Theorems · Theorem · order theory
IsCompactlyGenerated.exists_sSup_eq
∀ {α : Type u_3} {inst : CompleteLattice α} [self : IsCompactlyGenerated α] (x : α),
∃ s, (∀ x ∈ s, IsCompactElement x) ∧ sSup s = xIn a compactly generated complete lattice,
every element is the sSup of some set of compact elements.
- Defined in
- Mathlib.Order.CompactlyGenerated.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses no axioms
- Assumes
- IsCompactlyGenerated
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement · cited by 954
- IsCompactlyGeneratedstatement and proof · cited by 37
- IsCompactElementstatement · cited by 34
Cited by2
Results whose statement or proof uses this declaration.
- IsCompactlyGenerated.BooleanGenerators.atomisticproof · cited by 4
- sSup_compact_le_eqproof · cited by 2