closure_le — Mathlib

English

For a family of substructures, membership in the infimum is equivalent to membership in all members.

Русский

Для семейства подструктур членство в инфимуемумe эквивалентно членству во всех его членах.

LaTeX

$$$$ x \in iInf( S ) \iff ∀ i, x \in S_i $$$$

Lean4

/-- A substructure `S` includes `closure L s` if and only if it includes `s`. -/
@[simp]
theorem closure_le : closure L s ≤ S ↔ s ⊆ S :=
  (closure L).closure_le_closed_iff_le s S.closed

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