Power Function Strictly Preserves Ordering in Ordered Group

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Theorem

Let $\left({S,\circ, \preceq}\right)$ be an ordered Group.

Let $x,y \in S$.

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

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


Then the following hold:

If $x \preceq y$ then $x^n \preceq y^n$
If $x \prec y$ then $x^n \prec y^n$

where $x^n$ is the $n$th power of $x$.


Proof

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

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

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

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

By the definition of an ordered group, $\left({S,\circ}\right)$ is a group, and therefore a semigroup.

Thus the theorem holds by Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements.

$\blacksquare$