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

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

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

Then $\struct {\Z, +}$ and $\struct {2 \Z, +}$ are isomorphic.

Proof
Let $f: \Z \to 2 \Z$ be the mapping:
 * $\forall n \in \Z: \map f n = 2 n$

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

Let $m, n \in \Z$.

Then:

Thus $f$ is an isomorphism by definition.