Complex Numbers under Addition form Infinite Abelian Group

Theorem
Let $\C$ be the set of complex numbers.

The structure $\left({\C, +}\right)$ is an infinite abelian group.

Proof
Taking the group axioms in turn:

G0: Closure
Complex Addition is Closed.

G1: Associativity
Complex Addition is Associative.

G2: Identity
From Complex Addition Identity is Zero, we have that the identity element of $\left({\C, +}\right)$ is the complex number $0 + 0 i$:
 * $\left({x + i y}\right) + \left({0 + 0 i}\right) = \left({x + 0}\right) + i \left({y + 0}\right) = x + i y$

and similarly for $\left({0 + 0 i}\right) + \left({x + i y}\right)$.

G3: Inverses
From Inverse for Complex Addition, the inverse of $x + i y \in \left({\C, +}\right)$ is $-x - i y$.

C: Commutativity
Complex Addition is Commutative.

Infinite
Complex Numbers are Uncountable.