Praeclarum Theorema for Meet Semilattices

Theorem
Let $(S, \wedge, \preceq)$ be a meet semilattice.

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

Let $a \preceq b$.

Let $c \preceq d$.

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

Proof
By Meet Semilattice is Ordered Structure, $\preceq$ is compatible with $\wedge$.

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

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

Also See

 * Praeclarum Theorema, an analogous result in logic