Inverses of Elements Related by Compatible Relation

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\mathcal R$ be a relation compatible with $\circ$.

Let $x, y \in G$.

Then:
 * $x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x^{-1}$

Proof
Let $e$ be the group identity of $G$.

By User:Dfeuer/CRG2 $(1)$:


 * $(1): \quad x \mathrel{\mathcal R} y \iff e \mathrel{\mathcal R} y \circ x^{-1}$

By User:Dfeuer/CRG2 $(2)$, also:


 * $(2): \quad y^{-1} \mathrel{\mathcal R} x^{-1} \iff e \mathrel{\mathcal R} \left({y^{-1}}\right)^{-1} \circ x^{-1}$

By Inverse of Group Inverse $\left({y^{-1}}\right)^{-1} = y$.

Thus, we can rewrite $(2)$ as:


 * $(3): \quad y^{-1} \mathrel{\mathcal R} x^{-1} \iff e \mathrel {\mathcal R} y \circ x^{-1}$

Now note that the RHS of $(3)$ is the same as the RHS in $(1)$.

We conclude that:


 * $x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x$