nonUnitalCommSemiringTopologicalClosure

English

If a nonunital subsemiring is commutative, its topological closure is a commutative semiring as well.

Русский

Если неполное подпол semiring коммутативно, то его топологическое замыкание тоже коммутативное полусемiring.

LaTeX

$$$\\mathrm{NonUnitalCommSemiring}(\\mathrm{Subtype}\\big( s.topologicalClosure\\big))$, under commutativity of multiplication$$

Lean4

/-- If a non-unital subsemiring of a non-unital topological semiring is commutative, then so is its
topological closure.

See note [reducible non-instances] -/
abbrev nonUnitalCommSemiringTopologicalClosure [T2Space R] (s : NonUnitalSubsemiring R)
    (hs : ∀ x y : s, x * y = y * x) : NonUnitalCommSemiring s.topologicalClosure :=
  { NonUnitalSubsemiringClass.toNonUnitalSemiring s.topologicalClosure,
    s.toSubsemigroup.commSemigroupTopologicalClosure hs with }

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