Algebra Precalculus – Proof of the Rules of Elementary Algebra

Where can I find proofs of the rules of basic elementary algebra?

Here, when I say "elementary algebra," I'm not talking about Abstract Algebra/Modern Algebra. I'm talking about arithmetic operations and exponentiation over the real numbers. Grade school algebra.
I've long-since internalized the rules of arithmetic; namely:

$a(bc) = (ab)c $ and $a + (b + c) = (a + b) + c$
$ab = ba$ and $a + b = b + a$
Subtraction being the additive inverse
$a \div b = a * \frac{1}{b}$
$-(-a) = a$
$a(b + c) = ab + ac$
$-a < -b \implies a > b$

(and the like for all other relational operators)

$a^{-b}=\frac{1}{a^b}$
$a^{\frac{1}{b}}=\sqrt[b]{a}$
$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}*\frac{d}{c}$
$a^b * a^c = a^{b + c}$
$(a^b)^c = a^(b*c)$
$(ab)^c = a^c * b^c$
$(\frac{a}{b})^c = \frac{a^c}{b^c}$
$a^0 = 1$

Some more (perhaps too obvious) ones:

Zero as the additive and subtractive identity
One as the identity for multiplication, division, and exponentiation
$\frac{a}{a} = 1$ (i.e. multiplication of something by its multiplicative inverse yields the multiplicative identity)
The transitivity, reflexivity, and symmetry of the equality operator
(Those last few seem the most obvious/basic to me, but you can't take anything for granted if you want true rigor.)

(See this link for where I got most of these delineated: http://faculty.ung.edu/mgoodroe/PriorCourses/Math_0999_General/The%20Formal%20Rules%20of%20Algebra.pdf)

All of these rules just seem obvious to me, but what I've begun to realize is that, although I'm more than proficient in using them, I have no idea why they're true. I can easily manipulate expressions with any of them, but I'm not actually certain which ones are basic (i.e. the axioms) and which ones must be proven (whether by induction, contradiction, contraposition, or direct proof).

Furthermore, the greater question in my mind is why these rules should apply to quantities/volumes in the real world in the first place. Things like, "which axioms listed above prove that both quotative and partitive division valid uses of the division operator?"

I know that, since the real line is continuous and algebraically complete over everything except roots (see: complex numbers), its properties only hold true completely for approximately continuous substances and situations (like time or water).

In the abstract, the beauty of mathematics is that we can posit any axioms for any abstract structure we want, and as long as they are internally consistent, the actual content of the math consists of finding the consequences of these axioms. However, I'm not sure which of these axioms are the ones that necessary to connect to real-world examples before generalizations can be made that verify the validity of the rest of them by implication.

Obviously, at some step, a generalization has to be made that cannot be proven based on experience, because we cannot experience all situations or confirm all relationships—we can only posit some structure that fits the rules that seem to hold true so far.

I know that we can basically say "these are nothing but the properties of the structure we've posited, but if we can prove that some concrete examples possess the properties of this structure, then these rules apply to those structures as well" (i.e. the structure is isomorphic to certain conceptualizations of the real world). But that necessarily reminds me that I must question which rules that I verified in the abstract hold true in the concrete, and whether those observations are sufficient for the application of the rest of our abstract results to the analysis of the concrete.

Would the formal proof/axiomatization of each of these rules be found (from the ground up) in Field Theory? Peano Arithmetic? Or where would I find a rigorous treatment?

Algebra Precalculus – Proof of the Rules of Elementary Algebra

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