English
A set s is sSup independent iff every finite subset t of s is sSup independent.
Русский
Множество s является sSup независимым тогда и только тогда, когда всякое конечное подп мн组ение t ⊆ s также является независимым.
LaTeX
$$$ sSupIndep s \iff \forall t : Finset α, t \subseteq s \to sSupIndep t $$$
Lean4
theorem sSupIndep_iff_finite {s : Set α} : sSupIndep s ↔ ∀ t : Finset α, ↑t ⊆ s → sSupIndep (↑t : Set α) :=
⟨fun hs _ ht => hs.mono ht, fun h a ha =>
by
rw [disjoint_iff, inf_sSup_eq_iSup_inf_sup_finset, iSup_eq_bot]
intro t
rw [iSup_eq_bot, Finset.sup_id_eq_sSup]
intro ht
classical
have h' := (h (insert a t) ?_ (t.mem_insert_self a)).eq_bot
· rwa [Finset.coe_insert, Set.insert_diff_self_of_notMem] at h'
exact fun con => ((Set.mem_diff a).1 (ht con)).2 (Set.mem_singleton a)
· rw [Finset.coe_insert, Set.insert_subset_iff]
exact ⟨ha, Set.Subset.trans ht diff_subset⟩⟩
