How to remove italic lettering from the proof, definition and remark environments?

Question

I have the following document:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{proof}{Proof}
\numberwithin{equation}{section}
\begin{document}
\section{Section 1}
\begin{thm}
Theorem.
\end{thm}
\begin{rem}
Remark.
\end{rem}
\begin{lem}
Lemma
\end{lem}
\begin{prop}
Proposition.
\end{prop}
\begin{cor}
Corollary.
\end{cor}
\begin{defn}
Definition.
\end{defn}
\begin{thm}
Theorem.
\end{thm}
\begin{proof}
Proof.
\end{proof}
\end{document}

which provides the following:

How do I remove \textit from the content of the proof (in my example that is: some proof here), definition and remark? I have tried to change the \newtheorem command but it didn't work.

Answer 1

Just load amsthm and use \theoremstyle{definition} for those environments in which you don't want italics.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
%\newtheorem{proof}{Proof}
\numberwithin{equation}{section}
\begin{document}
\section{Section 1}
\begin{thm}
Theorem.
\end{thm}
\begin{rem}
Remark.
\end{rem}
\begin{lem}
Lemma
\end{lem}
\begin{prop}
Proposition.
\end{prop}
\begin{cor}
Corollary.
\end{cor}
\begin{defn}
Definition.
\end{defn}
\begin{thm}
Theorem.
\end{thm}
\begin{proof}
Proof.
\end{proof}
\end{document}

Answer 2

(I modified my answer after realizing that the OP desires to change the body font of just a few, but not all, theorem-like environments.)

I suggest you load the ntheorem package and issue the instruction \theorembodyfont{\upshape} right before \newtheorem{defn}[thm]{Definition}. That way, only the final three \newtheorem declarations will be affected by the font change.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb} % 'amssymb' loads 'amsfonts' automatically
\usepackage{ntheorem} % for '\theorembodyfont' macro
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\theorembodyfont{\upshape}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{proof}{Proof}
\begin{document}
\section{Introduction}
\begin{thm}   Theorem. \end{thm}
\begin{rem}   Remark.  \end{rem}
\begin{lem}   Lemma.   \end{lem}
\begin{prop}  Proposition. \end{prop}
\begin{cor}   Corollary.   \end{cor}
\begin{defn}  Definition.  \end{defn}
\begin{thm}   Theorem. \end{thm}
\begin{proof} Proof.   \end{proof}
\end{document}