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[Merged by Bors] - feat: interaction between StarSubalgebra.map and topologicalClosure #5166

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[Merged by Bors] - feat: interaction between StarSubalgebra.map and topologicalClosure #5166

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feat: interaction between `StarSubalgebra.map` and `topologicalClosure`
j-loreaux Jun 17, 2023
8f9997a
rename decls
j-loreaux Jun 17, 2023
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15 changes: 15 additions & 0 deletions Mathlib/Topology/Algebra/StarSubalgebra.lean
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Expand Up @@ -107,6 +107,21 @@ theorem topologicalClosure_mono : Monotone (topologicalClosure : _ → StarSubal
topologicalClosure_minimal (h.trans <| le_topologicalClosure S₂) (isClosed_topologicalClosure S₂)
#align star_subalgebra.topological_closure_mono StarSubalgebra.topologicalClosure_mono

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 :=
hφ.closure_image_subset _

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 :=
image_closure_subset_closure_image hφ

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 :=
SetLike.coe_injective <| hφ.closure_image_eq _

/-- If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. -/
@[reducible]
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