Modulus of Exponential is Exponential of Real Part

Theorem
Let $z\in \C$ be a complex number.

Let $\exp z$ denote the complex exponential function.

Let $\cmod {\, \cdot \,}$ denote the complex modulus

Then:


 * $\cmod {\exp z} = \exp \paren {\Re z}$

where $\Re z$ denotes the real part of $z$.

Proof
Let $z = x + iy$.