Constructive Dilemma for Join Semilattices

Theorem
Let $\struct {S, \vee, \preceq}$ be a join semilattice.

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

Let $a \preceq b$.

Let $c \preceq d$.

Then $\paren {a \vee c} \preceq \paren {b \vee d}$.

Proof
By Join Semilattice is Ordered Structure, $\preceq$ is compatible with $\vee$.

By the definition of ordering, $\preceq$ is transitive.

Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.