Theorems · Theorem · order theory

DirectedOn.sSup_inf_eq

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

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

Defined in

Mathlib.Order.CompactlyGenerated.Basic

Cited by

2 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLatticeIsCompactlyGenerated