Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition

Theorem
Let $\left({\Z, +}\right)$ be the algebraic structure formed by the set of integers under the operation of addition.

Let $\left({2 \Z, +}\right)$ be the algebraic structure formed by the set of even integers under the operation of addition.

Then $\left({\Z, +}\right)$ and $\left({2 \Z, +}\right)$ are isomorphic.

Proof
Let $f: \Z \to 2\Z$ be the mapping:
 * $\forall n \in \Z: f \left({n}\right) = 2 n$

From Bijection between Integers and Even Integers, $f$ is a bijection.

Let $m, n \in \Z$.

Then:

Thus $f$ is an isomorphism by definition.