Conjugate of Commuting Elements

Theorem
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$. Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then $x \circ y \circ x^{-1} = y$ iff $x$ and $y$ commute.

Proof
As $\left({S, \circ}\right)$ is a monoid, it is by definition a semigroup.

Therefore it is taken for granted that $\circ$ is associative, so we can dispense with parentheses.

We also take for granted the fact that $x$ and $y$ are cancellable from Invertible Elements of Semigroup Also Cancellable.

So: