iSup_mem_of_nonempty — Mathlib

English

Complement image preserves lattice closure: compl '' latticeClosure s = latticeClosure (compl '' s).

Русский

Образ дополнения через комплемента сохраняет замыкание: compl '' latticeClosure s = latticeClosure (compl '' s).

LaTeX

$$$\\\\operatorname{compl}''\\\\operatorname{latticeClosure}(s) =\\\\operatorname{latticeClosure}(\\\\operatorname{compl}''(s)).$$$

Lean4

theorem iSup_mem_of_nonempty [Finite ι] [Nonempty ι] (hs : SupClosed s) (hf : ∀ i, f i ∈ s) : ⨆ i, f i ∈ s :=
  by
  cases nonempty_fintype (PLift ι)
  rw [← iSup_plift_down, ← Finset.sup'_univ_eq_ciSup]
  exact hs.finsetSup'_mem Finset.univ_nonempty fun _ _ ↦ hf _

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