Non-Zero Natural Numbers under Multiplication with Divisibility forms Ordered Semigroup

Theorem
Let $\N_{>0}$ be the set of natural numbers without zero, that is, $\N_{>0} = \N \setminus \set 0$.

Let $\divides$ denote the divisibility relation on $\N_{>0}$:
 * $\forall a, b \in \N_{>0}: a \divides b \iff \exists k \in \Z: k \times a = b$

where $\times$ denotes conventional integer multiplication.

The ordered structure $\struct {\N_{>0}, \times, \divides}$ forms an ordered semigroup.

Proof
First we note that from Non-Zero Natural Numbers under Multiplication form Commutative Semigroup, $\struct {\N_{>0}, \times}$ is a semigroup.

From Divisor Relation on Positive Integers is Partial Ordering, $\struct {\N_{>0}, \divides}$ is an ordered set.

It remains to be shown that $\divides$ is compatible with $\times$.

Let $a, b \in \N_{>0}$ such that $a \divides b$.

We have:

Similarly:

Hence the result.