Theorems · Theorem · order theory

DirectedOn.inf_sSup_eq

∀ {α : Type u_2} [inst : CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α},
  DirectedOn (fun x1 x2 => x1 ≤ x2) s → a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b

This property is sometimes referred to as α being upper continuous.

Defined in

Mathlib.Order.CompactlyGenerated.Basic

Cited by

3 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLatticeIsCompactlyGenerated