Theorems · Theorem · order theory
directedOn_of_sup_mem
∀ {α : Type u_1} [inst : SemilatticeSup α] {S : Set α},
(∀ ⦃i j : α⦄, i ∈ S → j ∈ S → i ⊔ j ∈ S) → DirectedOn (fun x1 x2 => x1 ≤ x2) SA set stable by supremum is ≤-directed.
- Defined in
- Mathlib.Order.Directed
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
- Assumes
- SemilatticeSup
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 and proof · cited by 53,352
- SemilatticeSupstatement and proof · cited by 785
- DirectedOnstatement · cited by 271
- le_sup_leftproof · cited by 265
- le_sup_rightproof · cited by 242
Cited by7
Results whose statement or proof uses this declaration.
- ContinuousMultilinearMap.hasBasis_nhds_zero_of_basisproof · cited by 3
- UniformOnFun.isUniformEmbedding_toFun_finiteproof · cited by 2
- Filter.HasBasis.compactConvergenceUniformityproof · cited by 2
- PointwiseConvergenceCLM.hasBasis_nhds_zero_of_basisproof · cited by 1
- CompactConvergenceCLM.hasBasis_nhds_zero_of_basisproof · cited by 1
- ContinuousLinearMap.hasBasis_nhds_zero_of_basisproof · cited by 1
- directedOn_of_inf_memproof · cited by 0