Power Function Preserves Ordering in Ordered Group/Proof 1
Jump to navigation
Jump to search
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
By definition of ordered group:
- $\preccurlyeq$ is compatible with $\circ$.
By definition of ordering:
- $\preccurlyeq$ is transitive.
From Reflexive Reduction of Relation Compatible with Group Operation is Compatible:
- $\prec$ is also compatible with $\circ$.
From Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering:
- $\prec$ is also transitive.
By definition of ordered group:
- $\struct {S, \circ}$ is a ordered group, and therefore a fortiori a semigroup.
The result follows from Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements.
$\blacksquare$