Decrease horizontal space in enumitem lists

Question

To subdivide longer proofs of theorems, I use two list environments defined using enumitem: pfsteps for numbered steps and pfparts for labeled parts.

Horizontal space problems

1) In the pfsteps enumerate list, there seems to be a bit too much horizontal space after the colon that follows the step number — more space than would follow a colon in ordinary text.
2) In both the pfsteps enumerate list and the pfparts description list, there is definitely too much horizontal space between "Proof" and the first label of the list.

Questions

Problem 1) can be fixed by using something like the commented-out line \hspace{-1pt} after the colon in the definition of label for pfsteps. But is there a nicer way to decrease this space?

How can Problem 2) be fixed, for both the pfsteps and the pfparts list environments.

Sample source

\documentclass[12pt]{memoir}

\usepackage{amsthm}

% Indent ``Proof.''
\usepackage{xpatch}
\newlength{\normalparindent}
\AtBeginDocument{\setlength{\normalparindent}{\parindent}}
\xpatchcmd{\proof}{\hskip\labelsep}{\hskip\labelsep\hskip\normalparindent}{}{}

\let\oldproofname=\proofname
\renewcommand{\proofname}{\sffamily\bfseries{\oldproofname}}


\usepackage{enumitem}

\newlist{pfsteps}{enumerate}{3}
\setlist[pfsteps,1]{%
  %label=\sffamily{Step {\arabic*}:\hspace{-1pt}},
  label=\sffamily{Step {\arabic*}:},
  ref=\normalfont{\arabic**},
  wide,itemsep=0pt,topsep=0pt
}

\newlist{pfparts}{description}{1}
\setlist[pfparts,1]{%
  font=\normalfont\textsf,
  itemindent=0pt,
  wide,
  itemsep=0pt,topsep=2pt,
}

\begin{document}

Proof using \verb!pfsteps! enumerate list.

\begin{proof}

\begin{pfsteps}

\item do this first. The way to begin is to start at the very beginning and use the definitions involved to get going.

\item do this next. After the initial step, we proceed by showing something interesting next.

\item conclude with this. That's it! \qedhere

\end{pfsteps}

\end{proof}

\bigskip

Proof using \verb!pfparts! description list.

\begin{proof}

\begin{pfparts}

\item[Sufficiency.] First we show that the stated condition is sufficient for the conclusion to hold.

\item[Necessity.] To conclude the proof we show that the stated condition is a necessary condition for the conclusion. \qedhere

\end{pfparts}

\end{proof}

\bigskip

For comparison, a proof that is not divided into steps or parts.

\begin{proof}

Here is the proof of the main result.
%
\end{proof}


\end{document}

Output

In the output shown below, the red arrows point to excessive space. (And for comparison, the green arrow points to the normal space after a "Proof."

Answer 1

The space before the label item (Step) is controlled with labelindent and the space between label and text is controlled with labelsep -- choose some values appropiate for you, my given values are just guesses.

\documentclass[12pt]{memoir}

\usepackage{amsthm}

% Indent ``Proof.''
\usepackage{xpatch}
\newlength{\normalparindent}
\AtBeginDocument{\setlength{\normalparindent}{\parindent}}
\xpatchcmd{\proof}{\hskip\labelsep}{\hskip\labelsep\hskip\normalparindent}{}{}

\let\oldproofname=\proofname
\renewcommand{\proofname}{\sffamily\bfseries{\oldproofname}}


\usepackage{enumitem}

\newlist{pfsteps}{enumerate}{3}
\setlist[pfsteps,1]{%
  %label=\sffamily{Step {\arabic*}:\hspace{-1pt}},
  label=\sffamily{Step {\arabic*}:},
  ref=\normalfont{\arabic**},
  wide,itemsep=0pt,topsep=0pt,
  labelindent=0pt,
  labelsep=0.5ex,
}

\newlist{pfparts}{description}{1}
\setlist[pfparts,1]{%
  font=\normalfont\textsf,
  itemindent=0pt,
  wide,
  itemsep=0pt,topsep=2pt,
  labelindent=0pt,
  labelsep=0.5ex,
}

\begin{document}

Proof using \verb!pfsteps! enumerate list.

\begin{proof}

\begin{pfsteps}

\item do this first. The way to begin is to start at the very beginning and use the definitions involved to get going.

\item do this next. After the initial step, we proceed by showing something interesting next.

\item conclude with this. That's it! \qedhere

\end{pfsteps}

\end{proof}

\bigskip

Proof using \verb!pfparts! description list.

\begin{proof}

\begin{pfparts}

\item[Sufficiency.] First we show that the stated condition is sufficient for the conclusion to hold.

\item[Necessity.] To conclude the proof we show that the stated condition is a necessary condition for the conclusion. \qedhere

\end{pfparts}

\end{proof}

\bigskip

For comparison, a proof that is not divided into steps or parts.

\begin{proof}

Here is the proof of the main result.
%
\end{proof}


\end{document}