Isomorphism Preserves Left Cancellability

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an isomorphism.

Then:
 * $a \in S$ is left cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is left cancellable in $\left({T, *}\right)$.

Proof
Let $a$ be left cancellable in $\left({S, \circ}\right)$.

Let $x \in S$ and $y \in S$ be arbitrary.

Then:

That is, $\phi \left({a}\right)$ is left cancellable in $T$.

As $\phi$ is an isomorphism, then so is $\phi^{-1}$.

So the same proof works in reverse in exactly the same way.