Inversion Mapping Reverses Ordering in Ordered Group/Corollary

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

Let $x \in G$.

Then the following equivalences hold:


 * $(1):\quad x \le e \iff e \le x^{-1}$
 * $(2):\quad e \le x \iff x^{-1} \le e$
 * $(1'):\quad x < e \iff e < x^{-1}$
 * $(2'):\quad e < x \iff x^{-1} < e$

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

Thus by User:Dfeuer/CRG4, we obtain the first two results:


 * $(1):\quad x \le e \iff e \le x^{-1}$
 * $(2):\quad e \le x \iff x^{-1} \le e$

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

Thus by User:Dfeuer/CRG4, we obtain the remaining results:


 * $(1'):\quad x < e \iff e < x^{-1}$
 * $(2'):\quad e < x \iff x^{-1} < e$