Isomorphism Preserves Cancellability

Theorem
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

Then:
 * $a \in S$ is cancellable in $\struct {S, \circ}$ $\map \phi a \in T$ is cancellable in $\struct {T, *}$.

Proof
Let $\struct {S, \circ}$ be an algebraic structure in which $a$ is cancellable.

From Isomorphism Preserves Left Cancellability and Isomorphism Preserves Right Cancellability:
 * $a \in S$ is left cancellable in $\struct {S, \circ}$ $\map \phi a \in T$ is left cancellable in $\struct {T, *}$

and
 * $a \in S$ is right cancellable in $\struct {S, \circ}$ $\map \phi a \in T$ is right cancellable in $\struct {T, *}$.

It follows that:
 * $a \in S$ is cancellable in $\struct {S, \circ}$


 * $\map \phi a \in T$ is cancellable in $\struct {T, *}$.
 * $\map \phi a \in T$ is cancellable in $\struct {T, *}$.