Power Function Preserves Ordering in Ordered Group/Corollary

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

Let $x \in G$.

Let $n \in \N_{>0}$.

Then the following hold:


 * $x \le e \implies x^n \le e$
 * $e \le x \implies e \le x^n$
 * $x < e \implies x^n < e$
 * $e < x \implies e < x^n$

Proof
By the definition of the identity element of a group, $e$ is idempotent with respect to $\circ$.

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

Thus by User:Dfeuer/CTR5, we obtain the first two results:


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

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

By Reflexive Reduction of Transitive Antisymmetric Relation is Transitive, $<$ is transitive.

Thus by User:Dfeuer/CTR5, we obtain the remaining results:
 * $x < e \implies x^n < e$
 * $e < x \implies e < x^n$