Theorems · Theorem · order theory
isLUB_iff_sSup_eq
∀ {α : Type u_1} [inst : CompleteSemilatticeSup α] {s : Set α} {a : α}, IsLUB s a ↔ sSup s = a- Defined in
- Mathlib.Order.CompleteLattice.Defs
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
- Assumes
- CompleteSemilatticeSup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- SupSet.sSupstatement and proof · cited by 954
- IsLUBstatement · cited by 280
- IsLUB.uniqueproof · cited by 21
- isLUB_sSupproof · cited by 21
- CompleteSemilatticeSupstatement and proof · cited by 18
Cited by3
Results whose statement or proof uses this declaration.
- IsLUB.sSup_eqproof · cited by 17
- CompleteLattice.isCompactElement_iff_le_of_directed_sSup_leproof · cited by 6
- scottContinuous_iff_map_sSupproof · cited by 2