File: Mathlib/Topology/Algebra/StarSubalgebra.lean

full_name: StarSubalgebra.instTopologicalSemiringSubtypeMem
code: instance [TopologicalSemiring A] (s : StarSubalgebra R A) : TopologicalSemiring s 
goal: TopologicalSemiring ↥s

full_name: StarSubalgebra.embedding_inclusion
code: /-- The `StarSubalgebra.inclusion` of a star subalgebra is an `Embedding`. -/
theorem embedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂) : Embedding (inclusion h) 
goal: Embedding ⇑(StarSubalgebra.inclusion h)

full_name: StarSubalgebra.closedEmbedding_inclusion
code: /-- The `StarSubalgebra.inclusion` of a closed star subalgebra is a `ClosedEmbedding`. -/
theorem closedEmbedding_inclusion {S₁ S₂ : StarSubalgebra R A} (h : S₁ ≤ S₂)
    (hS₁ : IsClosed (S₁ : Set A)) : ClosedEmbedding (inclusion h) 
goal: ClosedEmbedding ⇑(StarSubalgebra.inclusion h)

full_name: StarSubalgebra.topologicalClosure
code: /-- The closure of a star subalgebra in a topological star algebra as a star subalgebra. -/
def topologicalClosure (s : StarSubalgebra R A) : StarSubalgebra R A 
goal: StarSubalgebra R A

full_name: StarSubalgebra.topologicalClosure_toSubalgebra_comm
code: theorem topologicalClosure_toSubalgebra_comm (s : StarSubalgebra R A) :
    s.topologicalClosure.toSubalgebra = s.toSubalgebra.topologicalClosure 
goal: s.topologicalClosure.toSubalgebra = s.topologicalClosure

full_name: StarSubalgebra.topologicalClosure_coe
code: @[simp]
theorem topologicalClosure_coe (s : StarSubalgebra R A) :
    (s.topologicalClosure : Set A) = closure (s : Set A) 
goal: ↑s.topologicalClosure = closure ↑s

full_name: StarSubalgebra.le_topologicalClosure
code: theorem le_topologicalClosure (s : StarSubalgebra R A) : s ≤ s.topologicalClosure 
goal: s ≤ s.topologicalClosure

full_name: StarSubalgebra.isClosed_topologicalClosure
code: theorem isClosed_topologicalClosure (s : StarSubalgebra R A) :
    IsClosed (s.topologicalClosure : Set A) 
goal: IsClosed ↑s.topologicalClosure

full_name: StarSubalgebra.instCompleteSpaceSubtypeMemTopologicalClosure
code: instance {A : Type*} [UniformSpace A] [CompleteSpace A] [Semiring A] [StarRing A]
    [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A]
    {S : StarSubalgebra R A} : CompleteSpace S.topologicalClosure 
goal: CompleteSpace ↥S.topologicalClosure

full_name: StarSubalgebra.topologicalClosure_minimal
code: theorem topologicalClosure_minimal {s t : StarSubalgebra R A} (h : s ≤ t)
    (ht : IsClosed (t : Set A)) : s.topologicalClosure ≤ t 
goal: s.topologicalClosure ≤ t

full_name: StarSubalgebra.topologicalClosure_mono
code: theorem topologicalClosure_mono : Monotone (topologicalClosure : _ → StarSubalgebra R A) 
goal: Monotone StarSubalgebra.topologicalClosure

full_name: StarSubalgebra.topologicalClosure_map_le
code: theorem topologicalClosure_map_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : IsClosedMap φ) :
    (map φ s).topologicalClosure ≤ map φ s.topologicalClosure 
goal: (StarSubalgebra.map φ s).topologicalClosure ≤ StarSubalgebra.map φ s.topologicalClosure

full_name: StarSubalgebra.map_topologicalClosure_le
code: theorem map_topologicalClosure_le [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : Continuous φ) :
    map φ s.topologicalClosure ≤ (map φ s).topologicalClosure 
goal: StarSubalgebra.map φ s.topologicalClosure ≤ (StarSubalgebra.map φ s).topologicalClosure

full_name: StarSubalgebra.topologicalClosure_map
code: theorem topologicalClosure_map [StarModule R B] [TopologicalSemiring B] [ContinuousStar B]
    (s : StarSubalgebra R A) (φ : A →⋆ₐ[R] B) (hφ : ClosedEmbedding φ) :
    (map φ s).topologicalClosure = map φ s.topologicalClosure 
goal: (StarSubalgebra.map φ s).topologicalClosure = StarSubalgebra.map φ s.topologicalClosure

full_name: Subalgebra.topologicalClosure_star_comm
code: theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
    (star s).topologicalClosure = star s.topologicalClosure 
goal: (star s).topologicalClosure = star s.topologicalClosure

full_name: StarSubalgebra.commSemiringTopologicalClosure
code: /-- If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. -/
abbrev commSemiringTopologicalClosure [T2Space A] (s : StarSubalgebra R A)
    (hs : ∀ x y : s, x * y = y * x) : CommSemiring s.topologicalClosure 
goal: CommSemiring ↥s.topologicalClosure

full_name: StarSubalgebra.commRingTopologicalClosure
code: /-- If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. -/
abbrev commRingTopologicalClosure {R A} [CommRing R] [StarRing R] [TopologicalSpace A] [Ring A]
    [Algebra R A] [StarRing A] [StarModule R A] [TopologicalRing A] [ContinuousStar A] [T2Space A]
    (s : StarSubalgebra R A) (hs : ∀ x y : s, x * y = y * x) : CommRing s.topologicalClosure 
goal: CommRing ↥s.topologicalClosure

full_name: StarAlgHom.ext_topologicalClosure
code: /-- Continuous `StarAlgHom`s from the topological closure of a `StarSubalgebra` whose
compositions with the `StarSubalgebra.inclusion` map agree are, in fact, equal. -/
theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
    {φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
    (h :
      φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
    φ = ψ 
goal: φ = ψ

full_name: StarAlgHomClass.ext_topologicalClosure
code: theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
    {S : StarSubalgebra R A} [FunLike F S.topologicalClosure B]
    [AlgHomClass F R S.topologicalClosure B] [StarAlgHomClass F R S.topologicalClosure B] {φ ψ : F}
    (hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S,
        φ (inclusion (le_topologicalClosure S) x) = ψ ((inclusion (le_topologicalClosure S)) x)) :
    φ = ψ 
goal: φ = ψ

full_name: elementalStarAlgebra
code: /-- The topological closure of the subalgebra generated by a single element. -/
def elementalStarAlgebra (x : A) : StarSubalgebra R A 
goal: StarSubalgebra R A

full_name: elementalStarAlgebra.self_mem
code: @[aesop safe apply (rule_sets := [SetLike])]
theorem self_mem (x : A) : x ∈ elementalStarAlgebra R x 
goal: x ∈ elementalStarAlgebra R x

full_name: elementalStarAlgebra.star_self_mem
code: theorem star_self_mem (x : A) : star x ∈ elementalStarAlgebra R x 
goal: star x ∈ elementalStarAlgebra R x

full_name: elementalStarAlgebra.instCommSemiringSubtypeMemStarSubalgebraOfT2SpaceOfIsStarNormal
code: /-- The `elementalStarAlgebra` generated by a normal element is commutative. -/
instance [T2Space A] {x : A} [IsStarNormal x] : CommSemiring (elementalStarAlgebra R x) 
goal: CommSemiring ↥(elementalStarAlgebra R x)

full_name: elementalStarAlgebra.instCommRingSubtypeMemStarSubalgebraOfT2SpaceOfIsStarNormal
code: /-- The `elementalStarAlgebra` generated by a normal element is commutative. -/
instance {R A} [CommRing R] [StarRing R] [TopologicalSpace A] [Ring A] [Algebra R A] [StarRing A]
    [StarModule R A] [TopologicalRing A] [ContinuousStar A] [T2Space A] {x : A} [IsStarNormal x] :
    CommRing (elementalStarAlgebra R x) 
goal: CommRing ↥(elementalStarAlgebra R x)

full_name: elementalStarAlgebra.isClosed
code: theorem isClosed (x : A) : IsClosed (elementalStarAlgebra R x : Set A) 
goal: IsClosed ↑(elementalStarAlgebra R x)

full_name: elementalStarAlgebra.instCompleteSpaceSubtypeMemStarSubalgebra
code: instance {A : Type*} [UniformSpace A] [CompleteSpace A] [Semiring A] [StarRing A]
    [TopologicalSemiring A] [ContinuousStar A] [Algebra R A] [StarModule R A] (x : A) :
    CompleteSpace (elementalStarAlgebra R x) 
goal: CompleteSpace ↥(elementalStarAlgebra R x)

full_name: elementalStarAlgebra.le_of_isClosed_of_mem
code: theorem le_of_isClosed_of_mem {S : StarSubalgebra R A} (hS : IsClosed (S : Set A)) {x : A}
    (hx : x ∈ S) : elementalStarAlgebra R x ≤ S 
goal: elementalStarAlgebra R x ≤ S

full_name: elementalStarAlgebra.closedEmbedding_coe
code: /-- The coercion from an elemental algebra to the full algebra as a `ClosedEmbedding`. -/
theorem closedEmbedding_coe (x : A) : ClosedEmbedding ((↑) : elementalStarAlgebra R x → A) 
goal: ClosedEmbedding Subtype.val

full_name: elementalStarAlgebra.induction_on
code: @[elab_as_elim]
theorem induction_on {x y : A}
    (hy : y ∈ elementalStarAlgebra R x) {P : (u : A) → u ∈ elementalStarAlgebra R x → Prop}
    (self : P x (self_mem R x)) (star_self : P (star x) (star_self_mem R x))
    (algebraMap : ∀ r, P (algebraMap R A r) (_root_.algebraMap_mem _ r))
    (add : ∀ u hu v hv, P u hu → P v hv → P (u + v) (add_mem hu hv))
    (mul : ∀ u hu v hv, P u hu → P v hv → P (u * v) (mul_mem hu hv))
    (closure : ∀ s : Set A, (hs : s ⊆ elementalStarAlgebra R x) → (∀ u, (hu : u ∈ s) →
      P u (hs hu)) → ∀ v, (hv : v ∈ closure s) → P v (closure_minimal hs (isClosed R x) hv)) :
    P y hy 
goal: P y hy

full_name: elementalStarAlgebra.starAlgHomClass_ext
code: theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
    [FunLike F (elementalStarAlgebra R a) B] [AlgHomClass F R _ B] [StarAlgHomClass F R _ B]
    {φ ψ : F} (hφ : Continuous φ)
    (hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ 
goal: φ = ψ