Non-Zero Complex Numbers are Closed under Multiplication/Proof 1

Theorem

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

Proof

Recall that Complex Numbers form Field under the operations of addition and multiplication.

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

Thus, by definition, $\times$ is closed in $\struct {\C_{\ne 0}, \times}$.

$\blacksquare$