Non-Zero Real Numbers under Multiplication form Abelian Group/Proof 2

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

We have Real Numbers under Multiplication form Monoid.

From Inverse for Real Multiplication, the non-zero numbers are exactly the invertible elements of real multiplication.

Thus from Invertible Elements of Monoid form Subgroup of Cancellable Elements, the non-zero real numbers under multiplication form a group.

From:

Real Multiplication is Commutative

Subset Product within Commutative Structure is Commutative

it follows that this group is also Abelian.

$\blacksquare$