Preceding is Auxiliary Relation

Theorem

Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Then

$\preceq$ is an auxiliary relation.

Proof

$\forall x, y \in S: x \preceq y \implies x \preceq y$

Then condition $(1)$ of auxiliary relation is satisfied.

By definition of transitivity:

$\forall x, y, z, u \in S: x \preceq y \preceq z \preceq u \implies x \preceq u$

Then the condition $(2)$ of auxiliary relation is satisfied.

By definition of supremum:

$\forall x, y, z \in S: x \preceq z \land y \preceq z \implies x \vee y \preceq z$

Then the condition $(3)$ of auxiliary relation is satisfied.

By definition of smallest element:

$\forall x \in S: \bot \preceq x$

Then the condition $(4)$ of auxiliary relation is satisfied.

Thus the result by definition auxiliary relation.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_4:funcreg 4