Theorems · Theorem · order theory
Directed.ciSup_le_iff
∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompletePartialOrderSup α] [Nonempty ι] {f : ι → α} {a : α},
Directed (fun x1 x2 => x1 ≤ x2) f → BddAbove (Set.range f) → (iSup f ≤ a ↔ ∀ (i : ι), f i ≤ a)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Set.rangestatement and proof · cited by 4,705
- iSupstatement · cited by 2,415
- BddAbovestatement and proof · cited by 620
- Directedstatement and proof · cited by 213
- Set.forall_mem_rangeproof · cited by 135
- ConditionallyCompletePartialOrderSupstatement and proof · cited by 52
- isLUB_le_iffproof · cited by 24
- Directed.isLUB_ciSupproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- ciSup_Iicproof · cited by 1
- Directed.Ici_ciSupproof · cited by 0