Properties of Ordered Group/OG5/Proof 2

Theorem
Let $\left({G, \circ, \preceq}\right)$ be an ordered group with identity $e$.

Let $\prec$ be the reflexive reduction of $\preceq$.

Let $x \in G$.

Let $n \in \N_{>0}$ be a strictly positive integer.

Then the following hold:


 * $x \preceq e \implies x^n \preceq e$
 * $e \preceq x \implies e \preceq x^n$
 * $x \prec e \implies x^n \prec e$
 * $e \prec x \implies e \prec x^n$

Proof
By the definition of an ordered group, $\preceq$ is a transitive relation compatible with $\circ$.

By Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements:
 * $x \preceq e \implies x^n \preceq e^n$
 * $e \preceq x \implies e^n \preceq x^n$

By Identity Element is Idempotent, $e$ is idempotent with respect to $\circ$.

Thus we obtain the first two results:
 * $x \preceq e \implies x^n \preceq e$
 * $e \preceq x \implies e \preceq x^n$

By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is compatible with $\circ$.

By Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering, $\prec$ is transitive.

Thus by the same method as above, we obtain the remaining results:


 * $x \prec e \implies x^n \prec e$
 * $e \prec x \implies e \prec x^n$