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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
coe_subtype : (S.subtype : S → A) = coe | rfl | lemma | star_subalgebra.coe_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_apply (x : S) : S.subtype x = (x : A) | rfl | lemma | star_subalgebra.subtype_apply | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subalgebra_subtype : S.to_subalgebra.val = S.subtype.to_alg_hom | rfl | lemma | star_subalgebra.to_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inclusion {S₁ S₂ : star_subalgebra R A} (h : S₁ ≤ S₂) : S₁ →⋆ₐ[R] S₂ | { to_fun := subtype.map id h,
map_one' := rfl,
map_mul' := λ x y, rfl,
map_zero' := rfl,
map_add' := λ x y, rfl,
commutes' := λ z, rfl,
map_star' := λ x, rfl } | def | star_subalgebra.inclusion | 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",
"subtype.map"
] | The inclusion map between `star_subalgebra`s given by `subtype.map id` as a `star_alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion_injective {S₁ S₂ : star_subalgebra R A} (h : S₁ ≤ S₂) :
function.injective $ inclusion h | set.inclusion_injective h | lemma | star_subalgebra.inclusion_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"
] | [
"set.inclusion_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_comp_inclusion {S₁ S₂ : star_subalgebra R A} (h : S₁ ≤ S₂) :
S₂.subtype.comp (inclusion h) = S₁.subtype | rfl | lemma | star_subalgebra.subtype_comp_inclusion | 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 | |
map (f : A →⋆ₐ[R] B) (S : star_subalgebra R A) : star_subalgebra R B | { star_mem' :=
begin
rintro _ ⟨a, ha, rfl⟩,
exact map_star f a ▸ set.mem_image_of_mem _ (S.star_mem' ha),
end,
.. S.to_subalgebra.map f.to_alg_hom } | def | star_subalgebra.map | 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.mem_image_of_mem",
"star_subalgebra"
] | Transport a star subalgebra via a star algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono {S₁ S₂ : star_subalgebra R A} {f : A →⋆ₐ[R] B} :
S₁ ≤ S₂ → S₁.map f ≤ S₂.map f | set.image_subset f | lemma | star_subalgebra.map_mono | 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.image_subset",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective {f : A →⋆ₐ[R] B} (hf : function.injective f) :
function.injective (map f) | λ S₁ S₂ ih, ext $ set.ext_iff.1 $ set.image_injective.2 hf $ set.ext $ set_like.ext_iff.mp ih | lemma | star_subalgebra.map_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"
] | [
"ih",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (S : star_subalgebra R A) : S.map (star_alg_hom.id R A) = S | set_like.coe_injective $ set.image_id _ | lemma | star_subalgebra.map_id | 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"
] | [
"map_id",
"set.image_id",
"set_like.coe_injective",
"star_alg_hom.id",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (S : star_subalgebra R A) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B) :
(S.map f).map g = S.map (g.comp f) | set_like.coe_injective $ set.image_image _ _ _ | lemma | star_subalgebra.map_map | 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.image_image",
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {S : star_subalgebra R A} {f : A →⋆ₐ[R] B} {y : B} :
y ∈ map f S ↔ ∃ x ∈ S, f x = y | subsemiring.mem_map | lemma | star_subalgebra.mem_map | 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"
] | [
"mem_map",
"star_subalgebra",
"subsemiring.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_to_subalgebra {S : star_subalgebra R A} {f : A →⋆ₐ[R] B} :
(S.map f).to_subalgebra = S.to_subalgebra.map f.to_alg_hom | set_like.coe_injective rfl | lemma | star_subalgebra.map_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"
] | [
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_map (S : star_subalgebra R A) (f : A →⋆ₐ[R] B) :
(S.map f : set B) = f '' S | rfl | lemma | star_subalgebra.coe_map | 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 | |
comap (f : A →⋆ₐ[R] B) (S : star_subalgebra R B) : star_subalgebra R A | { star_mem' := λ a ha, show f (star a) ∈ S, from (map_star f a).symm ▸ star_mem ha,
.. S.to_subalgebra.comap f.to_alg_hom } | def | star_subalgebra.comap | 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"
] | Preimage of a star subalgebra under an star algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_le_iff_le_comap {S : star_subalgebra R A} {f : A →⋆ₐ[R] B} {U : star_subalgebra R B} :
map f S ≤ U ↔ S ≤ comap f U | set.image_subset_iff | theorem | star_subalgebra.map_le_iff_le_comap | 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.image_subset_iff",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap (f : A →⋆ₐ[R] B) : galois_connection (map f) (comap f) | λ S U, map_le_iff_le_comap | lemma | star_subalgebra.gc_map_comap | 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"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono {S₁ S₂ : star_subalgebra R B} {f : A →⋆ₐ[R] B} :
S₁ ≤ S₂ → S₁.comap f ≤ S₂.comap f | set.preimage_mono | lemma | star_subalgebra.comap_mono | 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.preimage_mono",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_injective {f : A →⋆ₐ[R] B} (hf : function.surjective f) :
function.injective (comap f) | λ S₁ S₂ h, ext $ λ b, let ⟨x, hx⟩ := hf b in let this := set_like.ext_iff.1 h x in hx ▸ this | lemma | star_subalgebra.comap_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id (S : star_subalgebra R A) : S.comap (star_alg_hom.id R A) = S | set_like.coe_injective $ set.preimage_id | lemma | star_subalgebra.comap_id | 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.preimage_id",
"set_like.coe_injective",
"star_alg_hom.id",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap (S : star_subalgebra R C) (g : B →⋆ₐ[R] C) (f : A →⋆ₐ[R] B) :
(S.comap g).comap f = S.comap (g.comp f) | set_like.coe_injective $ set.preimage_preimage | lemma | star_subalgebra.comap_comap | 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.preimage_preimage",
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap (S : star_subalgebra R B) (f : A →⋆ₐ[R] B) (x : A) :
x ∈ S.comap f ↔ f x ∈ S | iff.rfl | lemma | star_subalgebra.mem_comap | 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_comap (S : star_subalgebra R B) (f : A →⋆ₐ[R] B) :
(S.comap f : set A) = f ⁻¹' (S : set B) | rfl | lemma | star_subalgebra.coe_comap | 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 | |
_root_.set.star_mem_centralizer {a : A} {s : set A}
(h : ∀ (a : A), a ∈ s → star a ∈ s) (ha : a ∈ set.centralizer s) :
star a ∈ set.centralizer s | λ y hy, by simpa using congr_arg star (ha _ (h _ hy)).symm | lemma | set.star_mem_centralizer | 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.centralizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer
(s : set A) (w : ∀ (a : A), a ∈ s → star a ∈ s) : star_subalgebra R A | { star_mem' := λ x, set.star_mem_centralizer w,
..subalgebra.centralizer R s, } | def | star_subalgebra.centralizer | 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.star_mem_centralizer",
"star_subalgebra",
"subalgebra.centralizer"
] | The centralizer, or commutant, of a *-closed set as star subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_centralizer (s : set A) (w : ∀ (a : A), a ∈ s → star a ∈ s) :
(centralizer R s w : set A) = s.centralizer | rfl | lemma | star_subalgebra.coe_centralizer | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_centralizer_iff {s : set A} {w} {z : A} :
z ∈ centralizer R s w ↔ ∀ g ∈ s, g * z = z * g | iff.rfl | lemma | star_subalgebra.mem_centralizer_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
centralizer_le (s t : set A)
(ws : ∀ (a : A), a ∈ s → star a ∈ s) (wt : ∀ (a : A), a ∈ t → star a ∈ t) (h : s ⊆ t) :
centralizer R t wt ≤ centralizer R s ws | set.centralizer_subset h | lemma | star_subalgebra.centralizer_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"
] | [
"set.centralizer_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_star_iff (S : subalgebra R A) (x : A) : x ∈ star S ↔ star x ∈ S | iff.rfl | lemma | subalgebra.mem_star_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"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mem_star_iff (S : subalgebra R A) (x : A) : star x ∈ star S ↔ x ∈ S | by simpa only [star_star] using mem_star_iff S (star x) | lemma | subalgebra.star_mem_star_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_star",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star (S : subalgebra R A) : ((star S : subalgebra R A) : set A) = star S | rfl | lemma | subalgebra.coe_star | 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"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mono : monotone (star : subalgebra R A → subalgebra R A) | λ _ _ h _ hx, h hx | lemma | subalgebra.star_mono | 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"
] | [
"monotone",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_adjoin_comm (s : set A) : star (algebra.adjoin R s) = algebra.adjoin R (star s) | have this : ∀ t : set A, algebra.adjoin R (star t) ≤ star (algebra.adjoin R t),
from λ t, algebra.adjoin_le (λ x hx, algebra.subset_adjoin hx),
le_antisymm (by simpa only [star_star] using subalgebra.star_mono (this (star s))) (this s) | lemma | subalgebra.star_adjoin_comm | 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.adjoin",
"algebra.adjoin_le",
"algebra.subset_adjoin",
"star_star",
"subalgebra.star_mono"
] | The star operation on `subalgebra` commutes with `algebra.adjoin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_closure (S : subalgebra R A) : star_subalgebra R A | { star_mem' := λ a ha,
begin
simp only [subalgebra.mem_carrier, ←(@algebra.gi R A _ _ _).l_sup_u _ _] at *,
rw [←mem_star_iff _ a, star_adjoin_comm],
convert ha,
simp [set.union_comm],
end,
.. S ⊔ star S } | def | subalgebra.star_closure | 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.gi",
"set.union_comm",
"star_subalgebra",
"subalgebra",
"subalgebra.mem_carrier"
] | The `star_subalgebra` obtained from `S : subalgebra R A` by taking the smallest subalgebra
containing both `S` and `star S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_closure_le {S₁ : subalgebra R A} {S₂ : star_subalgebra R A} (h : S₁ ≤ S₂.to_subalgebra) :
S₁.star_closure ≤ S₂ | star_subalgebra.to_subalgebra_le_iff.1 $ sup_le h $
λ x hx, (star_star x ▸ star_mem (show star x ∈ S₂, from h $ (S₁.mem_star_iff _).1 hx) : x ∈ S₂) | lemma | subalgebra.star_closure_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"
] | [
"star_star",
"star_subalgebra",
"subalgebra",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_closure_le_iff {S₁ : subalgebra R A} {S₂ : star_subalgebra R A} :
S₁.star_closure ≤ S₂ ↔ S₁ ≤ S₂.to_subalgebra | ⟨λ h, le_sup_left.trans h, star_closure_le⟩ | lemma | subalgebra.star_closure_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",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin (s : set A) : star_subalgebra R A | { star_mem' := λ x hx, by rwa [subalgebra.mem_carrier, ←subalgebra.mem_star_iff,
subalgebra.star_adjoin_comm, set.union_star, star_star, set.union_comm],
.. (algebra.adjoin R (s ∪ star s)) } | def | star_subalgebra.adjoin | 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.adjoin",
"set.union_comm",
"set.union_star",
"star_star",
"star_subalgebra",
"subalgebra.mem_carrier",
"subalgebra.star_adjoin_comm"
] | The minimal star subalgebra that contains `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_eq_star_closure_adjoin (s : set A) : adjoin R s = (algebra.adjoin R s).star_closure | to_subalgebra_injective $
show algebra.adjoin R (s ∪ star s) = algebra.adjoin R s ⊔ star (algebra.adjoin R s),
from (subalgebra.star_adjoin_comm R s).symm ▸ algebra.adjoin_union s (star s) | lemma | star_subalgebra.adjoin_eq_star_closure_adjoin | 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.adjoin",
"algebra.adjoin_union",
"subalgebra.star_adjoin_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_to_subalgebra (s : set A) :
(adjoin R s).to_subalgebra = (algebra.adjoin R (s ∪ star s)) | rfl | lemma | star_subalgebra.adjoin_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"
] | [
"algebra.adjoin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_adjoin (s : set A) : s ⊆ adjoin R s | (set.subset_union_left s (star s)).trans algebra.subset_adjoin | lemma | star_subalgebra.subset_adjoin | 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.subset_adjoin",
"set.subset_union_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_subset_adjoin (s : set A) : star s ⊆ adjoin R s | (set.subset_union_right s (star s)).trans algebra.subset_adjoin | lemma | star_subalgebra.star_subset_adjoin | 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.subset_adjoin",
"set.subset_union_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_mem_adjoin_singleton (x : A) : x ∈ adjoin R ({x} : set A) | algebra.subset_adjoin $ set.mem_union_left _ (set.mem_singleton x) | lemma | star_subalgebra.self_mem_adjoin_singleton | 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.subset_adjoin",
"set.mem_singleton",
"set.mem_union_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_self_mem_adjoin_singleton (x : A) : star x ∈ adjoin R ({x} : set A) | star_mem $ self_mem_adjoin_singleton R x | lemma | star_subalgebra.star_self_mem_adjoin_singleton | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc : galois_connection (adjoin R : set A → star_subalgebra R A) coe | begin
intros s S,
rw [←to_subalgebra_le_iff, adjoin_to_subalgebra, algebra.adjoin_le_iff, coe_to_subalgebra],
exact ⟨λ h, (set.subset_union_left s _).trans h,
λ h, set.union_subset h $ λ x hx, star_star x ▸ star_mem (show star x ∈ S, from h hx)⟩,
end | lemma | star_subalgebra.gc | 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.adjoin_le_iff",
"galois_connection",
"set.subset_union_left",
"set.union_subset",
"star_star",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion (adjoin R : set A → star_subalgebra R A) coe | { choice := λ s hs, (adjoin R s).copy s $ le_antisymm (star_subalgebra.gc.le_u_l s) hs,
gc := star_subalgebra.gc,
le_l_u := λ S, (star_subalgebra.gc (S : set A) (adjoin R S)).1 $ le_rfl,
choice_eq := λ _ _, star_subalgebra.copy_eq _ _ _ } | def | star_subalgebra.gi | 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"
] | [
"galois_insertion",
"le_rfl",
"star_subalgebra",
"star_subalgebra.copy_eq",
"star_subalgebra.gc"
] | Galois insertion between `adjoin` and `coe`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_le {S : star_subalgebra R A} {s : set A} (hs : s ⊆ S) : adjoin R s ≤ S | star_subalgebra.gc.l_le hs | lemma | star_subalgebra.adjoin_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"
] | [
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_le_iff {S : star_subalgebra R A} {s : set A} : adjoin R s ≤ S ↔ s ⊆ S | star_subalgebra.gc _ _ | lemma | star_subalgebra.adjoin_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",
"star_subalgebra.gc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.subalgebra.star_closure_eq_adjoin (S : subalgebra R A) :
S.star_closure = adjoin R (S : set A) | le_antisymm (subalgebra.star_closure_le_iff.2 $ subset_adjoin R (S : set A))
(adjoin_le (le_sup_left : S ≤ S ⊔ star S)) | lemma | subalgebra.star_closure_eq_adjoin | 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"
] | [
"le_sup_left",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_induction {s : set A} {p : A → Prop} {a : A} (h : a ∈ adjoin R s)
(Hs : ∀ (x : A), x ∈ s → p x) (Halg : ∀ (r : R), p (algebra_map R A r))
(Hadd : ∀ (x y : A), p x → p y → p (x + y)) (Hmul : ∀ (x y : A), p x → p y → p (x * y))
(Hstar : ∀ (x : A), p x → p (star x)) : p a | algebra.adjoin_induction h (λ x hx, hx.elim (λ hx, Hs x hx) (λ hx, star_star x ▸ Hstar _ (Hs _ hx)))
Halg Hadd Hmul | lemma | star_subalgebra.adjoin_induction | 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.adjoin_induction",
"algebra_map",
"star_star"
] | If some predicate holds for all `x ∈ (s : set A)` and this predicate is closed under the
`algebra_map`, addition, multiplication and star operations, then it holds for `a ∈ adjoin R s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_induction₂ {s : set A} {p : A → A → Prop} {a b : A} (ha : a ∈ adjoin R s)
(hb : b ∈ adjoin R s) (Hs : ∀ (x : A), x ∈ s → ∀ (y : A), y ∈ s → p x y)
(Halg : ∀ (r₁ r₂ : R), p (algebra_map R A r₁) (algebra_map R A r₂))
(Halg_left : ∀ (r : R) (x : A), x ∈ s → p (algebra_map R A r) x)
(Halg_right : ∀ (r : R) (... | begin
refine algebra.adjoin_induction₂ ha hb (λ x hx y hy, _) Halg (λ r x hx, _) (λ r x hx, _)
Hadd_left Hadd_right Hmul_left Hmul_right,
{ cases hx; cases hy,
exacts [Hs x hx y hy, star_star y ▸ Hstar_right _ _ (Hs _ hx _ hy),
star_star x ▸ Hstar_left _ _ (Hs _ hx _ hy),
star_star x ▸ star_star... | lemma | star_subalgebra.adjoin_induction₂ | 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.adjoin_induction₂",
"algebra_map",
"star_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_induction' {s : set A} {p : adjoin R s → Prop} (a : adjoin R s)
(Hs : ∀ x (h : x ∈ s), p ⟨x, subset_adjoin R s h⟩)
(Halg : ∀ r, p (algebra_map R _ r)) (Hadd : ∀ x y, p x → p y → p (x + y))
(Hmul : ∀ x y, p x → p y → p (x * y)) (Hstar : ∀ x, p x → p (star x)) : p a | subtype.rec_on a $ λ b hb,
begin
refine exists.elim _ (λ (hb : b ∈ adjoin R s) (hc : p ⟨b, hb⟩), hc),
apply adjoin_induction hb,
exacts [λ x hx, ⟨subset_adjoin R s hx, Hs x hx⟩,
λ r, ⟨star_subalgebra.algebra_map_mem _ r, Halg r⟩,
(λ x y hx hy, exists.elim hx $ λ hx' hx,
exists.elim hy $ λ hy' hy, ⟨a... | lemma | star_subalgebra.adjoin_induction' | 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"
] | The difference with `star_subalgebra.adjoin_induction` is that this acts on the subtype. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_comm_semiring_of_comm {s : set A}
(hcomm : ∀ (a : A), a ∈ s → ∀ (b : A), b ∈ s → a * b = b * a)
(hcomm_star : ∀ (a : A), a ∈ s → ∀ (b : A), b ∈ s → a * star b = star b * a) :
comm_semiring (adjoin R s) | { mul_comm :=
begin
rintro ⟨x, hx⟩ ⟨y, hy⟩,
ext,
simp only [set_like.coe_mk, mul_mem_class.mk_mul_mk],
rw [←mem_to_subalgebra, adjoin_to_subalgebra] at hx hy,
letI : comm_semiring (algebra.adjoin R (s ∪ star s)) := algebra.adjoin_comm_semiring_of_comm R
begin
intros a ha b hb,
case... | def | star_subalgebra.adjoin_comm_semiring_of_comm | 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.adjoin",
"algebra.adjoin_comm_semiring_of_comm",
"comm_semiring",
"mul_comm",
"mul_mem_class.mk_mul_mk",
"set_like.coe_mk",
"star_star"
] | If all elements of `s : set A` commute pairwise and also commute pairwise with elements of
`star s`, then `star_subalgebra.adjoin R s` is commutative. See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_comm_ring_of_comm (R : Type u) {A : Type v} [comm_ring R] [star_ring R]
[ring A] [algebra R A] [star_ring A] [star_module R A] {s : set A}
(hcomm : ∀ (a : A), a ∈ s → ∀ (b : A), b ∈ s → a * b = b * a)
(hcomm_star : ∀ (a : A), a ∈ s → ∀ (b : A), b ∈ s → a * star b = star b * a) :
comm_ring (adjoin R s) | { ..star_subalgebra.adjoin_comm_semiring_of_comm R hcomm hcomm_star,
..(adjoin R s).to_subalgebra.to_ring } | def | star_subalgebra.adjoin_comm_ring_of_comm | 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_ring",
"ring",
"star_module",
"star_ring",
"star_subalgebra.adjoin_comm_semiring_of_comm"
] | If all elements of `s : set A` commute pairwise and also commute pairwise with elements of
`star s`, then `star_subalgebra.adjoin R s` is commutative. See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_comm_semiring_of_is_star_normal (x : A) [is_star_normal x] :
comm_semiring (adjoin R ({x} : set A)) | adjoin_comm_semiring_of_comm R
(λ a ha b hb, by { rw [set.mem_singleton_iff] at ha hb, rw [ha, hb] })
(λ a ha b hb,
by { rw [set.mem_singleton_iff] at ha hb, simpa only [ha, hb] using (star_comm_self' x).symm }) | instance | star_subalgebra.adjoin_comm_semiring_of_is_star_normal | 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"
] | [
"comm_semiring",
"is_star_normal",
"set.mem_singleton_iff",
"star_comm_self'"
] | The star subalgebra `star_subalgebra.adjoin R {x}` generated by a single `x : A` is commutative
if `x` is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_comm_ring_of_is_star_normal (R : Type u) {A : Type v} [comm_ring R] [star_ring R]
[ring A] [algebra R A] [star_ring A] [star_module R A] (x : A) [is_star_normal x] :
comm_ring (adjoin R ({x} : set A)) | { mul_comm := mul_comm, ..(adjoin R ({x} : set A)).to_subalgebra.to_ring } | instance | star_subalgebra.adjoin_comm_ring_of_is_star_normal | 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_ring",
"is_star_normal",
"mul_comm",
"ring",
"star_module",
"star_ring"
] | The star subalgebra `star_subalgebra.adjoin R {x}` generated by a single `x : A` is commutative
if `x` is normal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_top : (↑(⊤ : star_subalgebra R A) : set A) = set.univ | rfl | lemma | star_subalgebra.coe_top | 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 | |
mem_top {x : A} : x ∈ (⊤ : star_subalgebra R A) | set.mem_univ x | lemma | star_subalgebra.mem_top | 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.mem_univ",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_to_subalgebra : (⊤ : star_subalgebra R A).to_subalgebra = ⊤ | rfl | lemma | star_subalgebra.top_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_eq_top {S : star_subalgebra R A} : S.to_subalgebra = ⊤ ↔ S = ⊤ | star_subalgebra.to_subalgebra_injective.eq_iff' top_to_subalgebra | lemma | star_subalgebra.to_subalgebra_eq_top | 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 | |
mem_sup_left {S T : star_subalgebra R A} : ∀ {x : A}, x ∈ S → x ∈ S ⊔ T | show S ≤ S ⊔ T, from le_sup_left | lemma | star_subalgebra.mem_sup_left | 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"
] | [
"le_sup_left",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_right {S T : star_subalgebra R A} : ∀ {x : A}, x ∈ T → x ∈ S ⊔ T | show T ≤ S ⊔ T, from le_sup_right | lemma | star_subalgebra.mem_sup_right | 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"
] | [
"le_sup_right",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_sup {S T : star_subalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈ T) :
x * y ∈ S ⊔ T | mul_mem (mem_sup_left hx) (mem_sup_right hy) | lemma | star_subalgebra.mul_mem_sup | 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 | |
map_sup (f : A →⋆ₐ[R] B) (S T : star_subalgebra R A) : map f (S ⊔ T) = map f S ⊔ map f T | (star_subalgebra.gc_map_comap f).l_sup | lemma | star_subalgebra.map_sup | 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",
"star_subalgebra.gc_map_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (S T : star_subalgebra R A) : (↑(S ⊓ T) : set A) = S ∩ T | rfl | lemma | star_subalgebra.coe_inf | 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 | |
mem_inf {S T : star_subalgebra R A} {x : A} : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T | iff.rfl | lemma | star_subalgebra.mem_inf | 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 | |
inf_to_subalgebra (S T : star_subalgebra R A) :
(S ⊓ T).to_subalgebra = S.to_subalgebra ⊓ T.to_subalgebra | rfl | lemma | star_subalgebra.inf_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_Inf (S : set (star_subalgebra R A)) : (↑(Inf S) : set A) = ⋂ s ∈ S, ↑s | Inf_image | lemma | star_subalgebra.coe_Inf | 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"
] | [
"Inf_image",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (star_subalgebra R A)} {x : A} : x ∈ Inf S ↔ ∀ p ∈ S, x ∈ p | by simp only [← set_like.mem_coe, coe_Inf, set.mem_Inter₂] | lemma | star_subalgebra.mem_Inf | 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.mem_Inter₂",
"set_like.mem_coe",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_to_subalgebra (S : set (star_subalgebra R A)) :
(Inf S).to_subalgebra = Inf (star_subalgebra.to_subalgebra '' S) | set_like.coe_injective $ by simp | lemma | star_subalgebra.Inf_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"
] | [
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι : Sort*} {S : ι → star_subalgebra R A} : (↑(⨅ i, S i) : set A) = ⋂ i, S i | by simp [infi] | lemma | star_subalgebra.coe_infi | 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"
] | [
"infi",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι : Sort*} {S : ι → star_subalgebra R A} {x : A} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i | by simp only [infi, mem_Inf, set.forall_range_iff] | lemma | star_subalgebra.mem_infi | 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"
] | [
"infi",
"set.forall_range_iff",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_to_subalgebra {ι : Sort*} (S : ι → star_subalgebra R A) :
(⨅ i, S i).to_subalgebra = ⨅ i, (S i).to_subalgebra | set_like.coe_injective $ by simp | lemma | star_subalgebra.infi_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"
] | [
"set_like.coe_injective",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_to_subalgebra : (⊥ : star_subalgebra R A).to_subalgebra = ⊥ | by { change algebra.adjoin R (∅ ∪ star ∅) = algebra.adjoin R ∅, simp } | lemma | star_subalgebra.bot_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"
] | [
"algebra.adjoin",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : A} : x ∈ (⊥ : star_subalgebra R A) ↔ x ∈ set.range (algebra_map R A) | by rw [←mem_to_subalgebra, bot_to_subalgebra, algebra.mem_bot] | theorem | star_subalgebra.mem_bot | 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.mem_bot",
"algebra_map",
"set.range",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : star_subalgebra R A) : set A) = set.range (algebra_map R A) | by simp [set.ext_iff, mem_bot] | theorem | star_subalgebra.coe_bot | 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",
"set.ext_iff",
"set.range",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff {S : star_subalgebra R A} :
S = ⊤ ↔ ∀ x : A, x ∈ S | ⟨λ h x, by rw h; exact mem_top, λ h, by ext x; exact ⟨λ _, mem_top, λ _, h x⟩⟩ | theorem | star_subalgebra.eq_top_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"
] | [
"eq_top_iff",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equalizer : star_subalgebra R A | { carrier := {a | f a = g a},
mul_mem' := λ a b (ha : f a = g a) (hb : f b = g b),
by rw [set.mem_set_of_eq, map_mul f, map_mul g, ha, hb],
add_mem' := λ a b (ha : f a = g a) (hb : f b = g b),
by rw [set.mem_set_of_eq, map_add f, map_add g, ha, hb],
algebra_map_mem' := λ r, by simp only [set.mem_set_of_eq... | def | star_alg_hom.equalizer | 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"
] | [
"map_mul",
"star_subalgebra"
] | The equalizer of two star `R`-algebra homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_equalizer (x : A) : x ∈ star_alg_hom.equalizer f g ↔ f x = g x | iff.rfl | lemma | star_alg_hom.mem_equalizer | 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_alg_hom.equalizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_le_equalizer {s : set A} (h : s.eq_on f g) : adjoin R s ≤ star_alg_hom.equalizer f g | adjoin_le h | lemma | star_alg_hom.adjoin_le_equalizer | 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_alg_hom.equalizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_of_adjoin_eq_top {s : set A}
(h : adjoin R s = ⊤) ⦃f g : F⦄ (hs : s.eq_on f g) : f = g | fun_like.ext f g $ λ x, star_alg_hom.adjoin_le_equalizer f g hs $ h.symm ▸ trivial | lemma | star_alg_hom.ext_of_adjoin_eq_top | 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"
] | [
"fun_like.ext",
"star_alg_hom.adjoin_le_equalizer"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_adjoin [star_module R B] (f : A →⋆ₐ[R] B) (s : set A) :
map f (adjoin R s) = adjoin R (f '' s) | galois_connection.l_comm_of_u_comm set.image_preimage (gc_map_comap f) star_subalgebra.gc
star_subalgebra.gc (λ _, rfl) | lemma | star_alg_hom.map_adjoin | 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"
] | [
"galois_connection.l_comm_of_u_comm",
"set.image_preimage",
"star_module",
"star_subalgebra.gc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_adjoin {s : set A} [star_alg_hom_class F R (adjoin R s) B] {f g : F}
(h : ∀ x : adjoin R s, (x : A) ∈ s → f x = g x) : f = g | begin
refine fun_like.ext f g (λ a, adjoin_induction' a (λ x hx, _) (λ r, _) (λ x y hx hy, _)
(λ x y hx hy, _) (λ x hx, _)),
{ exact h ⟨x, subset_adjoin R s hx⟩ hx },
{ simp only [alg_hom_class.commutes] },
{ rw [map_add, map_add, hx, hy] },
{ rw [map_mul, map_mul, hx, hy] },
{ rw [map_star, map_star, h... | lemma | star_alg_hom.ext_adjoin | 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"
] | [
"fun_like.ext",
"map_mul",
"star_alg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_adjoin_singleton {a : A} [star_alg_hom_class F R (adjoin R ({a} : set A)) B] {f g : F}
(h : f ⟨a, self_mem_adjoin_singleton R a⟩ = g ⟨a, self_mem_adjoin_singleton R a⟩) : f = g | ext_adjoin $ λ x hx, (show x = ⟨a, self_mem_adjoin_singleton R a⟩,
from subtype.ext $ set.mem_singleton_iff.mp hx).symm ▸ h | lemma | star_alg_hom.ext_adjoin_singleton | 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_alg_hom_class",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unitary (R : Type*) [monoid R] [star_semigroup R] : submonoid R | { carrier := {U | star U * U = 1 ∧ U * star U = 1},
one_mem' := by simp only [mul_one, and_self, set.mem_set_of_eq, star_one],
mul_mem' := λ U B ⟨hA₁, hA₂⟩ ⟨hB₁, hB₂⟩,
begin
refine ⟨_, _⟩,
{ calc star (U * B) * (U * B) = star B * star U * U * B : by simp only [mul_assoc, star_mul]
... | def | unitary | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"monoid",
"mul_assoc",
"mul_one",
"star_one",
"star_semigroup",
"submonoid"
] | In a *-monoid, `unitary R` is the submonoid consisting of all the elements `U` of
`R` such that `star U * U = 1` and `U * star U = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_iff {U : R} : U ∈ unitary R ↔ star U * U = 1 ∧ U * star U = 1 | iff.rfl | lemma | unitary.mem_iff | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self_of_mem {U : R} (hU : U ∈ unitary R) : star U * U = 1 | hU.1 | lemma | unitary.star_mul_self_of_mem | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_self_of_mem {U : R} (hU : U ∈ unitary R) : U * star U = 1 | hU.2 | lemma | unitary.mul_star_self_of_mem | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mem {U : R} (hU : U ∈ unitary R) : star U ∈ unitary R | ⟨by rw [star_star, mul_star_self_of_mem hU], by rw [star_star, star_mul_self_of_mem hU]⟩ | lemma | unitary.star_mem | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"star_star",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mem_iff {U : R} : star U ∈ unitary R ↔ U ∈ unitary R | ⟨λ h, star_star U ▸ star_mem h, star_mem⟩ | lemma | unitary.star_mem_iff | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"star_star",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star {U : unitary R} : ↑(star U) = (star U : R) | rfl | lemma | unitary.coe_star | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star_mul_self (U : unitary R) : (star U : R) * U = 1 | star_mul_self_of_mem U.prop | lemma | unitary.coe_star_mul_self | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_star_self (U : unitary R) : (U : R) * star U = 1 | mul_star_self_of_mem U.prop | lemma | unitary.coe_mul_star_self | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self (U : unitary R) : star U * U = 1 | subtype.ext $ coe_star_mul_self U | lemma | unitary.star_mul_self | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"subtype.ext",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_self (U : unitary R) : U * star U = 1 | subtype.ext $ coe_mul_star_self U | lemma | unitary.mul_star_self | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"subtype.ext",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_inv (U : unitary R) : star U = U⁻¹ | rfl | lemma | unitary.star_eq_inv | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_inv' : (star : unitary R → unitary R) = has_inv.inv | rfl | lemma | unitary.star_eq_inv' | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_units : unitary R →* Rˣ | { to_fun := λ x, ⟨x, ↑(x⁻¹), coe_mul_star_self x, coe_star_mul_self x⟩,
map_one' := units.ext rfl,
map_mul' := λ x y, units.ext rfl } | def | unitary.to_units | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"to_units",
"unitary",
"units.ext"
] | The unitary elements embed into the units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_units_injective : function.injective (to_units : unitary R → Rˣ) | λ x y h, subtype.ext $ units.ext_iff.mp h | lemma | unitary.to_units_injective | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"subtype.ext",
"to_units",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_star_mul_self {U : R} : U ∈ unitary R ↔ star U * U = 1 | mem_iff.trans $ and_iff_left_of_imp $ λ h, mul_comm (star U) U ▸ h | lemma | unitary.mem_iff_star_mul_self | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"and_iff_left_of_imp",
"mul_comm",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_self_mul_star {U : R} : U ∈ unitary R ↔ U * star U = 1 | mem_iff.trans $ and_iff_right_of_imp $ λ h, mul_comm U (star U) ▸ h | lemma | unitary.mem_iff_self_mul_star | algebra.star | src/algebra/star/unitary.lean | [
"algebra.star.basic",
"group_theory.submonoid.operations"
] | [
"and_iff_right_of_imp",
"mul_comm",
"unitary"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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