\documentclass[12pt]{article}
\usepackage{amssymb,amsmath,amscd,concmath}
\pagestyle{empty}
\begin{document}
\begin{equation*}
\begin{array}{lcl}
|x> \left|\,u\right> 
&=& \frac{1}{2^{m/2}}|x> 
\sum_{y=0}^{2^{m}-1}\exp \left( -\frac{2\pi i}{2^{m}}y\right) |y> \\
&\mapsto& \frac{1}{2^{m/2}}|x>
\sum_{y=0}^{2^{m}-1}\exp \left( -\frac{2\pi i}{2^{m}}y\right) \left|f(x)+y\right>\\
&=& \frac{e^{ \frac{2\pi i}{2^{m}}f(x) }}{2^{m/2}}|x>
\sum_{y=0}^{2^{m}-1}\exp \left( -\frac{2\pi i}{2^{m}}(f(x)+y)\right)\left|f(x)+y\right> \\
&=& \frac{e^{ \frac{2\pi i}{2^{m}}f(x) }}{2^{m/2}}|x>
\sum_{y=0}^{2^{m}-1}\exp \left( -\frac{2\pi i}{2^{m}}y\right)|y> \\
&=& e^{\frac{2\pi i}{2^{m}}f(x)} |x> \left|\,u\right> ,
\end{array}
\end{equation*}
\end{document}