case intro.intro
ι : Sort ?u.86401
α : Type u_1
inst✝² : CompleteLattice α
f : ι → α
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a b : α
s✝ s : Set α
hs : ∀ (x : α), x ∈ s → CompleteLattice.IsCompactElement x
⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s
ι : Sort ?u.86401
α : Type u_1
inst✝² : CompleteLattice α
f : ι → α
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a b✝ : α
s : Set α
b : α
⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b
rcases IsCompactlyGenerated.exists_sSup_eq b with ⟨s, hs, rfl⟩
no goals
case intro.intro
ι : Sort ?u.86401
α : Type u_1
inst✝² : CompleteLattice α
f : ι → α
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a b : α
s✝ s : Set α
hs : ∀ (x : α), x ∈ s → CompleteLattice.IsCompactElement x
⊢ sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ sSup s} = sSup s
exact le_antisymm (sSup_le fun c hc => hc.2) (sSup_le_sSup fun c cs => ⟨hs c cs, le_sSup cs⟩)