infClosure — Mathlib

English

If s is SupClosed in a distributive lattice, then infClosure(s) is SupClosed.

Русский

Если s является SupClosed в распределённой решётке, то infClosure(s) тоже SupClosed.

LaTeX

$$$\text{SupClosed}(s) \Rightarrow \text{SupClosed}(\infClosure s)$$$

Lean4

protected theorem infClosure (hs : SupClosed s) : SupClosed (infClosure s) :=
  by
  rintro _ ⟨t, ht, hts, rfl⟩ _ ⟨u, hu, hus, rfl⟩
  rw [inf'_sup_inf']
  exact finsetInf'_mem_infClosure _ fun i hi ↦ hs (hts (mem_product.1 hi).1) (hus (mem_product.1 hi).2)

Похожие утверждения