supClosure — Mathlib

English

InfClosure is closed under SupClosure to yield latticeClosure: infClosure(supClosure s) = latticeClosure s.

Русский

InfClosure сохраняется под SupClosure и даёт latticeClosure: infClosure(supClosure s) = latticeClosure s.

LaTeX

$$$\infClosure(\supClosure(s)) = \mathrm{latticeClosure}(s)$$$

Lean4

protected theorem supClosure (hs : InfClosed s) : InfClosed (supClosure s) :=
  by
  rintro _ ⟨t, ht, hts, rfl⟩ _ ⟨u, hu, hus, rfl⟩
  rw [sup'_inf_sup']
  exact finsetSup'_mem_supClosure _ fun i hi ↦ hs (hts (mem_product.1 hi).1) (hus (mem_product.1 hi).2)

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