Modulus of Exponential is Exponential of Real Part

Theorem
Let $\exp z$ be the complex exponential.

Let $\left \vert {\,\cdot\,} \right \vert$ be the complex modulus

Then:


 * $\left \vert {\exp z} \right \vert = \left \vert {\exp \left({\operatorname{Re}z}\right)} \right \vert$

where $\operatorname{Re}z$ is the real part of $z$.

Proof
Let $z = x + iy$.