Positive Rational Numbers under Addition form Ordered Semigroup/Proof 2

Theorem

Let $\Q_{\ge 0}$ denote the set of positive rational numbers.

The algebraic structure:

$\struct {\Q_{\ge 0}, +, \le}$

forms an ordered semigroup.

Proof

From Rational Numbers form Ordered Field, $\struct {\Q, +, \times, \le}$ is an ordered field.

Hence $\struct {\Q, +, \le}$ is an ordered group, and so an ordered semigroup.

From Positive Rational Numbers are Closed under Addition we have that $\struct {\Q_{\ge 0}, +}$ is closed.

Hence from Subsemigroup Closure Test, $\struct {\Q_{\ge 0}, +}$ is a subsemigroup of $\struct {\Q, +}$.

From Subsemigroup of Ordered Semigroup is Ordered, $\struct {\Q_{\ge 0}, +, \le}$ is an ordered semigroup.

$\blacksquare$