Join Semilattice is Ordered Structure/Proof 2

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

Then $\left({S, \vee, \preceq}\right)$ is an ordered structure.

That is, $\preceq$ is compatible with $\vee$.

Proof
Let $a, b, c \in S$.

Let $a \preceq b$.

By the definition of join semilattice:
 * $a \vee b = b$

Thus:
 * $\left({a \vee b}\right) \vee c = b \vee c$

Since $\vee$ is associative, commutative, and idempotent:
 * $\left({a \vee c}\right) \vee \left({b \vee c}\right) = b \vee c$

Therefore, $a \vee c \preceq b \vee c$.

From Join is Commutative, we conclude that:
 * $c \vee a \preceq c \vee b$