Theorems · Theorem · order theory
le_iff_compact_le_imp
∀ {α : Type u_2} [inst : CompleteLattice α] [IsCompactlyGenerated α] {a b : α},
a ≤ b ↔ ∀ (c : α), IsCompactElement c → c ≤ a → c ≤ b- Defined in
- Mathlib.Order.CompactlyGenerated.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Set.ofPredproof · cited by 6,101
- CompleteLatticestatement and proof · cited by 1,048
- le_transproof · cited by 985
- SupSet.sSupproof · cited by 954
- IsCompactlyGeneratedstatement and proof · cited by 37
- IsCompactElementstatement and proof · cited by 34
- sSup_le_sSupproof · cited by 24
- sSup_compact_le_eqproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- inf_sSup_eq_iSup_inf_sup_finsetproof · cited by 4
- DirectedOn.inf_sSup_eqproof · cited by 3