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
The identity element of $$\left({\C, +}\right)$$ is the complex number $$0 + 0 i$$:

We have $$\left({x + i y}\right) + \left({0 + 0 i}\right) = \left({x + 0}\right) + i \left({y + 0}\right) = x + i y$$.

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

G3: Inverses
The inverse of $$x + i y \in \left({\C, +}\right)$$ is $$-x - i y$$:

We have $$\left({x + i y}\right) + \left({-x - i y}\right) = \left({x - x}\right) + i \left({y - y}\right) = 0 + 0 i$$.

Similarly for $$\left({-x - i y}\right) + \left({x + i y}\right)$$.

C: Commutativity
Complex Addition is Commutative.

Infinite
Complex Numbers are Uncountable.