Epimorphism from Integers to Cyclic Group

Theorem
Let $\left \langle {a} \right \rangle = \left({G, \circ}\right)$ be a cyclic group.

Let $f: \Z \to G$ be a mapping defined as:

$\forall n \in \Z: f \left({n}\right) = a^n$.

Then $f$ is a (group) epimorphism from $\left({\Z, +}\right)$ onto $\left \langle {a} \right \rangle$.

Proof
By Powers of Element form Subgroup:
 * $\forall n \in \N: a^n \in \left \langle {a} \right \rangle$

Hence by the Index Law for Monoids: Negative Index:
 * $\forall n \in \Z: a^n \in \left \langle {a} \right \rangle$

Also, by Index Law for Monoids: Sum of Indices, $f$ is a homomorphism from $\left({\Z, +}\right)$ into $\left({G, \circ}\right)$.

By Homomorphism Preserves Subsemigroups, its codomain $f \left({\Z}\right)$ is therefore a subgroup of $\left \langle {a} \right \rangle$ containing $a$.

By Existence of Unique Subgroup Generated by Subset: Singleton Generator, $\left \langle {a} \right \rangle$ is the smallest subgroup of $G$ containing $a$.

So $f \left({\Z}\right) = \left \langle {a} \right \rangle$.

Therefore $f$ is an epimorphism from $\left({\Z, +}\right)$ onto $\left({G, \circ}\right)$.