I've always heard the distributive law of multiplication described as an axiom, an assumed property of a given set of numbers of any kind.
That is not true. In our current number system up to complex number, we have a bottom up approach to define everything, including rules of addtion and multiplication. All the properties such as commutative law can be deduced and proven using their definition. The axioms of number system only contain the Peano's Axioms, with some usage of set theory so you can consider axioms of set theory is used.
This bottom up approach is like this:
Building natural numbers using Peano's Axioms. There are 5 of them. Addition(natural number limited) can be constructed using the successor notation(which is the 2nd Peano's Axiom). Within this range, we can prove some property of addtion(natural number limited), like communtation and association, using mathematical induction, which is the 5th Peano's Axiom. On this base we can define multiplication as repeated addtion, an prove some of its properties.
Then we use natural number and addtion to construct and define negative numbers. While doing this, we must prove that in this newly defined system, all the previously defined operations still hold. We must check and prove each of additive commutation and so on still hold for negative numbers, and we can say that expansion from natural number to integer is successful. This process will be somehow tedious and trivial, but if we want it to be must rigorous it will be necessary.
Then we use integers and multiplication to define rational numbers. Still, check if each existing rules still hold.
Finally, we use the concept of limit to define real number. This step is less trivial because it involves some new concept, but all these concept do not come from axioms but can be defined using rational numbers. Note that if we want to do this rigorously bottom up, we can only use sequences of rational numbers, and the concept of distance must be limited to rational. Once this step is done, we can easily see that all the operations, properties, concepts can be immediately apply to real numbers, and finally we have what we are very familiar with.
In a top down approach, we will say like this: We want to prove, say distribution law for real numbers, then we must know how real number is defined using rational numbers, and how to prove distribution law in rational numbers. Again for rational numbers, we will trace back to integers, and proof of integer distribution law is done by mathematical induction which is 5th Peano's Axiom. Here is the axiom really on the bottom.
Terry Tao in his Analysis book did what we just mentioned in huge and tireless details, which cost 3 or 4 chapters, so that we can completely see through the entire approach of how our number system is built. Actually I learned all this from his book.