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limit_cone_is_limit (F : J ⥤ Ring.{max v u}) : is_limit (limit_cone F) | lifted_limit_is_limit _ | def | Ring.limit_cone_is_limit | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | The chosen cone is a limit cone.
(Generally, you'll just want to use `limit.cone F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v v} Ring.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit_of_created F (forget₂ Ring SemiRing.{max v u}) } } | instance | Ring.has_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | The category of rings has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits Ring.{u} | Ring.has_limits_of_size.{u u} | instance | Ring.has_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_SemiRing_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ Ring SemiRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F, by apply_instance } } | instance | Ring.forget₂_SemiRing_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | The forgetful functor from rings to semirings preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_SemiRing_preserves_limits : preserves_limits (forget₂ Ring SemiRing.{u}) | Ring.forget₂_SemiRing_preserves_limits_of_size.{u u} | instance | Ring.forget₂_SemiRing_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_AddCommGroup_preserves_limits_aux (F : J ⥤ Ring.{max v u}) :
is_limit ((forget₂ Ring AddCommGroup).map_cone (limit_cone F)) | by apply AddCommGroup.limit_cone_is_limit (F ⋙ forget₂ Ring AddCommGroup.{max v u}) | def | Ring.forget₂_AddCommGroup_preserves_limits_aux | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | An auxiliary declaration to speed up typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommGroup_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ Ring AddCommGroup.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (forget₂_AddCommGroup_preserves_limits_aux F) } } | instance | Ring.forget₂_AddCommGroup_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | The forgetful functor from rings to additive commutative groups preserves all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_AddCommGroup_preserves_limits : preserves_limits (forget₂ Ring AddCommGroup.{u}) | Ring.forget₂_AddCommGroup_preserves_limits_of_size.{u u} | instance | Ring.forget₂_AddCommGroup_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size : preserves_limits_of_size.{v v} (forget Ring.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F,
limits.comp_preserves_limit (forget₂ Ring SemiRing) (forget SemiRing.{max v u}) } } | instance | Ring.forget_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"Ring",
"SemiRing"
] | The forgetful functor from rings to types preserves all limits. (That is, the underlying
types could have been computed instead as limits in the category of types.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget Ring.{u}) | Ring.forget_preserves_limits_of_size.{u u} | instance | Ring.forget_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_ring_obj (F : J ⥤ CommRing.{max v u}) (j) :
comm_ring ((F ⋙ forget CommRing).obj j) | by { change comm_ring (F.obj j), apply_instance } | instance | CommRing.comm_ring_obj | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_comm_ring (F : J ⥤ CommRing.{max v u}) :
comm_ring (types.limit_cone (F ⋙ forget CommRing.{max v u})).X | @subring.to_comm_ring (Π j, F.obj j) _
(Ring.sections_subring (F ⋙ forget₂ CommRing Ring.{max v u})) | instance | CommRing.limit_comm_ring | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing",
"Ring.sections_subring",
"comm_ring",
"subring.to_comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone (F : J ⥤ CommRing.{max v u}) : cone F | lift_limit (limit.is_limit (F ⋙ (forget₂ CommRing Ring.{max v u}))) | def | CommRing.limit_cone | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | A choice of limit cone for a functor into `CommRing`.
(Generally, you'll just want to use `limit F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
limit_cone_is_limit (F : J ⥤ CommRing.{max v u}) : is_limit (limit_cone F) | lifted_limit_is_limit _ | def | CommRing.limit_cone_is_limit | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | The chosen cone is a limit cone.
(Generally, you'll just want to use `limit.cone F`.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits_of_size : has_limits_of_size.{v v} CommRing.{max v u} | { has_limits_of_shape := λ J 𝒥, by exactI
{ has_limit := λ F, has_limit_of_created F (forget₂ CommRing Ring.{max v u}) } } | instance | CommRing.has_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | The category of commutative rings has all limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_limits : has_limits CommRing.{u} | CommRing.has_limits_of_size.{u u} | instance | CommRing.has_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_Ring_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ CommRing Ring.{max v u}) | { preserves_limits_of_shape := λ J 𝒥,
{ preserves_limit := λ F, by apply_instance } } | instance | CommRing.forget₂_Ring_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | The forgetful functor from commutative rings to rings preserves all limits.
(That is, the underlying rings could have been computed instead as limits in the category of rings.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_Ring_preserves_limits : preserves_limits (forget₂ CommRing Ring.{u}) | CommRing.forget₂_Ring_preserves_limits_of_size.{u u} | instance | CommRing.forget₂_Ring_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget₂_CommSemiRing_preserves_limits_aux (F : J ⥤ CommRing.{max v u}) :
is_limit ((forget₂ CommRing CommSemiRing).map_cone (limit_cone F)) | by apply CommSemiRing.limit_cone_is_limit (F ⋙ forget₂ CommRing CommSemiRing.{max v u}) | def | CommRing.forget₂_CommSemiRing_preserves_limits_aux | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing",
"CommSemiRing",
"CommSemiRing.limit_cone_is_limit"
] | An auxiliary declaration to speed up typechecking. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_CommSemiRing_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget₂ CommRing CommSemiRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, preserves_limit_of_preserves_limit_cone
(limit_cone_is_limit F) (forget₂_CommSemiRing_preserves_limits_aux F) } } | instance | CommRing.forget₂_CommSemiRing_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | The forgetful functor from commutative rings to commutative semirings preserves all limits.
(That is, the underlying commutative semirings could have been computed instead as limits
in the category of commutative semirings.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_CommSemiRing_preserves_limits :
preserves_limits (forget₂ CommRing CommSemiRing.{u}) | CommRing.forget₂_CommSemiRing_preserves_limits_of_size.{u u} | instance | CommRing.forget₂_CommSemiRing_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_preserves_limits_of_size :
preserves_limits_of_size.{v v} (forget CommRing.{max v u}) | { preserves_limits_of_shape := λ J 𝒥, by exactI
{ preserves_limit := λ F, limits.comp_preserves_limit (forget₂ CommRing Ring) (forget Ring) } } | instance | CommRing.forget_preserves_limits_of_size | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [
"CommRing",
"Ring"
] | The forgetful functor from commutative rings to types preserves all limits.
(That is, the underlying types could have been computed instead as limits in the category of types.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_preserves_limits : preserves_limits (forget CommRing.{u}) | CommRing.forget_preserves_limits_of_size.{u u} | instance | CommRing.forget_preserves_limits | algebra.category.Ring | src/algebra/category/Ring/limits.lean | [
"algebra.ring.pi",
"algebra.category.Ring.basic",
"algebra.category.Group.limits",
"ring_theory.subring.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Magma : Type (u+1) | bundled has_mul | def | Magma | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [] | The category of magmas and magma morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bundled_hom : bundled_hom @mul_hom | ⟨@mul_hom.to_fun, @mul_hom.id, @mul_hom.comp, @mul_hom.coe_inj⟩ | instance | Magma.bundled_hom | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"mul_hom",
"mul_hom.comp",
"mul_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of (M : Type u) [has_mul M] : Magma | bundled.of M | def | Magma.of | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma"
] | Construct a bundled `Magma` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {X Y : Type u} [has_mul X] [has_mul Y] (f : X →ₙ* Y) :
of X ⟶ of Y | f | def | Magma.of_hom | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [] | Typecheck a `mul_hom` as a morphism in `Magma`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {X Y : Type u} [has_mul X] [has_mul Y] (f : X →ₙ* Y)
(x : X) : of_hom f x = f x | rfl | lemma | Magma.of_hom_apply | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [has_mul R] : (Magma.of R : Type u) = R | rfl | lemma | Magma.coe_of | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma.of"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Semigroup : Type (u+1) | bundled semigroup | def | Semigroup | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"semigroup"
] | The category of semigroups and semigroup morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (M : Type u) [semigroup M] : Semigroup | bundled.of M | def | Semigroup.of | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup",
"semigroup"
] | Construct a bundled `Semigroup` from the underlying type and typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom {X Y : Type u} [semigroup X] [semigroup Y] (f : X →ₙ* Y) :
of X ⟶ of Y | f | def | Semigroup.of_hom | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"semigroup"
] | Typecheck a `mul_hom` as a morphism in `Semigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_hom_apply {X Y : Type u} [semigroup X] [semigroup Y] (f : X →ₙ* Y)
(x : X) : of_hom f x = f x | rfl | lemma | Semigroup.of_hom_apply | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_of (R : Type u) [semigroup R] : (Semigroup.of R : Type u) = R | rfl | lemma | Semigroup.coe_of | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup.of",
"semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Magma : has_forget₂ Semigroup Magma | bundled_hom.forget₂ _ _ | instance | Semigroup.has_forget_to_Magma | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma",
"Semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.to_Magma_iso (e : X ≃* Y) : Magma.of X ≅ Magma.of Y | { hom := e.to_mul_hom,
inv := e.symm.to_mul_hom } | def | mul_equiv.to_Magma_iso | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma.of"
] | Build an isomorphism in the category `Magma` from a `mul_equiv` between `has_mul`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_Semigroup_iso (e : X ≃* Y) : Semigroup.of X ≅ Semigroup.of Y | { hom := e.to_mul_hom,
inv := e.symm.to_mul_hom } | def | mul_equiv.to_Semigroup_iso | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup.of"
] | Build an isomorphism in the category `Semigroup` from a `mul_equiv` between `semigroup`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Magma_iso_to_mul_equiv {X Y : Magma} (i : X ≅ Y) : X ≃* Y | { to_fun := i.hom,
inv_fun := i.inv,
left_inv := λ x, by simp,
right_inv := λ y, by simp,
map_mul' := by simp } | def | category_theory.iso.Magma_iso_to_mul_equiv | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma",
"inv_fun"
] | Build a `mul_equiv` from an isomorphism in the category `Magma`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Semigroup_iso_to_mul_equiv {X Y : Semigroup} (i : X ≅ Y) : X ≃* Y | { to_fun := i.hom,
inv_fun := i.inv,
left_inv := λ x, by simp,
right_inv := λ y, by simp,
map_mul' := by simp } | def | category_theory.iso.Semigroup_iso_to_mul_equiv | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup",
"inv_fun"
] | Build a `mul_equiv` from an isomorphism in the category `Semigroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_Magma_iso {X Y : Type u} [has_mul X] [has_mul Y] :
(X ≃* Y) ≅ (Magma.of X ≅ Magma.of Y) | { hom := λ e, e.to_Magma_iso,
inv := λ i, i.Magma_iso_to_mul_equiv } | def | mul_equiv_iso_Magma_iso | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma.of"
] | multiplicative equivalences between `has_mul`s are the same as (isomorphic to) isomorphisms
in `Magma` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv_iso_Semigroup_iso {X Y : Type u} [semigroup X] [semigroup Y] :
(X ≃* Y) ≅ (Semigroup.of X ≅ Semigroup.of Y) | { hom := λ e, e.to_Semigroup_iso,
inv := λ i, i.Semigroup_iso_to_mul_equiv } | def | mul_equiv_iso_Semigroup_iso | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup.of",
"semigroup"
] | multiplicative equivalences between `semigroup`s are the same as (isomorphic to) isomorphisms
in `Semigroup` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Magma.forget_reflects_isos : reflects_isomorphisms (forget Magma.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget Magma).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Magma_iso).1⟩,
end } | instance | Magma.forget_reflects_isos | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Magma"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Semigroup.forget_reflects_isos : reflects_isomorphisms (forget Semigroup.{u}) | { reflects := λ X Y f _,
begin
resetI,
let i := as_iso ((forget Semigroup).map f),
let e : X ≃* Y := { ..f, ..i.to_equiv },
exact ⟨(is_iso.of_iso e.to_Semigroup_iso).1⟩,
end } | instance | Semigroup.forget_reflects_isos | algebra.category.Semigroup | src/algebra/category/Semigroup/basic.lean | [
"algebra.pempty_instances",
"algebra.hom.equiv.basic",
"category_theory.concrete_category.bundled_hom",
"category_theory.functor.reflects_isomorphisms",
"category_theory.elementwise"
] | [
"Semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p_of_injective_algebra_map {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]
(h : function.injective (algebra_map R A)) (p : ℕ) [char_p R p] : char_p A p | { cast_eq_zero_iff := λx,
begin
rw ←char_p.cast_eq_zero_iff R p x,
change algebra_map ℕ A x = 0 ↔ algebra_map ℕ R x = 0,
rw is_scalar_tower.algebra_map_apply ℕ R A x,
refine iff.trans _ h.eq_iff,
rw ring_hom.map_zero,
end } | lemma | char_p_of_injective_algebra_map | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra",
"algebra_map",
"char_p",
"comm_semiring",
"is_scalar_tower.algebra_map_apply",
"ring_hom.map_zero",
"semiring"
] | If the algebra map `R →+* A` is injective then `A` has the same characteristic as `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p_of_injective_algebra_map' (R A : Type*) [field R] [semiring A] [algebra R A]
[nontrivial A] (p : ℕ) [char_p R p] : char_p A p | char_p_of_injective_algebra_map (algebra_map R A).injective p | lemma | char_p_of_injective_algebra_map' | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra",
"algebra_map",
"char_p",
"char_p_of_injective_algebra_map",
"field",
"nontrivial",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_zero_of_injective_algebra_map {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]
(h : function.injective (algebra_map R A)) [char_zero R] : char_zero A | { cast_injective := λ x y hxy,
begin
change algebra_map ℕ A x = algebra_map ℕ A y at hxy,
rw is_scalar_tower.algebra_map_apply ℕ R A x at hxy,
rw is_scalar_tower.algebra_map_apply ℕ R A y at hxy,
exact char_zero.cast_injective (h hxy),
end } | lemma | char_zero_of_injective_algebra_map | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra",
"algebra_map",
"char_zero",
"comm_semiring",
"is_scalar_tower.algebra_map_apply",
"semiring"
] | If the algebra map `R →+* A` is injective and `R` has characteristic zero then so does `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_rat.char_p_zero [semiring R] [algebra ℚ R] : char_p R 0 | char_p_of_injective_algebra_map (algebra_map ℚ R).injective 0 | lemma | algebra_rat.char_p_zero | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra",
"algebra_map",
"char_p",
"char_p_of_injective_algebra_map",
"semiring"
] | A nontrivial `ℚ`-algebra has `char_p` equal to zero.
This cannot be a (local) instance because it would immediately form a loop with the
instance `algebra_rat`. It's probably easier to go the other way: prove `char_zero R` and
automatically receive an `algebra ℚ R` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_rat.char_zero [ring R] [algebra ℚ R] : char_zero R | @char_p.char_p_to_char_zero R _ (algebra_rat.char_p_zero R) | lemma | algebra_rat.char_zero | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra",
"algebra_rat.char_p_zero",
"char_p.char_p_to_char_zero",
"char_zero",
"ring"
] | A nontrivial `ℚ`-algebra has characteristic zero.
This cannot be a (local) instance because it would immediately form a loop with the
instance `algebra_rat`. It's probably easier to go the other way: prove `char_zero R` and
automatically receive an `algebra ℚ R` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.char_p_iff (p : ℕ) : char_p K p ↔ char_p L p | (algebra_map K L).char_p_iff_char_p p | lemma | algebra.char_p_iff | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra_map",
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.ring_char_eq : ring_char K = ring_char L | by { rw [ring_char.eq_iff, algebra.char_p_iff K L], apply ring_char.char_p } | lemma | algebra.ring_char_eq | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"algebra.char_p_iff",
"ring_char",
"ring_char.char_p",
"ring_char.eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p [char_p R p] : char_p (free_algebra R X) p | char_p_of_injective_algebra_map free_algebra.algebra_map_left_inverse.injective p | instance | free_algebra.char_p | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_p",
"char_p_of_injective_algebra_map",
"free_algebra"
] | If `R` has characteristic `p`, then so does `free_algebra R X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero [char_zero R] : char_zero (free_algebra R X) | char_zero_of_injective_algebra_map free_algebra.algebra_map_left_inverse.injective | instance | free_algebra.char_zero | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_zero",
"char_zero_of_injective_algebra_map",
"free_algebra"
] | If `R` has characteristic `0`, then so does `free_algebra R X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p_of_is_fraction_ring [char_p R p] : char_p K p | char_p_of_injective_algebra_map (is_fraction_ring.injective R K) p | lemma | is_fraction_ring.char_p_of_is_fraction_ring | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_p",
"char_p_of_injective_algebra_map",
"is_fraction_ring.injective"
] | If `R` has characteristic `p`, then so does Frac(R). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero_of_is_fraction_ring [char_zero R] : char_zero K | @char_p.char_p_to_char_zero K _ (char_p_of_is_fraction_ring R 0) | lemma | is_fraction_ring.char_zero_of_is_fraction_ring | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_p.char_p_to_char_zero",
"char_zero"
] | If `R` has characteristic `0`, then so does Frac(R). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p [char_p R p] : char_p (fraction_ring R) p | char_p_of_is_fraction_ring R p | instance | is_fraction_ring.char_p | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_p",
"fraction_ring"
] | If `R` has characteristic `p`, then so does `fraction_ring R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_zero [char_zero R] : char_zero (fraction_ring R) | char_zero_of_is_fraction_ring R | instance | is_fraction_ring.char_zero | algebra.char_p | src/algebra/char_p/algebra.lean | [
"algebra.char_p.basic",
"ring_theory.localization.fraction_ring",
"algebra.free_algebra"
] | [
"char_zero",
"fraction_ring"
] | If `R` has characteristic `0`, then so does `fraction_ring R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_pow_prime_pow_eq (hp : p.prime) (h : commute x y) (n : ℕ) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n +
p * ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑((p ^ n).choose k / p) | begin
transitivity
x ^ p ^ n + y ^ p ^ n + ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * (p ^ n).choose k,
{ simp_rw [h.add_pow, ←nat.Ico_zero_eq_range, nat.Ico_succ_right, Icc_eq_cons_Ico (zero_le _),
finset.sum_cons, Ico_eq_cons_Ioo (pow_pos hp.pos _), finset.sum_cons, tsub_self, tsub_zero,
pow_... | lemma | commute.add_pow_prime_pow_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_prime_pow_eq",
"commute",
"finset.mul_sum",
"mul_assoc",
"mul_one",
"nat.Ico_succ_right",
"nat.cast_comm",
"nat.cast_one",
"nat.choose_self",
"nat.choose_zero_right",
"one_mul",
"pow_pos",
"pow_zero",
"tsub_self",
"tsub_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_prime_eq (hp : p.prime) (h : commute x y) :
(x + y) ^ p = x ^ p + y ^ p +
p * ∑ k in finset.Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p) | by simpa using h.add_pow_prime_pow_eq hp 1 | lemma | commute.add_pow_prime_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_prime_eq",
"commute",
"finset.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_add_pow_prime_pow_eq (hp : p.prime) (h : commute x y) (n : ℕ) :
∃ r, (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + p * r | ⟨_, h.add_pow_prime_pow_eq hp n⟩ | lemma | commute.exists_add_pow_prime_pow_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute",
"exists_add_pow_prime_pow_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_add_pow_prime_eq (hp : p.prime) (h : commute x y) :
∃ r, (x + y) ^ p = x ^ p + y ^ p + p * r | ⟨_, h.add_pow_prime_eq hp⟩ | lemma | commute.exists_add_pow_prime_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute",
"exists_add_pow_prime_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_prime_pow_eq (hp : p.prime) (x y : R) (n : ℕ) :
(x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n +
p * ∑ k in finset.Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑((p ^ n).choose k / p) | (commute.all x y).add_pow_prime_pow_eq hp n | lemma | add_pow_prime_pow_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute.all",
"finset.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_prime_eq (hp : p.prime) (x y : R) :
(x + y) ^ p = x ^ p + y ^ p +
p * ∑ k in finset.Ioo 0 p, x ^ k * y ^ (p - k) * ↑(p.choose k / p) | (commute.all x y).add_pow_prime_eq hp | lemma | add_pow_prime_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute.all",
"finset.Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_add_pow_prime_pow_eq (hp : p.prime) (x y : R) (n : ℕ) :
∃ r, (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + p * r | (commute.all x y).exists_add_pow_prime_pow_eq hp n | lemma | exists_add_pow_prime_pow_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_add_pow_prime_eq (hp : p.prime) (x y : R) :
∃ r, (x + y) ^ p = x ^ p + y ^ p + p * r | (commute.all x y).exists_add_pow_prime_eq hp | lemma | exists_add_pow_prime_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p [add_monoid_with_one R] (p : ℕ) : Prop | (cast_eq_zero_iff [] : ∀ x:ℕ, (x:R) = 0 ↔ p ∣ x) | class | char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one"
] | The generator of the kernel of the unique homomorphism ℕ → R for a semiring R.
*Warning*: for a semiring `R`, `char_p R 0` and `char_zero R` need not coincide.
* `char_p R 0` asks that only `0 : ℕ` maps to `0 : R` under the map `ℕ → R`;
* `char_zero R` requires an injection `ℕ ↪ R`.
For instance, endowing `{0, 1}` wi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_p.cast_eq_zero [add_monoid_with_one R] (p : ℕ) [char_p R p] :
(p:R) = 0 | (char_p.cast_eq_zero_iff R p p).2 (dvd_refl p) | theorem | char_p.cast_eq_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p",
"dvd_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.cast_card_eq_zero [add_group_with_one R] [fintype R] :
(fintype.card R : R) = 0 | by rw [← nsmul_one, card_nsmul_eq_zero] | lemma | char_p.cast_card_eq_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_group_with_one",
"fintype",
"fintype.card",
"nsmul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.add_order_of_one (R) [semiring R] : char_p R (add_order_of (1 : R)) | ⟨λ n, by rw [← nat.smul_one_eq_coe, add_order_of_dvd_iff_nsmul_eq_zero]⟩ | lemma | char_p.add_order_of_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"nat.smul_one_eq_coe",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.int_cast_eq_zero_iff [add_group_with_one R] (p : ℕ) [char_p R p]
(a : ℤ) :
(a : R) = 0 ↔ (p:ℤ) ∣ a | begin
rcases lt_trichotomy a 0 with h|rfl|h,
{ rw [← neg_eq_zero, ← int.cast_neg, ← dvd_neg],
lift -a to ℕ using neg_nonneg.mpr (le_of_lt h) with b,
rw [int.cast_coe_nat, char_p.cast_eq_zero_iff R p, int.coe_nat_dvd] },
{ simp only [int.cast_zero, eq_self_iff_true, dvd_zero] },
{ lift a to ℕ using (le_o... | lemma | char_p.int_cast_eq_zero_iff | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_group_with_one",
"char_p",
"dvd_neg",
"dvd_zero",
"int.cast_coe_nat",
"int.cast_neg",
"int.cast_zero",
"int.coe_nat_dvd",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.int_cast_eq_int_cast [add_group_with_one R] (p : ℕ) [char_p R p] {a b : ℤ} :
(a : R) = b ↔ a ≡ b [ZMOD p] | by rw [eq_comm, ←sub_eq_zero, ←int.cast_sub, char_p.int_cast_eq_zero_iff R p, int.modeq_iff_dvd] | lemma | char_p.int_cast_eq_int_cast | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_group_with_one",
"char_p",
"char_p.int_cast_eq_zero_iff",
"int.modeq_iff_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.nat_cast_eq_nat_cast [add_group_with_one R] (p : ℕ) [char_p R p] {a b : ℕ} :
(a : R) = b ↔ a ≡ b [MOD p] | begin
rw [←int.cast_coe_nat, ←int.cast_coe_nat b],
exact (char_p.int_cast_eq_int_cast _ _).trans int.coe_nat_modeq_iff,
end | lemma | char_p.nat_cast_eq_nat_cast | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_group_with_one",
"char_p",
"char_p.int_cast_eq_int_cast",
"int.coe_nat_modeq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.eq [add_monoid_with_one R] {p q : ℕ} (c1 : char_p R p) (c2 : char_p R q) :
p = q | nat.dvd_antisymm
((char_p.cast_eq_zero_iff R p q).1 (char_p.cast_eq_zero _ _))
((char_p.cast_eq_zero_iff R q p).1 (char_p.cast_eq_zero _ _)) | theorem | char_p.eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p",
"char_p.cast_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.of_char_zero [add_monoid_with_one R] [char_zero R] : char_p R 0 | ⟨λ x, by rw [zero_dvd_iff, ← nat.cast_zero, nat.cast_inj]⟩ | instance | char_p.of_char_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p",
"char_zero",
"nat.cast_inj",
"nat.cast_zero",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.exists [non_assoc_semiring R] : ∃ p, char_p R p | by letI := classical.dec_eq R; exact
classical.by_cases
(assume H : ∀ p:ℕ, (p:R) = 0 → p = 0, ⟨0,
⟨λ x, by rw [zero_dvd_iff]; exact ⟨H x, by rintro rfl; simp⟩⟩⟩)
(λ H, ⟨nat.find (not_forall.1 H), ⟨λ x,
⟨λ H1, nat.dvd_of_mod_eq_zero (by_contradiction $ λ H2,
nat.find_min (not_forall.1 H)
(nat.m... | theorem | char_p.exists | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"by_contradiction",
"char_p",
"classical.dec_eq",
"nat.cast_add",
"nat.cast_mul",
"non_assoc_semiring",
"not_imp_of_and_not",
"not_not_of_not_imp",
"not_of_not_imp",
"of_not_not",
"zero_dvd_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.exists_unique [non_assoc_semiring R] : ∃! p, char_p R p | let ⟨c, H⟩ := char_p.exists R in ⟨c, H, λ y H2, char_p.eq R H2 H⟩ | theorem | char_p.exists_unique | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.eq",
"char_p.exists",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.congr {R : Type u} [add_monoid_with_one R] {p : ℕ} (q : ℕ) [hq : char_p R q]
(h : q = p) :
char_p R p | h ▸ hq | theorem | char_p.congr | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_monoid_with_one",
"char_p"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_char [non_assoc_semiring R] : ℕ | classical.some (char_p.exists_unique R) | def | ring_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p.exists_unique",
"non_assoc_semiring"
] | Noncomputable function that outputs the unique characteristic of a semiring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
spec : ∀ x:ℕ, (x:R) = 0 ↔ ring_char R ∣ x | by letI := (classical.some_spec (char_p.exists_unique R)).1;
unfold ring_char; exact char_p.cast_eq_zero_iff R (ring_char R) | theorem | ring_char.spec | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p.exists_unique",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq (p : ℕ) [C : char_p R p] : ring_char R = p | ((classical.some_spec (char_p.exists_unique R)).2 p C).symm | theorem | ring_char.eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.exists_unique",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p : char_p R (ring_char R) | ⟨spec R⟩ | instance | ring_char.char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq {p : ℕ} (h : ring_char R = p) : char_p R p | char_p.congr (ring_char R) h | theorem | ring_char.of_eq | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"char_p.congr",
"of_eq",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff {p : ℕ} : ring_char R = p ↔ char_p R p | ⟨of_eq, @eq R _ p⟩ | theorem | ring_char.eq_iff | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd {x : ℕ} (hx : (x : R) = 0) : ring_char R ∣ x | (spec R x).1 hx | theorem | ring_char.dvd | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero [char_zero R] : ring_char R = 0 | eq R 0 | lemma | ring_char.eq_zero | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_zero",
"ring_char"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.cast_ring_char : (ring_char R : R) = 0 | by rw ring_char.spec | lemma | ring_char.nat.cast_ring_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"ring_char",
"ring_char.spec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_char_of_commute [semiring R] {p : ℕ} [hp : fact p.prime]
[char_p R p] (x y : R) (h : commute x y) :
(x + y)^p = x^p + y^p | let ⟨r, hr⟩ := h.exists_add_pow_prime_eq hp.out in by simp [hr] | theorem | add_pow_char_of_commute | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"commute",
"fact",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_char_pow_of_commute [semiring R] {p n : ℕ} [hp : fact p.prime] [char_p R p]
(x y : R) (h : commute x y) :
(x + y) ^ (p ^ n) = x ^ (p ^ n) + y ^ (p ^ n) | let ⟨r, hr⟩ := h.exists_add_pow_prime_pow_eq hp.out n in by simp [hr] | theorem | add_pow_char_pow_of_commute | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"commute",
"fact",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_pow_char_of_commute [ring R] {p : ℕ} [fact p.prime]
[char_p R p] (x y : R) (h : commute x y) :
(x - y)^p = x^p - y^p | begin
rw [eq_sub_iff_add_eq, ← add_pow_char_of_commute _ _ _ (commute.sub_left h rfl)],
simp, repeat {apply_instance},
end | theorem | sub_pow_char_of_commute | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char_of_commute",
"char_p",
"commute",
"commute.sub_left",
"fact",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_pow_char_pow_of_commute [ring R] {p : ℕ} [fact p.prime]
[char_p R p] {n : ℕ} (x y : R) (h : commute x y) :
(x - y) ^ (p ^ n) = x ^ (p ^ n) - y ^ (p ^ n) | begin
induction n, { simp, },
rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih],
apply sub_pow_char_of_commute, apply commute.pow_pow h,
end | theorem | sub_pow_char_pow_of_commute | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"commute",
"commute.pow_pow",
"fact",
"pow_mul",
"pow_succ'",
"ring",
"sub_pow_char_of_commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_char [comm_semiring R] {p : ℕ} [fact p.prime]
[char_p R p] (x y : R) : (x + y)^p = x^p + y^p | add_pow_char_of_commute _ _ _ (commute.all _ _) | theorem | add_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char_of_commute",
"char_p",
"comm_semiring",
"commute.all",
"fact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_char_pow [comm_semiring R] {p : ℕ} [fact p.prime]
[char_p R p] {n : ℕ} (x y : R) :
(x + y) ^ (p ^ n) = x ^ (p ^ n) + y ^ (p ^ n) | add_pow_char_pow_of_commute _ _ _ (commute.all _ _) | theorem | add_pow_char_pow | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char_pow_of_commute",
"char_p",
"comm_semiring",
"commute.all",
"fact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_pow_char [comm_ring R] {p : ℕ} [fact p.prime]
[char_p R p] (x y : R) : (x - y)^p = x^p - y^p | sub_pow_char_of_commute _ _ _ (commute.all _ _) | theorem | sub_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"comm_ring",
"commute.all",
"fact",
"sub_pow_char_of_commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_pow_char_pow [comm_ring R] {p : ℕ} [fact p.prime]
[char_p R p] {n : ℕ} (x y : R) :
(x - y) ^ (p ^ n) = x ^ (p ^ n) - y ^ (p ^ n) | sub_pow_char_pow_of_commute _ _ _ (commute.all _ _) | theorem | sub_pow_char_pow | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"comm_ring",
"commute.all",
"fact",
"sub_pow_char_pow_of_commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.neg_one_ne_one [ring R] (p : ℕ) [char_p R p] [fact (2 < p)] :
(-1 : R) ≠ (1 : R) | begin
suffices : (2 : R) ≠ 0,
{ symmetry, rw [ne.def, ← sub_eq_zero, sub_neg_eq_add], exact this },
assume h,
rw [show (2 : R) = (2 : ℕ), by norm_cast] at h,
have := (char_p.cast_eq_zero_iff R p 2).mp h,
have := nat.le_of_dvd dec_trivial this,
rw fact_iff at *, linarith,
end | lemma | char_p.neg_one_ne_one | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"fact",
"fact_iff",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.neg_one_pow_char [comm_ring R] (p : ℕ) [char_p R p] [fact p.prime] :
(-1 : R) ^ p = -1 | begin
rw eq_neg_iff_add_eq_zero,
nth_rewrite 1 ← one_pow p,
rw [← add_pow_char, add_left_neg, zero_pow (fact.out (nat.prime p)).pos],
end | lemma | char_p.neg_one_pow_char | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char",
"char_p",
"comm_ring",
"fact",
"nat.prime",
"one_pow",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
char_p.neg_one_pow_char_pow [comm_ring R] (p n : ℕ) [char_p R p] [fact p.prime] :
(-1 : R) ^ p ^ n = -1 | begin
rw eq_neg_iff_add_eq_zero,
nth_rewrite 1 ← one_pow (p ^ n),
rw [← add_pow_char_pow, add_left_neg, zero_pow (pow_pos (fact.out (nat.prime p)).pos _)],
end | lemma | char_p.neg_one_pow_char_pow | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char_pow",
"char_p",
"comm_ring",
"fact",
"nat.prime",
"one_pow",
"pow_pos",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.char_p_iff_char_p {K L : Type*} [division_ring K] [semiring L] [nontrivial L]
(f : K →+* L) (p : ℕ) :
char_p K p ↔ char_p L p | by simp only [char_p_iff, ← f.injective.eq_iff, map_nat_cast f, f.map_zero] | lemma | ring_hom.char_p_iff_char_p | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"char_p",
"division_ring",
"map_nat_cast",
"nontrivial",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frobenius : R →+* R | { to_fun := λ x, x^p,
map_one' := one_pow p,
map_mul' := λ x y, mul_pow x y p,
map_zero' := zero_pow (fact.out (nat.prime p)).pos,
map_add' := add_pow_char R } | def | frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"add_pow_char",
"mul_pow",
"nat.prime",
"one_pow",
"zero_pow"
] | The frobenius map that sends x to x^p | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frobenius_def : frobenius R p x = x ^ p | rfl | theorem | frobenius_def | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_frobenius (n : ℕ) : (frobenius R p)^[n] x = x ^ p ^ n | begin
induction n, {simp},
rw [function.iterate_succ', pow_succ', pow_mul, function.comp_apply, frobenius_def, n_ih]
end | theorem | iterate_frobenius | algebra.char_p | src/algebra/char_p/basic.lean | [
"data.int.modeq",
"data.nat.multiplicity",
"group_theory.order_of_element",
"ring_theory.nilpotent"
] | [
"frobenius",
"frobenius_def",
"function.comp_apply",
"function.iterate_succ'",
"pow_mul",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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