Non-Zero Real Numbers under Multiplication form Abelian Group

Theorem
Let $\R_{\ne 0}$ be the set of real numbers without zero, i.e. $\R_{\ne 0} = \R \setminus \left\{{0}\right\}$.

The structure $\left({\R_{\ne 0}, \times}\right)$ is an uncountable abelian group.

Proof
Taking the group axioms in turn:

G0: Closure
Real Multiplication is Closed.

G1: Associativity
Real Multiplication is Associative.

C: Commutativity
Real Multiplication is Commutative.

Infinite
Real Numbers are Uncountably Infinite.