You surely can define your shortcuts; indeed you should.
Let's make some examples. Suppose your document is full of Fourier transforms, for which you need a fancy F. Instead of writing every time
$\mathcal{F}(f)$
it's surely better to define a new command, say
\newcommand{\FT}{\mathcal{F}}
(choose any name you like), so that you can type
$\FT(f)$
and get the same result as before, with a big bonus! If you change your mind about the notation, you can simply modify the definition.
Another example. The "gradient" operator is not predefined; so you might want to have a command for it:
\DeclareMathOperator{\grad}{grad}
A different one; my preferred notation for vectors is, say, \mathbf{v}. However, since conventions are different, I never type vectors in that way, but prefer to have
\newcommand{\vect}[1]{\mathbf{#1}}
for the same reason as before; I might change my mind and want to modify the appearance, say for using bold italic; this would be accomplished just by saying
\usepackage{bm}
and changing the above into
\newcommand{\vect}[1]{\bm{#1}}
How do you organize this? Here's an example:
\documentclass[a4paper]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % input encoding UTF-8
% Useful packages
\usepackage{amsmath,amssymb,amsthm}
% add all the packages you need
% Personal definitions
\newcommand{\FT}{\mathcal{F}} % Fourier transform operator
\DeclareMathOperator{\grad}{grad} % gradient
\newcommand{\vect}[1]{\mathbf{#1}} % vectors and matrices
\begin{document}
...
\end{document}
Add definitions while you find that they are useful for distinguishing logical units of your document.
Best Answer
As @egreg said in the comments, you can use the
proofenvironment fromamsthm, which will automatically place a QED symbol at the end of the proof. You can also use the command\qedsymbol, also fromamsthm, to place a QED symbol anywhere. Here's an example of both options, with the preamble you provided.In fact,
amsthmuses the command\qedsymbolto place the symbol at the end of theproofenvironment, so if you want to use something else as a QED symbol (for example a black square), you can simply redefine\qedsymbol, and theproofenvironment will use it.