Theorems · Theorem · order theory
csInf_upperBounds_eq_csSup
∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α},
BddAbove s → s.Nonempty → sInf (upperBounds s) = sSup s- Cited by
- 2 results in Mathlib
- Foundations
- Depth 10 from the axioms · uses no axioms
- Assumes
- ConditionallyCompleteLattice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Set.Nonemptystatement and proof · cited by 2,627
- SupSet.sSupstatement · cited by 954
- InfSet.sInfstatement · cited by 935
- BddAbovestatement and proof · cited by 620
- ConditionallyCompleteLatticestatement and proof · cited by 364
- upperBoundsstatement and proof · cited by 263
- Set.Nonempty.monoproof · cited by 88
- isLUB_csSupproof · cited by 34
- isGLB_csInfproof · cited by 24
- IsLeast.isGLBproof · cited by 23
- IsGLB.uniqueproof · cited by 13
Cited by2
Results whose statement or proof uses this declaration.
- Filter.limsSup_principal_eq_csSupproof · cited by 2
- csInf_upperBounds_rangeproof · cited by 1