Non-Zero Complex Numbers are Closed under Multiplication

Theorem
The set of non-zero complex numbers is closed under multiplication.

Proof
We have that the complex numbers form a field under the operations of addition and multiplication.

By definition of a field, the algebraic structure $\left({\C_{\ne 0}, \times}\right)$ is a group.

Thus, by definition, $\times$ is closed in $\left({\C_{\ne 0}, \times}\right)$.