Power Function Preserves Ordering in Ordered Group/Corollary/Proof 1
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Corollary to Power Function Preserves Ordering in Ordered Group
Let $\struct {G, \circ, \preccurlyeq}$ 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:
\(\ds \forall x \in S: \, \) | \(\ds x \preccurlyeq e\) | \(\implies\) | \(\ds x^n \preccurlyeq e\) | |||||||||||
\(\ds e \preccurlyeq x\) | \(\implies\) | \(\ds e \preccurlyeq x^n\) | ||||||||||||
\(\ds x \prec e\) | \(\implies\) | \(\ds x^n \prec e\) | ||||||||||||
\(\ds e \prec x\) | \(\implies\) | \(\ds e \prec x^n\) |
Proof
By Power Function Preserves Ordering in Ordered Group:
\(\ds \forall x \in S: \, \) | \(\ds x \preccurlyeq e\) | \(\implies\) | \(\ds x^n \preccurlyeq e^n\) | |||||||||||
\(\ds e \preccurlyeq x\) | \(\implies\) | \(\ds e^n \preccurlyeq x^n\) | ||||||||||||
\(\ds x \prec e\) | \(\implies\) | \(\ds x^n \prec e^n\) | ||||||||||||
\(\ds e \prec x\) | \(\implies\) | \(\ds e^n \prec x^n\) |
By Identity Element is Idempotent, $e$ is idempotent with respect to $\circ$.
Therefore by the definition of an idempotent element, $e^n = e$.
Thus the theorem holds.
$\blacksquare$