As a relative beginner trying to understand math more deeply, I'm trying to learn more about the mathematical laws (the laws of the operations $+, -, \times, \div$)
For example, I know the basic laws (the ones that are just taken to be true) -- the commutative, associative, and distributive laws. What area of math (what books to read) would I find things like how subtraction is defined from addition (and how division is defined from multiplication), and the presentation of the various laws of subtraction and division as they relate to the basic laws and how to prove them. Some examples:
$n-(m+k) = (n-m)-k$
$\frac{k}{m}+\frac{n}{m}=\frac{k+n}{m}$
$\frac{\frac{n}{m}}{\frac{k}{l}}=\frac{n\times l}{m\times k}$
How to derive and prove such laws (and many others like them)?
Thank you