Power Function Preserves Ordering in Ordered Group/Proof 2
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Theorem
Let $n \in \N_{>0}$ be a strictly positive integer.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Then the following hold:
\(\ds \forall x, y \in S: \, \) | \(\ds x \preccurlyeq y\) | \(\implies\) | \(\ds x^n \preccurlyeq y^n\) | |||||||||||
\(\ds \forall x, y \in S: \, \) | \(\ds x \prec y\) | \(\implies\) | \(\ds x^n \prec y^n\) |
where $x^n$ denotes the $n$th power of $x$.
Proof
An ordered group is an ordered structure which is also a group.
Hence an ordered group is a fortiori an ordered semigroup.
From Power Function Preserves Ordering in Ordered Semigroup:
- $\forall x, y \in S: x \preccurlyeq y \implies x^n \preccurlyeq y^n$
From the Cancellation Laws, every element of a group is cancellable.
Hence from Power Function with Cancellable Element Preserves Strict Ordering in Ordered Semigroup:
- $\forall x, y \in S: x \prec y \implies x^n \prec y^n$
$\blacksquare$