Product of Complex Conjugates/Examples/3 Arguments/Proof 1

Theorem

Let $z_1, z_2, z_3 \in \C$ be complex numbers.

Let $\overline z$ denote the complex conjugate of the complex number $z$.

Then:

$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$

$\ds \overline {z_1 z_2 z_3}$

$=$

$\ds \overline {\paren {z_1 z_2} \paren {z_3} }$

$\ds $

$=$

$\ds \paren {\overline {z_1 z_2} } \cdot \overline {z_3}$

$\ds $

$=$

$\ds \paren {\overline {z_1} \cdot \overline {z_2} } \cdot \overline {z_3}$

$\ds $

$=$

$\ds \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$

$\blacksquare$