Lower Closure is Prime Ideal for Prime Element

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

Let $p \in S$ be a prime element.

Then $p^\preceq$ is a prime ideal.

Proof
Let $x, y \in S$ such that
 * $x \wedge y \in p^\preceq$

By definition of lower closure of element:
 * $x \wedge y \preceq p$

By Characterization of Prime Ideal:
 * $x \preceq p$ or $y \preceq p$

Thus by definition of lower closure of element:
 * $x \in p^\preceq$ or $y \in p^\preceq$