Using \gather and \align together

taking a hint from @PeterGrill, here's a method of combining all the parts into one, using gather* since the "single" line is to be center-aligned, and the sub-environment aligned to take care of the multi-line expression.

since you're using amsthm, i've also taken the liberty to use \qedhere to position the "tombstone" on the last line of the display.

\documentclass{article}
\usepackage{mathtools}
\usepackage{amsthm}
\begin{document}
\begin{proof}
\begin{gather*}
 \begin{aligned}
10 &= 2+2+2+2+2 \\
   &= 3+3+3+1
\end{aligned}
\shortintertext{No annoying large space above this line...}
  5=5
\qedhere
\end{gather*}
\end{proof}
\end{document}

As per Adjust vspace between multiple align environments you should never use two consecutive display math environments.

Single align environment:

You could just use a single align* environment. I would also recommend using \shortintertext form the \mathtools package as that provides tighter vertical spacing:

\documentclass{article}
\usepackage{showframe}
\usepackage{mathtools}
\usepackage{amsthm}
\begin{document}
\begin{proof}
\begin{align*}
    10 &= 2+2+2+2+2\\
       &= 3+3+3+1
    \shortintertext{Now this no longer puts an annoying large space above this line...}
    5 &=5 \qedhere
\end{align*}
\end{proof}
\end{document}

Using align within gather:

Alternatively, you could also use gather to center some equations and also align other portions, but this dot not look very good, especially with short text snippets in between.

\documentclass{article}
\usepackage{showframe}
\usepackage{mathtools}
\usepackage{amsthm}
\begin{document}
\begin{proof}
\begin{gather*}
\begin{align*}
10 &= 2+2+2+2+2 &\\
   &= 3+3+3+1
\end{align*}
    \shortintertext{Now this no longer puts an annoying large space above this line...}
5=5 \qedhere
\end{gather*}
\end{proof}
\end{document}

[Note: the improvement below, actually does not work.]

[Note: there is an important improvement to this code below.]

The closest I can get to a solution is

\documentclass[10pt,a4paper]{article}

\usepackage{amsmath,amsthm,mathtools}

\begin{document}
\begin{proof}
  \begin{align*}
    15 &= 5+4+3+2+1 = \sum_{i=1}^5 i \\
    120&=5\cdot4\cdot3\cdot2\cdot1 = 5! \\
    \shortintertext{
      \[ e^{\pi i} + 1 = 0 \]
      \[ x^n + y^n = z^n \]
    } 
    27 &= 3^3 \qedhere
  \end{align*}
\end{proof}
\end{document}

which gives as result

and actually the spacing is quite nice, I think.

\documentclass[10pt,a4paper]{article}

\usepackage{amsmath,amsthm,mathtools}

\begin{document}
\begin{proof}
  \begin{align*}
    15 &= 5+4+3+2+1 = \sum_{i=1}^5 i \\
    120&=5\cdot4\cdot3\cdot2\cdot1 = 5! \\
    \begin{gathered}
      e^{\pi i} + 1 = 0 \\
      x^n + y^n = z^n
    \end{gathered} 
    27 &= 3^3 \qedhere
  \end{align*}
\end{proof}
\end{document}

Credits go to barbara beeton (please vote up her comment on her own answer).