Conjugate of Commuting Elements

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

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

Proof
As $\struct {S, \circ}$ 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 Element of Monoid is Cancellable.

So: