Commutation with Inverse in Monoid

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

Then $$x$$ commutes with $$y$$ iff $$x$$ commutes with $$y^{-1}$$.

Necessary Condition
Let $$x$$ commute with $$y$$. Then:

$$ $$ $$ $$ $$ $$

So $$x$$ commutes with $$y^{-1}$$.

Sufficient Condition
Now let $$x$$ commute with $$y^{-1}$$.

From the above, it follows that $$x$$ commutes with $$\left({y^{-1}}\right)^{-1}$$.

From Inverse of an Inverse, $$\left({y^{-1}}\right)^{-1} = y$$

Thus $$x$$ commutes with $$y$$.