Meet-Continuous implies Shift Mapping Preserves Directed Suprema

Theorem
Let $\mathscr S = \left({S, \vee, \wedge, \preceq}\right)$ be a meet-continuous lattice.

Let $x \in S$.

Let $f: S \to S$ be a mapping such that
 * $\forall y \in S: f\left({y}\right) = x \wedge y$

Then
 * $f$ preserves directed suprema.

Proof
Let $D$ be a directed subset of $S$ such that
 * $D$ admits a supremum.

By Singleton is Directed and Filtered Subset:
 * $\left\{ {x}\right\}$ is directed.

By Up-Complete Product/Lemma 1:
 * $\left\{ {x}\right\} \times D$ is directed in Cartesian product $\left({S \times S, \precsim}\right)$ of $\mathscr S$ and $\mathscr S$.

Define a mapping $g: S \times S \to S$:
 * $\forall s, t \in S: g\left({s, t}\right) = s \wedge t$

By Meet-Continuous iff Meet Preserves Directed Suprema:
 * $g$ preserves directed suprema.

By definition of meet-continuous:
 * $\mathscr S$ is up-complete.

By Up-Complete Product:
 * $\left({S \times S, \precsim}\right)$ is up-complete.

By definition of up-complete:
 * $\left\{ {x}\right\} \times D$ admits a supremum.

Thus