Non-Zero Real Numbers under Multiplication form Abelian Group/Proof 3

Theorem

Let $\R_{\ne 0}$ be the set of real numbers without zero:

$\R_{\ne 0} = \R \setminus \set 0$

The structure $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.

Proof

From Non-Zero Real Numbers under Multiplication form Group, $\struct {\R_{\ne 0}, \times}$ forms a group.

$\Box$

From Real Multiplication is Commutative it follows that $\struct {\R_{\ne 0}, \times}$ is abelian.

$\Box$

From Real Numbers are Uncountably Infinite it follows that $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.

$\blacksquare$