Inversion Mapping Reverses Ordering in Ordered Group/Corollary/Proof 2

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

Let $x \in G$.

Then the following equivalences hold:

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

Thus by Properties of Relation Compatible with Group Operation/CRG4, we obtain the first two results:


 * $(\operatorname{OG}4.1):\quad x \preceq e \iff e \preceq x^{-1}$
 * $(\operatorname{OG}4.2):\quad e \preceq x \iff x^{-1} \preceq e$

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

Thus by again Properties of Relation Compatible with Group Operation/CRG4, we obtain the remaining results:


 * $(\operatorname{OG}4.1'):\quad x \prec e \iff e \prec x^{-1}$
 * $(\operatorname{OG}4.2'):\quad e \prec x \iff x^{-1} \prec e$