Ordering Compatible with Group Operation is Strongly Compatible/Corollary

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

Let $x, y \in \left({G, \circ, \le}\right)$.

Then the following equivalences hold:


 * $x \le y \iff I \le y \circ x^{-1}$
 * $x \le y \iff I \le x^{-1} \circ y$
 * $x < y \iff I < y \circ x^{-1}$
 * $x < y \iff I < x^{-1} \circ y$

Proof
Since a group is closed under inverses, we can apply User:Dfeuer/OG1 to $x$, $y$, and $-x$ to obtain the following equivalences:


 * $x \le y \iff x \circ x^{-1} \le y \circ x^{-1}$
 * $x \le y \iff x^{-1} \circ x \le x^{-1} \circ y$
 * $x < y \iff x \circ x^{-1} < y \circ x^{-1}$
 * $x < y \iff x^{-1} \circ x < x^{-1} \circ y$

By the definition of group inverse, $x \circ x^{-1} = x^{-1} \circ x = I$.

Making these substitutions proves the theorem.