subsemiringClosure_toNonUnitalSubsemiring

English

The Subsemiring closure of a multiplicative submonoid M, viewed as a nonunital subsemiring, equals the closure of M in the nonunital sense.

Русский

Замыкание подsemiring для умноженного надмножества M, рассмотренное как ненулевой подsemiring, равно замыканию M в ненулевом смысле.

LaTeX

$$M.subsemiringClosure.toNonUnitalSubsemiring = .closure M$$

Lean4

@[simp]
theorem subsemiringClosure_toNonUnitalSubsemiring (M : Submonoid R) :
    M.subsemiringClosure.toNonUnitalSubsemiring = .closure M :=
  by
  refine Eq.symm (NonUnitalSubsemiring.closure_eq_of_le ?_ fun _ hx ↦ ?_)
  · simp [Submonoid.subsemiringClosure_coe]
  · simp only [Subsemiring.mem_toNonUnitalSubsemiring, subsemiringClosure_mem] at hx
    induction hx using AddSubmonoid.closure_induction <;> aesop

Похожие утверждения