Theorems · Theorem · order theory

Directed.le_ciInf_iff

∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompletePartialOrderInf α] [Nonempty ι] {f : ι → α} {a : α},
  Directed (fun x1 x2 => x2 ≤ x1) f → BddBelow (Set.range f) → (a ≤ iInf f ↔ ∀ (i : ι), a ≤ f i)

Defined in: Mathlib.Order.ConditionallyCompletePartialOrder.Indexed
Cited by: 2 results in Mathlib
Foundations: Depth 15 from the axioms · uses propext, Quot.sound
Assumes: ConditionallyCompletePartialOrderInf Nonempty

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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
iInfstatement · cited by 1,690
BddBelowstatement and proof · cited by 401
Directedstatement and proof · cited by 213
Set.forall_mem_rangeproof · cited by 135
ConditionallyCompletePartialOrderInfstatement and proof · cited by 51
le_isGLB_iffproof · cited by 17
Directed.isGLB_ciInfproof · cited by 1

Cited by2

Results whose statement or proof uses this declaration.

ciInf_Iciproof · cited by 0
Directed.Iic_ciInfproof · cited by 0