English
Every set in a lattice generates a set closed under join; the closure operator yields a Sublattice containing the original set.
Русский
Каждое множество в решётке порождает множество, замкнутое относительно объединения; замыкание образует подпредмет-теорему.
LaTeX
$$$\mathrm{IsSublattice}(\mathrm{latticeClosure}(s))$$$
Lean4
@[simp]
theorem infClosure_prod (s : Set α) (t : Set β) : infClosure (s ×ˢ t) = infClosure s ×ˢ infClosure t :=
le_antisymm
(infClosure_min (Set.prod_mono subset_infClosure subset_infClosure) <|
infClosed_infClosure.prod infClosed_infClosure) <|
by
rintro ⟨_, _⟩ ⟨⟨u, hu, hus, rfl⟩, v, hv, hvt, rfl⟩
refine ⟨u ×ˢ v, hu.product hv, ?_, ?_⟩
· simpa only [coe_product] using Set.prod_mono hus hvt
· simp [prodMk_inf'_inf']
