Constructive Dilemma for Join Semilattices

Theorem
Let $(S, \vee, \preceq)$ be a join semilattice.

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

Let $a \preceq b$.

Let $c \preceq d$.

Then $(a \vee c) \preceq (b \vee d)$.

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.