Aligning equations when the number of columns change

Question

I want to have some equations aligned in two columns followed by a single centered equation, i.e. something that looks like multiple align enviroments, one after another, but with no excessive space between them. Ideally, there would be an align-like environment where the number of columns may be different in different rows, but all rows that have the same number of columns would be horizontally aligned together.

Also, in the last equation I have cases and it doesn't work well with proof. I have found no place to put \qedhere that produces the result I want – the qed symbol on the righthand end, but with no extra line.

Finally, I'd appretiate any general advice how to treat issues like this – I'm writing some math, I want to do something natural that is a slight generalization of what I do in TeX regularly, but it is a big problem to explain this to TeX. It feels like (La)TeX doesn't work on the level of abstraction I desire, and issues like this interrupt my work.

An example follows.

\documentclass{article}

\usepackage{amsmath}
\usepackage{amsthm}

\begin{document}
    \begin{proof}
        We have the following equalities.
        \begin{align*}
            f^\leftrightarrow(x) &= f(-x) & f^{\mathrel{\text{\ooalign{$\swarrow$\cr$\nearrow$}}}}(x) &= -f(-x) \\
            f^+(x) &= \max(f, 0) & f^-(x) &= \min(f, 0) \\
            |x| = \begin{cases}
                +x \quad \text{if $x \ge 0$} \\
                -x \quad \text{if $x \le 0$} \\
            \end{cases} % \qedhere
        \end{align*} \qedhere
    \end{proof}
\end{document}

Answer 1

Use gather*, and nest an aligned enviroinment within it:

\documentclass{article}

\usepackage{amsmath}
 \usepackage{amsthm}

\begin{document}

    \begin{proof}
        We have the following equalities.
        \begin{gather*}
            \begin{aligned}
            f^\leftrightarrow(x) &= f(-x) & f^{\mathrel{\text{\ooalign{$\swarrow$\cr$\nearrow$}}}}(x) &= -f(-x) \\
            f^+(x) &= \max(f, 0) & f^-(x) &= \min(f, 0) \\
            \end{aligned}\\
            |x| = \begin{cases}
                +x \quad \text{if $x \ge 0$} \\
                -x \quad \text{if $x \le 0$}
            \end{cases}
            \qedhere%
        \end{gather*}
    \end{proof}

\end{document}

Using ntheorem you can have a better alignment of the end-of-proof symbol, as it allows the local replacement of a symbol with another symbol:

\documentclass{article}

\usepackage{stackengine}
\usepackage{amsmath, amssymb}
\usepackage[thmmarks, amsmath]{ntheorem}
\theoremstyle{nonumberplain}
\theoremheaderfont{\normalfont\itshape}
\theorembodyfont{\normalfont}
\theoremsymbol{$\square$}
\newtheorem{proof}{Proof}
\qedsymbol{\bllap[1.75ex]{$\square$}}

\begin{document}

    \begin{proof}%
        We have the following equalities.
        \qed%
        \begin{gather*}
            \begin{alignedat}{3}
            f^\leftrightarrow(x) &= f(-x) & f^{\mathrel{\text{\ooalign{$\swarrow$\cr$\nearrow$}}}}(x) &= -f(-x) \\
            f^+(x) &= \max(f, 0) &\hspace{6em} f^-(x) &= \min(f, 0) \\
            \end{alignedat}\\
           |x| = \begin{cases}
                +x \quad \text{if $x \ge 0$} \\
                -x \quad \text{if $x \le 0$}
            \end{cases}
        \end{gather*}
    \end{proof}

\end{document}