statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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non_unital_star_alg_hom (R A B : Type*) [monoid R]
[non_unital_non_assoc_semiring A] [distrib_mul_action R A] [has_star A]
[non_unital_non_assoc_semiring B] [distrib_mul_action R B] [has_star B]
extends A →ₙₐ[R] B | (map_star' : ∀ a : A, to_fun (star a) = star (to_fun a)) | structure | non_unital_star_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"distrib_mul_action",
"has_star",
"monoid",
"non_unital_non_assoc_semiring"
] | A *non-unital ⋆-algebra homomorphism* is a non-unital algebra homomorphism between
non-unital `R`-algebras `A` and `B` equipped with a `star` operation, and this homomorphism is
also `star`-preserving. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_star_alg_hom_class (F : Type*) (R : out_param Type*) (A : out_param Type*)
(B : out_param Type*) [monoid R] [has_star A] [has_star B]
[non_unital_non_assoc_semiring A] [non_unital_non_assoc_semiring B]
[distrib_mul_action R A] [distrib_mul_action R B]
extends non_unital_alg_hom_class F R A B, star_ho... | class | non_unital_star_alg_hom_class | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"distrib_mul_action",
"has_star",
"monoid",
"non_unital_alg_hom_class",
"non_unital_non_assoc_semiring",
"star_hom_class"
] | `non_unital_star_alg_hom_class F R A B` asserts `F` is a type of bundled non-unital ⋆-algebra
homomorphisms from `A` to `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe {F : Type*} [non_unital_star_alg_hom_class F R A B] (f : F) :
⇑(f : A →⋆ₙₐ[R] B) = f | rfl | lemma | non_unital_star_alg_hom.coe_coe | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"coe_coe",
"non_unital_star_alg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_non_unital_alg_hom {f : A →⋆ₙₐ[R] B} :
(f.to_non_unital_alg_hom : A → B) = f | rfl | lemma | non_unital_star_alg_hom.coe_to_non_unital_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : A →⋆ₙₐ[R] B} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | non_unital_star_alg_hom.ext | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = f) : A →⋆ₙₐ[R] B | { to_fun := f',
map_smul' := h.symm ▸ map_smul f,
map_zero' := h.symm ▸ map_zero f,
map_add' := h.symm ▸ map_add f,
map_mul' := h.symm ▸ map_mul f,
map_star' := h.symm ▸ map_star f } | def | non_unital_star_alg_hom.copy | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"map_mul"
] | Copy of a `non_unital_star_alg_hom` with a new `to_fun` equal to the old one. Useful
to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | non_unital_star_alg_hom.coe_copy | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : A →⋆ₙₐ[R] B) (f' : A → B) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | non_unital_star_alg_hom.copy_eq | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : A → B) (h₁ h₂ h₃ h₄ h₅) :
((⟨f, h₁, h₂, h₃, h₄, h₅⟩ : A →⋆ₙₐ[R] B) : A → B) = f | rfl | lemma | non_unital_star_alg_hom.coe_mk | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : A →⋆ₙₐ[R] B) (h₁ h₂ h₃ h₄ h₅) :
(⟨f, h₁, h₂, h₃, h₄, h₅⟩ : A →⋆ₙₐ[R] B) = f | by { ext, refl, } | lemma | non_unital_star_alg_hom.mk_coe | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : A →⋆ₙₐ[R] A | { map_star' := λ x, rfl, .. (1 : A →ₙₐ[R] A) } | def | non_unital_star_alg_hom.id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | The identity as a non-unital ⋆-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(non_unital_star_alg_hom.id R A) = id | rfl | lemma | non_unital_star_alg_hom.coe_id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : A →⋆ₙₐ[R] C | { map_star' := by simp only [map_star, non_unital_alg_hom.to_fun_eq_coe, eq_self_iff_true,
non_unital_alg_hom.coe_comp, coe_to_non_unital_alg_hom, function.comp_app, forall_const],
.. f.to_non_unital_alg_hom.comp g.to_non_unital_alg_hom } | def | non_unital_star_alg_hom.comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"forall_const",
"non_unital_alg_hom.coe_comp",
"non_unital_alg_hom.to_fun_eq_coe"
] | The composition of non-unital ⋆-algebra homomorphisms, as a non-unital ⋆-algebra
homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) : ⇑(comp f g) = f ∘ g | rfl | lemma | non_unital_star_alg_hom.coe_comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : B →⋆ₙₐ[R] C) (g : A →⋆ₙₐ[R] B) (a : A) : comp f g a = f (g a) | rfl | lemma | non_unital_star_alg_hom.comp_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : C →⋆ₙₐ[R] D) (g : B →⋆ₙₐ[R] C) (h : A →⋆ₙₐ[R] B) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | non_unital_star_alg_hom.comp_assoc | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : A →⋆ₙₐ[R] B) : (non_unital_star_alg_hom.id _ _).comp f = f | ext $ λ _, rfl | lemma | non_unital_star_alg_hom.id_comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : A →⋆ₙₐ[R] B) : f.comp (non_unital_star_alg_hom.id _ _) = f | ext $ λ _, rfl | lemma | non_unital_star_alg_hom.comp_id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : A →⋆ₙₐ[R] A) : A → A) = id | rfl | lemma | non_unital_star_alg_hom.coe_one | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : A) : (1 : A →⋆ₙₐ[R] A) a = a | rfl | lemma | non_unital_star_alg_hom.one_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : A →⋆ₙₐ[R] B) : A → B) = 0 | rfl | lemma | non_unital_star_alg_hom.coe_zero | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (a : A) : (0 : A →⋆ₙₐ[R] B) a = 0 | rfl | lemma | non_unital_star_alg_hom.zero_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_alg_hom (R A B: Type*) [comm_semiring R] [semiring A] [algebra R A] [has_star A]
[semiring B] [algebra R B] [has_star B] extends alg_hom R A B | (map_star' : ∀ x : A, to_fun (star x) = star (to_fun x)) | structure | star_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom",
"algebra",
"comm_semiring",
"has_star",
"semiring"
] | A *⋆-algebra homomorphism* is an algebra homomorphism between `R`-algebras `A` and `B`
equipped with a `star` operation, and this homomorphism is also `star`-preserving. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_alg_hom_class (F : Type*) (R : out_param Type*) (A : out_param Type*)
(B : out_param Type*) [comm_semiring R] [semiring A] [algebra R A] [has_star A]
[semiring B] [algebra R B] [has_star B] extends alg_hom_class F R A B, star_hom_class F A B | class | star_alg_hom_class | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom_class",
"algebra",
"comm_semiring",
"has_star",
"semiring",
"star_hom_class"
] | `star_alg_hom_class F R A B` states that `F` is a type of ⋆-algebra homomorphisms.
You should also extend this typeclass when you extend `star_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_star_alg_hom_class : non_unital_star_alg_hom_class F R A B | { map_smul := map_smul,
.. star_alg_hom_class.to_alg_hom_class F R A B,
.. star_alg_hom_class.to_star_hom_class F R A B, } | instance | star_alg_hom_class.to_non_unital_star_alg_hom_class | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_star_alg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe {F : Type*} [star_alg_hom_class F R A B] (f : F) :
⇑(f : A →⋆ₐ[R] B) = f | rfl
initialize_simps_projections star_alg_hom (to_fun → apply) | lemma | star_alg_hom.coe_coe | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"coe_coe",
"star_alg_hom",
"star_alg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_alg_hom {f : A →⋆ₐ[R] B} :
(f.to_alg_hom : A → B) = f | rfl | lemma | star_alg_hom.coe_to_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : A →⋆ₐ[R] B} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | star_alg_hom.ext | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = f) : A →⋆ₐ[R] B | { to_fun := f',
map_one' := h.symm ▸ map_one f ,
map_mul' := h.symm ▸ map_mul f,
map_zero' := h.symm ▸ map_zero f,
map_add' := h.symm ▸ map_add f,
commutes' := h.symm ▸ alg_hom_class.commutes f,
map_star' := h.symm ▸ map_star f } | def | star_alg_hom.copy | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"map_mul",
"map_one"
] | Copy of a `star_alg_hom` with a new `to_fun` equal to the old one. Useful
to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | star_alg_hom.coe_copy | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : A →⋆ₐ[R] B) (f' : A → B) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | star_alg_hom.copy_eq | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : A → B) (h₁ h₂ h₃ h₄ h₅ h₆) :
((⟨f, h₁, h₂, h₃, h₄, h₅, h₆⟩ : A →⋆ₐ[R] B) : A → B) = f | rfl | lemma | star_alg_hom.coe_mk | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : A →⋆ₐ[R] B) (h₁ h₂ h₃ h₄ h₅ h₆) :
(⟨f, h₁, h₂, h₃, h₄, h₅, h₆⟩ : A →⋆ₐ[R] B) = f | by { ext, refl, } | lemma | star_alg_hom.mk_coe | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : A →⋆ₐ[R] A | { map_star' := λ x, rfl, .. alg_hom.id _ _ } | def | star_alg_hom.id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom.id"
] | The identity as a `star_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(star_alg_hom.id R A) = id | rfl | lemma | star_alg_hom.coe_id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) : A →⋆ₐ[R] C | { map_star' := by simp only [map_star, alg_hom.to_fun_eq_coe, alg_hom.coe_comp, coe_to_alg_hom,
function.comp_app, eq_self_iff_true, forall_const],
.. f.to_alg_hom.comp g.to_alg_hom } | def | star_alg_hom.comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom.coe_comp",
"alg_hom.to_fun_eq_coe",
"forall_const"
] | The composition of ⋆-algebra homomorphisms, as a ⋆-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) : ⇑(comp f g) = f ∘ g | rfl | lemma | star_alg_hom.coe_comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : B →⋆ₐ[R] C) (g : A →⋆ₐ[R] B) (a : A) : comp f g a = f (g a) | rfl | lemma | star_alg_hom.comp_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : C →⋆ₐ[R] D) (g : B →⋆ₐ[R] C) (h : A →⋆ₐ[R] B) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | star_alg_hom.comp_assoc | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : A →⋆ₐ[R] B) : (star_alg_hom.id _ _).comp f = f | ext $ λ _, rfl | lemma | star_alg_hom.id_comp | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : A →⋆ₐ[R] B) : f.comp (star_alg_hom.id _ _) = f | ext $ λ _, rfl | lemma | star_alg_hom.comp_id | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_non_unital_star_alg_hom (f : A →⋆ₐ[R] B) : A →⋆ₙₐ[R] B | { map_smul' := map_smul f, .. f, } | def | star_alg_hom.to_non_unital_star_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | A unital morphism of ⋆-algebras is a `non_unital_star_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_non_unital_star_alg_hom (f : A →⋆ₐ[R] B) :
(f.to_non_unital_star_alg_hom : A → B) = f | rfl | lemma | star_alg_hom.coe_to_non_unital_star_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : A × B →⋆ₙₐ[R] A | { map_star' := λ x, rfl, .. non_unital_alg_hom.fst R A B } | def | non_unital_star_alg_hom.fst | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_alg_hom.fst"
] | The first projection of a product is a non-unital ⋆-algebra homomoprhism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : A × B →⋆ₙₐ[R] B | { map_star' := λ x, rfl, .. non_unital_alg_hom.snd R A B } | def | non_unital_star_alg_hom.snd | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"non_unital_alg_hom.snd"
] | The second projection of a product is a non-unital ⋆-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : (A →⋆ₙₐ[R] B × C) | { map_star' := λ x, by simp [map_star, prod.star_def],
.. f.to_non_unital_alg_hom.prod g.to_non_unital_alg_hom } | def | non_unital_star_alg_hom.prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"prod.star_def"
] | The `pi.prod` of two morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | non_unital_star_alg_hom.coe_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) :
(fst R B C).comp (prod f g) = f | by ext; refl | theorem | non_unital_star_alg_hom.fst_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_prod (f : A →⋆ₙₐ[R] B) (g : A →⋆ₙₐ[R] C) :
(snd R B C).comp (prod f g) = g | by ext; refl | theorem | non_unital_star_alg_hom.snd_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv : ((A →⋆ₙₐ[R] B) × (A →⋆ₙₐ[R] C)) ≃ (A →⋆ₙₐ[R] B × C) | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((fst _ _ _).comp f, (snd _ _ _).comp f),
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl } | def | non_unital_star_alg_hom.prod_equiv | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"inv_fun"
] | Taking the product of two maps with the same domain is equivalent to taking the product of
their codomains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl : A →⋆ₙₐ[R] A × B | prod 1 0 | def | non_unital_star_alg_hom.inl | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | The left injection into a product is a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr : B →⋆ₙₐ[R] A × B | prod 0 1 | def | non_unital_star_alg_hom.inr | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | The right injection into a product is a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst : A × B →⋆ₐ[R] A | { map_star' := λ x, rfl, .. alg_hom.fst R A B } | def | star_alg_hom.fst | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom.fst"
] | The first projection of a product is a ⋆-algebra homomoprhism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : A × B →⋆ₐ[R] B | { map_star' := λ x, rfl, .. alg_hom.snd R A B } | def | star_alg_hom.snd | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"alg_hom.snd"
] | The second projection of a product is a ⋆-algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : (A →⋆ₐ[R] B × C) | { map_star' := λ x, by simp [prod.star_def, map_star],
.. f.to_alg_hom.prod g.to_alg_hom } | def | star_alg_hom.prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"prod.star_def"
] | The `pi.prod` of two morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | star_alg_hom.coe_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) :
(fst R B C).comp (prod f g) = f | by ext; refl | theorem | star_alg_hom.fst_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_prod (f : A →⋆ₐ[R] B) (g : A →⋆ₐ[R] C) :
(snd R B C).comp (prod f g) = g | by ext; refl | theorem | star_alg_hom.snd_prod | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_equiv : ((A →⋆ₐ[R] B) × (A →⋆ₐ[R] C)) ≃ (A →⋆ₐ[R] B × C) | { to_fun := λ f, f.1.prod f.2,
inv_fun := λ f, ((fst _ _ _).comp f, (snd _ _ _).comp f),
left_inv := λ f, by ext; refl,
right_inv := λ f, by ext; refl } | def | star_alg_hom.prod_equiv | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"inv_fun"
] | Taking the product of two maps with the same domain is equivalent to taking the product of
their codomains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_alg_equiv (R A B : Type*) [has_add A] [has_mul A] [has_smul R A] [has_star A]
[has_add B] [has_mul B] [has_smul R B] [has_star B] extends A ≃+* B | (map_star' : ∀ a : A, to_fun (star a) = star (to_fun a))
(map_smul' : ∀ (r : R) (a : A), to_fun (r • a) = r • to_fun a) | structure | star_alg_equiv | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"has_smul",
"has_star"
] | A *⋆-algebra* equivalence is an equivalence preserving addition, multiplication, scalar
multiplication and the star operation, which allows for considering both unital and non-unital
equivalences with a single structure. Currently, `alg_equiv` requires unital algebras, which is
why this structure does not extend it. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_alg_equiv_class (F : Type*) (R : out_param Type*) (A : out_param Type*)
(B : out_param Type*) [has_add A] [has_mul A] [has_smul R A] [has_star A] [has_add B] [has_mul B]
[has_smul R B] [has_star B] extends ring_equiv_class F A B | (map_star : ∀ (f : F) (a : A), f (star a) = star (f a))
(map_smul : ∀ (f : F) (r : R) (a : A), f (r • a) = r • f a) | class | star_alg_equiv_class | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"has_smul",
"has_star",
"ring_equiv_class"
] | `star_alg_equiv_class F R A B` asserts `F` is a type of bundled ⋆-algebra equivalences between
`A` and `B`.
You should also extend this typeclass when you extend `star_alg_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {f g : A ≃⋆ₐ[R] B} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | star_alg_equiv.ext | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : A ≃⋆ₐ[R] B} : f = g ↔ ∀ a, f a = g a | fun_like.ext_iff | lemma | star_alg_equiv.ext_iff | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : A ≃⋆ₐ[R] A | { map_smul' := λ r a, rfl, map_star' := λ a, rfl, ..ring_equiv.refl A } | def | star_alg_equiv.refl | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"ring_equiv.refl"
] | Star algebra equivalences are reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_refl : ⇑(refl : A ≃⋆ₐ[R] A) = id | rfl | lemma | star_alg_equiv.coe_refl | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : A ≃⋆ₐ[R] B) : B ≃⋆ₐ[R] A | { map_star' := λ b, by simpa only [e.left_inv (star (e.inv_fun b)), e.right_inv b]
using congr_arg e.inv_fun (e.map_star' (e.inv_fun b)).symm,
map_smul' := λ r b, by simpa only [e.left_inv (r • e.inv_fun b), e.right_inv b]
using congr_arg e.inv_fun (e.map_smul' r (e.inv_fun b)).symm,
..e.to_ring_equiv.symm,... | def | star_alg_equiv.symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | Star algebra equivalences are symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (e : A ≃⋆ₐ[R] B) : B → A | e.symm
initialize_simps_projections star_alg_equiv (to_fun → apply, inv_fun → simps.symm_apply) | def | star_alg_equiv.simps.symm_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"inv_fun",
"star_alg_equiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_eq_symm {e : A ≃⋆ₐ[R] B} : e.inv_fun = e.symm | rfl | lemma | star_alg_equiv.inv_fun_eq_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (e : A ≃⋆ₐ[R] B) : e.symm.symm = e | by { ext, refl, } | lemma | star_alg_equiv.symm_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective : function.bijective (symm : (A ≃⋆ₐ[R] B) → (B ≃⋆ₐ[R] A)) | equiv.bijective ⟨symm, symm, symm_symm, symm_symm⟩ | lemma | star_alg_equiv.symm_bijective | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"equiv.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe' (e : A ≃⋆ₐ[R] B) (f h₁ h₂ h₃ h₄ h₅ h₆) :
(⟨f, e, h₁, h₂, h₃, h₄, h₅, h₆⟩ : B ≃⋆ₐ[R] A) = e.symm | symm_bijective.injective $ ext $ λ x, rfl | lemma | star_alg_equiv.mk_coe' | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk (f f') (h₁ h₂ h₃ h₄ h₅ h₆) :
(⟨f, f', h₁, h₂, h₃, h₄, h₅, h₆⟩ : A ≃⋆ₐ[R] B).symm =
{ to_fun := f', inv_fun := f,
..(⟨f, f', h₁, h₂, h₃, h₄, h₅, h₆⟩ : A ≃⋆ₐ[R] B).symm } | rfl | lemma | star_alg_equiv.symm_mk | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (star_alg_equiv.refl : A ≃⋆ₐ[R] A).symm = star_alg_equiv.refl | rfl | lemma | star_alg_equiv.refl_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_equiv_symm (f : A ≃⋆ₐ[R] B) : (f : A ≃+* B).symm = f.symm | rfl | lemma | star_alg_equiv.to_ring_equiv_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_ring_equiv (e : A ≃⋆ₐ[R] B) : (e.symm : B ≃+* A) = (e : A ≃+* B).symm | rfl | lemma | star_alg_equiv.symm_to_ring_equiv | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) : A ≃⋆ₐ[R] C | { map_smul' := λ r a, show e₂.to_fun (e₁.to_fun (r • a)) = r • e₂.to_fun (e₁.to_fun a),
by rw [e₁.map_smul', e₂.map_smul'],
map_star' := λ a, show e₂.to_fun (e₁.to_fun (star a)) = star (e₂.to_fun (e₁.to_fun a)),
by rw [e₁.map_star', e₂.map_star'],
..(e₁.to_ring_equiv.trans e₂.to_ring_equiv), } | def | star_alg_equiv.trans | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | Star algebra equivalences are transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (e : A ≃⋆ₐ[R] B) : ∀ x, e (e.symm x) = x | e.to_ring_equiv.apply_symm_apply | lemma | star_alg_equiv.apply_symm_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (e : A ≃⋆ₐ[R] B) : ∀ x, e.symm (e x) = x | e.to_ring_equiv.symm_apply_apply | lemma | star_alg_equiv.symm_apply_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : C) :
(e₁.trans e₂).symm x = e₁.symm (e₂.symm x) | rfl | lemma | star_alg_equiv.symm_trans_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) :
⇑(e₁.trans e₂) = e₂ ∘ e₁ | rfl | lemma | star_alg_equiv.coe_trans | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_apply (e₁ : A ≃⋆ₐ[R] B) (e₂ : B ≃⋆ₐ[R] C) (x : A) :
(e₁.trans e₂) x = e₂ (e₁ x) | rfl | lemma | star_alg_equiv.trans_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_symm (e : A ≃⋆ₐ[R] B) : function.left_inverse e.symm e | e.left_inv | theorem | star_alg_equiv.left_inverse_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inverse_symm (e : A ≃⋆ₐ[R] B) : function.right_inverse e.symm e | e.right_inv | theorem | star_alg_equiv.right_inverse_symm | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_star_alg_hom (f : F) (g : G) (h₁ : ∀ x, g (f x) = x) (h₂ : ∀ x, f (g x) = x) :
A ≃⋆ₐ[R] B | { to_fun := f,
inv_fun := g,
left_inv := h₁,
right_inv := h₂,
map_add' := map_add f,
map_mul' := map_mul f,
map_smul' := map_smul f,
map_star' := map_star f } | def | star_alg_equiv.of_star_alg_hom | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"inv_fun",
"map_mul"
] | If a (unital or non-unital) star algebra morphism has an inverse, it is an isomorphism of
star algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_bijective (f : F) (hf : function.bijective f) : A ≃⋆ₐ[R] B | { to_fun := f,
map_star' := map_star f,
map_smul' := map_smul f,
.. ring_equiv.of_bijective f (hf : function.bijective (f : A → B)), } | def | star_alg_equiv.of_bijective | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"ring_equiv.of_bijective"
] | Promote a bijective star algebra homomorphism to a star algebra equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_bijective {f : F} (hf : function.bijective f) :
(star_alg_equiv.of_bijective f hf : A → B) = f | rfl | lemma | star_alg_equiv.coe_of_bijective | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_equiv.of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective_apply {f : F} (hf : function.bijective f) (a : A) :
(star_alg_equiv.of_bijective f hf) a = f a | rfl | lemma | star_alg_equiv.of_bijective_apply | algebra.star | src/algebra/star/star_alg_hom.lean | [
"algebra.hom.non_unital_alg",
"algebra.star.prod",
"algebra.algebra.prod"
] | [
"star_alg_equiv.of_bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_subalgebra (R : Type u) (A : Type v) [comm_semiring R] [star_ring R]
[semiring A] [star_ring A] [algebra R A] [star_module R A] extends subalgebra R A : Type v | (star_mem' {a} : a ∈ carrier → star a ∈ carrier) | structure | star_subalgebra | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"algebra",
"comm_semiring",
"semiring",
"star_module",
"star_ring",
"subalgebra"
] | A *-subalgebra is a subalgebra of a *-algebra which is closed under *. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier {s : star_subalgebra R A} {x : A} : x ∈ s.carrier ↔ x ∈ s | iff.rfl | lemma | star_subalgebra.mem_carrier | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {S T : star_subalgebra R A} (h : ∀ x : A, x ∈ S ↔ x ∈ T) : S = T | set_like.ext h | theorem | star_subalgebra.ext | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"set_like.ext",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_subalgebra {S : star_subalgebra R A} {x} : x ∈ S.to_subalgebra ↔ x ∈ S | iff.rfl | lemma | star_subalgebra.mem_to_subalgebra | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_subalgebra (S : star_subalgebra R A) : (S.to_subalgebra : set A) = S | rfl | lemma | star_subalgebra.coe_to_subalgebra | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra_injective :
function.injective (to_subalgebra : star_subalgebra R A → subalgebra R A) | λ S T h, ext $ λ x, by rw [← mem_to_subalgebra, ← mem_to_subalgebra, h] | theorem | star_subalgebra.to_subalgebra_injective | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra_inj {S U : star_subalgebra R A} : S.to_subalgebra = U.to_subalgebra ↔ S = U | to_subalgebra_injective.eq_iff | theorem | star_subalgebra.to_subalgebra_inj | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra_le_iff {S₁ S₂ : star_subalgebra R A} :
S₁.to_subalgebra ≤ S₂.to_subalgebra ↔ S₁ ≤ S₂ | iff.rfl | lemma | star_subalgebra.to_subalgebra_le_iff | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (S : star_subalgebra R A) (s : set A) (hs : s = ↑S) : star_subalgebra R A | { carrier := s,
add_mem' := λ _ _, hs.symm ▸ S.add_mem',
mul_mem' := λ _ _, hs.symm ▸ S.mul_mem',
algebra_map_mem' := hs.symm ▸ S.algebra_map_mem',
star_mem' := λ _, hs.symm ▸ S.star_mem' } | def | star_subalgebra.copy | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | Copy of a star subalgebra with a new `carrier` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (S : star_subalgebra R A) (s : set A) (hs : s = ↑S) :
(S.copy s hs : set A) = s | rfl | lemma | star_subalgebra.coe_copy | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (S : star_subalgebra R A) (s : set A) (hs : s = ↑S) : S.copy s hs = S | set_like.coe_injective hs | lemma | star_subalgebra.copy_eq | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
srange_le : (algebra_map R A).srange ≤ S.to_subalgebra.to_subsemiring | λ x ⟨r, hr⟩, hr ▸ S.algebra_map_mem r | theorem | star_subalgebra.srange_le | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype : S →⋆ₐ[R] A | by refine_struct { to_fun := (coe : S → A) }; intros; refl | def | star_subalgebra.subtype | algebra.star | src/algebra/star/subalgebra.lean | [
"algebra.star.star_alg_hom",
"algebra.algebra.subalgebra.basic",
"algebra.star.pointwise",
"algebra.star.module",
"ring_theory.adjoin.basic"
] | [] | Embedding of a subalgebra into the algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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