submonoid_closure — Mathlib

English

For a submonoid S of M, the units of S form a subgroup of the unit group of M, i.e., S.units ≤ Units(M).

Русский

Для подмонома S моноида M едииницы S образуют подгруппу группы единиц M.

LaTeX

$$$S.{\\rm units} \\le \\mathrm{Units}(M)$$$

Lean4

@[to_additive]
theorem submonoid_closure (hpos : ∀ x : α, x ∈ s → 1 ≤ x) (h : s.IsPWO) : IsPWO (Submonoid.closure s : Set α) :=
  by
  rw [Submonoid.closure_eq_image_prod]
  refine (h.partiallyWellOrderedOn_sublistForall₂ (· ≤ ·)).image_of_monotone_on ?_
  exact fun l1 _ l2 hl2 h12 => h12.prod_le_prod' fun x hx => hpos x <| hl2 x hx

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