How to suppress vertical space between proof environment heads and itemize environments?

Question

The user 'egreg' recently helped me to suppress vertical space between theorem heads and enumitem environments (such as itemize; see the link below). I now have a similar problem: When a proof environment starts with an itemize, there is too much vertical space. How can the proof environment be redefined such that this space is (at least party) suppressed?

Ideally, it should be a "clean" solution, so that one can type \begin{proof} followed by \begin{itemize} and no complicated commands in-between.

By redefining the proof environment as a theorem, one could circumvent the problem, but how can the q.e.d. symbol be nicely placed? Not sure if that's a good approach.

\documentclass{scrartcl}

\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[american]{babel}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{etoolbox}
\usepackage{enumitem}

% theorems
\newif\ifstarttheorem
\newtheoremstyle{mythmstyle}%
{0.5em}%
{0.5em}%
{}% 
{}% 
{\sffamily\bfseries\global\starttheoremtrue}% 
{}% 
{\newline}% 
{\thmname{#1}\ \thmnumber{#2}\ \thmnote{(#3)}}% 
\theoremstyle{mythmstyle}% 

% proofs
\renewcommand*\proofname{Proof}
\makeatletter% q.e.d. adjustment (posted by ???)
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont\topsep2\p@\@plus2\p@\relax
  \trivlist
\item[\hskip\labelsep
  \sffamily\bfseries #1]\mbox{}\hfill\\*\ignorespaces
}{%
  \popQED\endtrivlist\@endpefalse
}
\makeatother

% suppress vertical spacing (for itemize) when newtheorems start with an environment
% see http://tex.stackexchange.com/questions/60001/nested-enumerate-enumitem-interferes-with-suppressing-vertical-space-after-the
\makeatletter
\preto\itemize{%
  \if@inlabel
    \ifstarttheorem
      \mbox{}\par\nobreak\vskip\glueexpr-\parskip-\baselineskip+0.3em\relax\hrule\@height\z@
      \global\starttheoremfalse
    \fi
  \fi}
\preto\enditemize{\global\starttheoremfalse}
\makeatother

% document
\begin{document}
Let's prove something:
\begin{proof}
  \begin{align*}
    x=y,\quad y=z\quad\Rightarrow\quad x=z.
  \end{align*}
\end{proof}
Here a nicer vertical spacing:
\begin{proof}
  Just some text:
  \begin{align*}
    x=y,\quad y=z\quad\Rightarrow\quad x=z.
  \end{align*}
\end{proof}
Here a version with itemize:
\begin{proof}
  \begin{itemize}
    \item $x=y,\quad y=z\quad\Rightarrow\quad x=z.$
  \end{itemize}
\end{proof}
\end{document}

Answer 1

You can try the following. Here the environment proof gets a new flag \starttheoremtrue and the name of the environment is saved inside the command \mycurrenvir.

\documentclass{scrartcl}

\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[american]{babel}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{etoolbox}
\usepackage{enumitem}

% theorems
\newif\ifstarttheorem
\newtheoremstyle{mythmstyle}%
{0.5em}%
{0.5em}%
{}% 
{}% 
{\sffamily\bfseries\global\starttheoremtrue}% 
{}% 
{\newline}% 
{\thmname{#1}\ \thmnumber{#2}\ \thmnote{(#3)}}% 
\theoremstyle{mythmstyle}% 

% proofs
\renewcommand*\proofname{Proof}
\makeatletter% q.e.d. adjustment (posted by ???)
\renewenvironment{proof}[1][\proofname]{\par
  \pushQED{\qed}%
  \normalfont\topsep2\p@\@plus2\p@\relax
  \trivlist
\item[\hskip\labelsep
  \sffamily\bfseries #1]\gdef\mycurrenvir{proof}\global\starttheoremtrue\mbox{}\hfill\\*\ignorespaces
}{%
 \gdef\mycurrenvir{\relax}
  \popQED\endtrivlist\@endpefalse
}
\makeatother

% suppress vertical spacing (for itemize) when newtheorems start with an environment
% see http://tex.stackexchange.com/questions/60001/nested-enumerate-enumitem-interferes-with-suppressing-vertical-space-after-the
\makeatletter
\preto\itemize{%
  \if@inlabel
    \ifstarttheorem
      \mbox{}\par\nobreak\vskip\glueexpr-\parskip-\baselineskip+0.3em\relax\hrule\@height\z@
      \global\starttheoremfalse%
    \fi%
  \fi%
 \def\tempa{proof}%
 \ifx\tempa\mycurrenvir
    \ifstarttheorem
      \mbox{}\par\nobreak\vskip\glueexpr-\parskip-\baselineskip+0.3em\relax\hrule\@height\z@
      \global\starttheoremfalse%
    \fi%
 \fi%
}
\preto\enditemize{\global\starttheoremfalse}
\makeatother

% document
\begin{document}
Let's prove something:
\begin{proof}
  \begin{align*}
    x=y,\quad y=z\quad\Rightarrow\quad x=z.
  \end{align*}
\end{proof}
Here a nicer vertical spacing:
\begin{proof}
  Just some text:
  \begin{align*}
    x=y,\quad y=z\quad\Rightarrow\quad x=z.
  \end{align*}
\end{proof}
Here a version with itemize:
\begin{proof}
  \begin{itemize}
    \item $x=y,\quad y=z\quad\Rightarrow\quad x=z.$
  \end{itemize}
\end{proof}
\end{document}