How can I use amsmath package and mathptmx package together? [duplicate]

Question

I use the below method.
But this method doesn't work for this situation. Why? I have the same results. This is my minimal file and I use texlive 2015. This problem is about \prod too. Thanks for your responsibility.

\documentclass[12pt,a4paper]{report}
\usepackage{amsmath}
\let\amssum=\sum
\usepackage{mathptmx}
\let\ptmxsum=\sum
\let\sum\amssum
\begin{document}
I want this one in ams style:       
\[
\sum
\]
text text text text text text text text text text text text  
and this one in mathabx style:  
\let\sum\ptmxsum
\[
\sum
\]  
But both of them are the same and I don't like it .....
\end{document}

Answer 1

Since it seems you added \prod to the question, I take my answer at Different Sum signs, and using the \fonttable (commented in the MWE), determine the slots for text and display style \prod as 81 and 89, respectively.

\documentclass{article}
\usepackage{mathptmx,mathtools}
% =============================================
%Import symbols from font cmex without importing the whole package
% =============================================
\DeclareFontFamily{U} {cmex}{}

\DeclareFontShape{U}{cmex}{m}{n}{
  <-8> cmex7
  <8-9> cmex8
  <9-10> cmex9
  <10-> cmex10}{}

\DeclareSymbolFont{Xcmex} {U} {cmex}{m}{n}

\DeclareMathSymbol{\Xdsum}{\mathop}{Xcmex}{88}
\DeclareMathSymbol{\Xtsum}{\mathop}{Xcmex}{80}
\DeclareMathOperator*{\Xsum}{\mathchoice{\Xdsum}{\Xtsum}{\Xtsum}{\Xtsum}}
\DeclareMathSymbol{\Xdprod}{\mathop}{Xcmex}{89}
\DeclareMathSymbol{\Xtprod}{\mathop}{Xcmex}{81}
\DeclareMathOperator*{\Xprod}{\mathchoice{\Xdprod}{\Xtprod}{\Xtprod}{\Xtprod}}
% =============================================
%\usepackage{fonttable}
\begin{document}
\centering
sum and prod under mathptmx:\par
$\sum_{i=1}^2 x_i \ne \prod_{i=1}^2 x_i$
\[\sum_{i=1}^2 x_i \ne \prod_{i=1}^2 x_i \]
Defined Xsum and Xprod from cmex:\par
$\Xsum_{i=1}^2 x_i \ne \Xprod_{i=1}^2 x_i$
\[\Xsum_{i=1}^2 x_i \ne \Xprod_{i=1}^2 x_i\]
%\tiny\fonttable{cmex8}
\end{document}