Additive Group of Integers is Normal Subgroup of Complex

Theorem
Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\C, +}\right)$ be the additive group of complex numbers.

Then $\left({\Z, +}\right)$ is a normal subgroup of $\left({\C, +}\right)$.

Proof
From Additive Group of Integers is Subgroup of Reals, $\left({\Z, +}\right) \lhd \left({\R, +}\right)$.

From Additive Group of Reals Subgroup of Complex, $\left({\R, +}\right) \lhd \left({\C, +}\right)$.

Thus $\left({\Z, +}\right) \le \left({\C, +}\right)$.

From Complex Numbers under Addition form Abelian Group, $\left({\C, +}\right)$ is abelian.

From Subgroup of Abelian Group is Normal, it follows that $\left({\Z, +}\right) \lhd \left({\C, +}\right)$.