoid, involutiveInv with
--pi_instance
mul_inv_rev := by intros; ext; exact mul_inv_rev _ _
inv_eq_of_mul := by
intros _ _ h; ext; | Mathlib.Algebra.Group.Pi.137_0.DFGfFEDon0PHcgt | @[to_additive Pi.subtractionMonoid]
instance divisionMonoid [∀ i, DivisionMonoid <| f i] : DivisionMonoid (∀ i, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Group (f i)
src✝ : DivInvMonoid ((i : I) → f i) := divInvMonoid
⊢ ∀ (a : (i : I) → f i), a⁻¹ * a = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) :=
{ divInvMonoid with
--pi_instance
mul_left_inv := by | Mathlib.Algebra.Group.Pi.150_0.DFGfFEDon0PHcgt | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Group (f i)
src✝ : DivInvMonoid ((i : I) → f i) := divInvMonoid
a✝ : (i : I) → f i
⊢ a✝⁻¹ * a✝ = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) :=
{ divInvMonoid with
--pi_instance
mul_left_inv := by intros; | Mathlib.Algebra.Group.Pi.150_0.DFGfFEDon0PHcgt | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → Group (f i)
src✝ : DivInvMonoid ((i : I) → f i) := divInvMonoid
a✝ : (i : I) → f i
x✝ : I
⊢ (a✝⁻¹ * a✝) x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_left_inv _ | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) :=
{ divInvMonoid with
--pi_instance
mul_left_inv := by intros; ext; | Mathlib.Algebra.Group.Pi.150_0.DFGfFEDon0PHcgt | @[to_additive]
instance group [∀ i, Group <| f i] : Group (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
⊢ ∀ (a : (i : I) → f i), 0 * a = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
a✝ : (i : I) → f i
⊢ 0 * a✝ = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by intros; | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
a✝ : (i : I) → f i
x✝ : I
⊢ (0 * a✝) x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact zero_mul _ | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by intros; ext; | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
⊢ ∀ (a : (i : I) → f i), a * 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intros | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by intros; ext; exact zero_mul _
mul_zero := by | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
a✝ : (i : I) → f i
⊢ a✝ * 0 = 0 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by intros; ext; exact zero_mul _
mul_zero := by intros; | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝ : (i : I) → MulZeroClass (f i)
a✝ : (i : I) → f i
x✝ : I
⊢ (a✝ * 0) x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact mul_zero _ | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) :=
{ zero := (0 : ∀ i, f i)
mul := (· * ·)
--pi_instance
zero_mul := by intros; ext; exact zero_mul _
mul_zero := by intros; ext; | Mathlib.Algebra.Group.Pi.221_0.DFGfFEDon0PHcgt | instance mulZeroClass [∀ i, MulZeroClass <| f i] : MulZeroClass (∀ i : I, f i) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → Mul (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : Mul α
inst✝ : Mul β
f : α →ₙ* β
I : Type u_5
x✝¹ x✝ : I → α
⊢ (fun h => ⇑f ∘ h) (x✝¹ * x✝) = (fun h => ⇑f ∘ h) x✝¹ * (fun h => ⇑f ∘ h) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | /-- Semigroup homomorphism between the function spaces `I → α` and `I → β`, induced by a semigroup
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps) "Additive semigroup homomorphism between the function spaces `I → α`
and `I → β`, induced by an additive semigroup homomorphism `f` between `α` and `β... | Mathlib.Algebra.Group.Pi.346_0.DFGfFEDon0PHcgt | /-- Semigroup homomorphism between the function spaces `I → α` and `I → β`, induced by a semigroup
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → Mul (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : Mul α
inst✝ : Mul β
f : α →ₙ* β
I : Type u_5
x✝² x✝¹ : I → α
x✝ : I
⊢ (fun h => ⇑f ∘ h) (x✝² * x✝¹) x✝ = ((fun h => ⇑f ∘ h) x✝² * (fun h => ⇑f ∘ h) x✝¹) x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp | /-- Semigroup homomorphism between the function spaces `I → α` and `I → β`, induced by a semigroup
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps) "Additive semigroup homomorphism between the function spaces `I → α`
and `I → β`, induced by an additive semigroup homomorphism `f` between `α` and `β... | Mathlib.Algebra.Group.Pi.346_0.DFGfFEDon0PHcgt | /-- Semigroup homomorphism between the function spaces `I → α` and `I → β`, induced by a semigroup
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → MulOneClass (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : MulOneClass α
inst✝ : MulOneClass β
f : α →* β
I : Type u_5
⊢ (fun h => ⇑f ∘ h) 1 = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps)
"Additive monoid homomorphism between the function spaces `I → α` and `I → β`, induced by an
additive monoid homomorphism `f` between `α` and `β`"]
prot... | Mathlib.Algebra.Group.Pi.407_0.DFGfFEDon0PHcgt | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → MulOneClass (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : MulOneClass α
inst✝ : MulOneClass β
f : α →* β
I : Type u_5
x✝ : I
⊢ (fun h => ⇑f ∘ h) 1 x✝ = OfNat.ofNat 1 x✝ | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | dsimp | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps)
"Additive monoid homomorphism between the function spaces `I → α` and `I → β`, induced by an
additive monoid homomorphism `f` between `α` and `β`"]
prot... | Mathlib.Algebra.Group.Pi.407_0.DFGfFEDon0PHcgt | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → MulOneClass (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : MulOneClass α
inst✝ : MulOneClass β
f : α →* β
I : Type u_5
x✝ : I
⊢ f 1 = 1 | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps)
"Additive monoid homomorphism between the function spaces `I → α` and `I → β`, induced by an
additive monoid homomorphism `f` between `α` and `β`"]
prot... | Mathlib.Algebra.Group.Pi.407_0.DFGfFEDon0PHcgt | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → MulOneClass (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : MulOneClass α
inst✝ : MulOneClass β
f : α →* β
I : Type u_5
x✝¹ x✝ : I → α
⊢ OneHom.toFun { toFun := fun h => ⇑f ∘ h, map_one' := (_ : (fun h => ⇑f ∘ h) 1 = 1)... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps)
"Additive monoid homomorphism between the function spaces `I → α` and `I → β`, induced by an
additive monoid homomorphism `f` between `α` and `β`"]
prot... | Mathlib.Algebra.Group.Pi.407_0.DFGfFEDon0PHcgt | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α✝ : Type u_2
I✝ : Type u
f✝ : I✝ → Type v
x y : (i : I✝) → f✝ i
i j : I✝
inst✝² : (i : I✝) → MulOneClass (f✝ i)
α : Type u_3
β : Type u_4
inst✝¹ : MulOneClass α
inst✝ : MulOneClass β
f : α →* β
I : Type u_5
x✝² x✝¹ : I → α
x✝ : I
⊢ OneHom.toFun { toFun := fun h => ⇑f ∘ h, map_one' := (_ : (fun h =>... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr := simps)
"Additive monoid homomorphism between the function spaces `I → α` and `I → β`, induced by an
additive monoid homomorphism `f` between `α` and `β`"]
prot... | Mathlib.Algebra.Group.Pi.407_0.DFGfFEDon0PHcgt | /-- Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. -/
@[to_additive (attr | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
⊢ Pairwise fun i j => ∀ (x : f i) (y : f j), Commute (mulSingle i x) (mulSingle j y) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | intro i j hij x y | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
⊢ Commute (mulSingle i x) (mulSingle j y) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext k | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
k : I
⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | by_cases h1 : i = k | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case pos
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
k : I
h1 : i = k
⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | subst h1 | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case pos
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
⊢ (mulSingle i x * mulSingle j y) i = (mulSingle j y * mulSingle i x) i | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp [hij] | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case neg
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
k : I
h1 : ¬i = k
⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | by_cases h2 : j = k | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case pos
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
k : I
h1 : ¬i = k
h2 : j = k
⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | subst h2 | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case pos
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
h1 : ¬i = j
⊢ (mulSingle i x * mulSingle j y) j = (mulSingle j y * mulSingle i x) j | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp [hij] | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
case neg
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y✝ : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
i j : I
hij : i ≠ j
x : f i
y : f j
k : I
h1 : ¬i = k
h2 : ¬j = k
⊢ (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp [h1, h2] | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib.Algebra.Group.Pi.568_0.DFGfFEDon0PHcgt | /-- The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `Commute.map` -/
@[to_additive
"The injection into an additive pi group at different indices commutes.
For injections of commuting elements at the same index, see `AddCommute.map`"]... | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
x : (i : I) → f i
i j : I
⊢ Commute (mulSingle i (x i)) (mulSingle j (x j)) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | obtain rfl | hij := Decidable.eq_or_ne i j | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) := by
| Mathlib.Algebra.Group.Pi.588_0.DFGfFEDon0PHcgt | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) | Mathlib_Algebra_Group_Pi |
case inl
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i✝ j : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
x : (i : I) → f i
i : I
⊢ Commute (mulSingle i (x i)) (mulSingle i (x i)) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | rfl | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) := by
obtain rfl | hij... | Mathlib.Algebra.Group.Pi.588_0.DFGfFEDon0PHcgt | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) | Mathlib_Algebra_Group_Pi |
case inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i✝ j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → MulOneClass (f i)
x : (i : I) → f i
i j : I
hij : i ≠ j
⊢ Commute (mulSingle i (x i)) (mulSingle j (x j)) | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | exact Pi.mulSingle_commute hij _ _ | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) := by
obtain rfl | hij... | Mathlib.Algebra.Group.Pi.588_0.DFGfFEDon0PHcgt | /-- The injection into a pi group with the same values commutes. -/
@[to_additive "The injection into an additive pi group with the same values commutes."]
theorem Pi.mulSingle_apply_commute [∀ i, MulOneClass <| f i] (x : ∀ i, f i) (i j : I) :
Commute (mulSingle i (x i)) (mulSingle j (x j)) | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → Group (f i)
g : (i : I) → f i
x : f i
⊢ Function.update g i x = g / mulSingle i (g i) * mulSingle i x | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | ext j | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x := by
| Mathlib.Algebra.Group.Pi.598_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x | Mathlib_Algebra_Group_Pi |
case h
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → Group (f i)
g : (i : I) → f i
x : f i
j : I
⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | rcases eq_or_ne i j with (rfl | h) | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x := by
ext j
| Mathlib.Algebra.Group.Pi.598_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x | Mathlib_Algebra_Group_Pi |
case h.inl
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → Group (f i)
g : (i : I) → f i
x : f i
⊢ Function.update g i x i = (g / mulSingle i (g i) * mulSingle i x) i | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x := by
ext j
rcases eq_or_ne i j with (rfl | h)
· | Mathlib.Algebra.Group.Pi.598_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x | Mathlib_Algebra_Group_Pi |
case h.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x✝ y : (i : I) → f i
i j✝ : I
inst✝¹ : DecidableEq I
inst✝ : (i : I) → Group (f i)
g : (i : I) → f i
x : f i
j : I
h : i ≠ j
⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | simp [Function.update_noteq h.symm, h] | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x := by
ext j
rcases eq_or_ne i j with (rfl | h)
· simp
· | Mathlib.Algebra.Group.Pi.598_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.update_eq_div_mul_mulSingle [∀ i, Group <| f i] (g : ∀ i : I, f i) (x : f i) :
Function.update g i x = g / mulSingle i (g i) * mulSingle i x | Mathlib_Algebra_Group_Pi |
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
⊢ mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | refine' ⟨fun h => _, _⟩ | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
| Mathlib.Algebra.Group.Pi.608_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | Mathlib_Algebra_Group_Pi |
case refine'_1
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | have hk := congr_fun h k | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
re... | Mathlib.Algebra.Group.Pi.608_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | Mathlib_Algebra_Group_Pi |
case refine'_1
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mu... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | have hl := congr_fun h l | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
re... | Mathlib.Algebra.Group.Pi.608_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | Mathlib_Algebra_Group_Pi |
case refine'_1
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mu... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | have hm := (congr_fun h m).symm | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
re... | Mathlib.Algebra.Group.Pi.608_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | Mathlib_Algebra_Group_Pi |
case refine'_1
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mu... | /-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Init.CCLemmas
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Pi.Algebra
import Mathlib.Data.Set.Function
import Mathlib.Logic... | have hn := (congr_fun h n).symm | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
re... | Mathlib.Algebra.Group.Pi.608_0.DFGfFEDon0PHcgt | @[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
{k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
(mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n | Mathlib_Algebra_Group_Pi |
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