closure_mono — Mathlib

English

Membership in sInf S is equivalent to membership in all members of S.

Русский

Членство в sInf S эквивалентно принадлежности всем элементам множества S.

LaTeX

$$x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p$$

Lean4

/-- Submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
then `closure s ≤ closure t`. -/
@[to_additive (attr := gcongr) /-- Additive submonoid closure of a set is monotone in its argument: if `s ⊆ t`,
      then `closure s ≤ closure t`. -/
    ]
theorem closure_mono ⦃s t : Set M⦄ (h : s ⊆ t) : closure s ≤ closure t :=
  closure_le.2 <| Subset.trans h subset_closure

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