Theorems · Theorem · order theory
csSup_lowerBounds_eq_csInf
∀ {α : Type u_1} [inst : ConditionallyCompleteLattice α] {s : Set α},
BddBelow s → s.Nonempty → sSup (lowerBounds s) = sInf s- Cited by
- 1 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
- BddBelowstatement and proof · cited by 401
- ConditionallyCompleteLatticestatement and proof · cited by 364
- lowerBoundsstatement and proof · cited by 212
- Set.Nonempty.monoproof · cited by 88
- isLUB_csSupproof · cited by 34
- IsGreatest.isLUBproof · cited by 25
- isGLB_csInfproof · cited by 24
- IsLUB.uniqueproof · cited by 21
Cited by1
Results whose statement or proof uses this declaration.
- csSup_lowerBounds_rangeproof · cited by 0