Which field property enables us to multiply on both sides by the same value, while preserving equality? I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off.
I was wondering which field property enables us to multiply on both sides of an equation and still preserve equality.
There is a very clear proposition stated in the book that gives me this for inequalities:
$$
\text{If} \ \ x > 0 \ \ \text{and} \ \ y < z \ \ \text{then} \ \ xy<xz.
$$
However, the only proposition that seems useful for this in the case of equalities, is stated as an implication and not an equivalence:
$$
\text{If} \ x\not= 0 \ \ \text{and} \ \ xy=xz \ \ \text{then} \ \ y=z.
$$
Any help would be much appreciated.
 A: This isn't a field property, it's a property of the underlying logical framework within which we're defining fields in the first place.
Specifically, the main property is that if $a=b$ then any sentence involving $a$ is equivalent to the same sentence gotten by replacing some of the $a$s with $b$s; we also use the simpler property that "$=$" is reflexive. From this we can argue:


*

*Suppose $a=b$.

*By reflexivity we have $ma=ma$.

*Now by the first bulletpoint we can substitute $b$ for the second $a$ in the second bulletpoint, which gives us $$ma=mb$$ as desired.

That logical framework is often swept under the rug. Some people find this helpful since it means that they don't have to worry about such "basic" facts and can focus on more interesting stuff. Others find this annoying since hiding assumptions really goes against the whole point of the "axiomatic" turn that the definition of fields is part of in the first place. Personally, I lean on the side of not sweeping this sort of thing under the rug, but that reflects my own logician-y biases.
Aside from basic rules for equality, our logical rules also tell us how to manipulate statements in general. E.g. the fact that you can prove "Every $x$ has property $P$" by introducing an arbitrary $x$ and showing it has property $P$ is just the rule of universal generalization.
However, there are some subtleties around this logical framework itself. Basically, "naive" mathematical reasoning takes place in second-order (or similar) logic, but that's truly terrible when we actually look at it. First-order logic turns out to be the right way to go, but with a twist: we study (for example) fields within the large first-order theory $\mathsf{ZFC}$, the latter of which serves as a general all-purpose framework for conducting mathematics.
A: Well, actually this is due to some field property (but one which is usually in the preamble of the definition and not in the list of axioms): The definition of a field states that multiplication is a map $mult$ taking two arguments of the ground set to another one.
And one of the inherent properties of maps is that they have a single output for any given input. That means that if $a=b$ than $mult(m,a)$ and $mult(m,b)$ have the same input and thus, their outputs $ma$ and $mb$ are the same.
