Theorems · Theorem · order theory

Directed.ciInf_le_ciSup

∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompletePartialOrder α] [Nonempty ι] {f : ι → α},
  Directed (fun x1 x2 => x1 ≥ x2) f →
    BddBelow (Set.range f) → Directed (fun x1 x2 => x1 ≤ x2) f → BddAbove (Set.range f) → ⨅ i, f i ≤ ⨆ i, f i

Defined in: Mathlib.Order.ConditionallyCompletePartialOrder.Indexed
Cited by: 0 results in Mathlib
Foundations: Depth 15 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: ConditionallyCompletePartialOrder Nonempty

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites11

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
LE.le.transproof · cited by 3,151
iSupstatement · cited by 2,415
iInfstatement · cited by 1,690
BddAbovestatement and proof · cited by 620
BddBelowstatement and proof · cited by 401
Directedstatement and proof · cited by 213
Classical.arbitraryproof · cited by 161
Directed.le_ciSupproof · cited by 4
ConditionallyCompletePartialOrderstatement and proof · cited by 4
Directed.ciInf_leproof · cited by 3

Cited by0

Results whose statement or proof uses this declaration.

Nothing cites this yet.