Theorems · Theorem · order theory
sSup_eq_iSup
∀ {α : Type u_1} [inst : CompleteLattice α] {s : Set α}, sSup s = ⨆ a ∈ s, a- Defined in
- Mathlib.Order.CompleteLattice.Basic
- Cited by
- 42 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses no axioms
- Assumes
- CompleteLattice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- iSupstatement · cited by 2,415
- le_antisymmproof · cited by 2,068
- CompleteLatticestatement and proof · cited by 1,048
- SupSet.sSupstatement · cited by 954
- iSup₂_leproof · cited by 96
- le_sSupproof · cited by 79
- le_iSup₂proof · cited by 56
- sSup_leproof · cited by 35
Cited by42
Results whose statement or proof uses this declaration.
- GaloisConnection.l_sSupproof · cited by 19
- Finset.sup_id_eq_sSupproof · cited by 12
- TopologicalSpace.Opens.mem_sSupproof · cited by 6
- GaloisInsertion.l_sSup_u_imageproof · cited by 4
- sSupIndep_iffproof · cited by 3
- OrderIso.map_sSupproof · cited by 3
- CompleteLattice.ωScottContinuous.sSupproof · cited by 3
- IsSemisimpleModule.finite_tfaeproof · cited by 2
- sSup_iUnionproof · cited by 2
- sSup_sUnionproof · cited by 2
- Submodule.le_linearEquiv_of_le_sSupproof · cited by 2
- Submodule.fg_iff_compactproof · cited by 1