Complex Numbers under Addition form Infinite Abelian Group

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

The structure $\struct {\C, +}$ 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 $\struct {\C, +}$ is the complex number $0 + 0 i$:
 * $\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$

and similarly for $\paren {0 + 0 i} + \paren {x + i y}$.

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

C: Commutativity
Complex Addition is Commutative.

Infinite
Complex Numbers are Uncountable.