I believe I have been able to construct in two ways, using the field axioms, that if $x \neq 0$ and $xy = xz$, then $y = z$. However, I've seen similar proofs like this assume that we can perform arithmetic operations, such as multiplying both sides by an inverse–which mirrors in some sense some proofs I've written in an abstract-algebra context–whereas others are more 'purist' in this sense. The similar proof in Rudin, for example, does not assume that we can use simple arithmetic.
My question, then, is which of these is 'more' standard in a first-year analysis course?
Proof 1: Assuming I can use arithmetic .
Since $x \neq 0$, $\exists x^{-1}$ s.t. $xx^{-1} = x^{-1} x = 1$ by the field axioms. Therefore,
\begin{align*}
xy = xz & & \text{By assumption} \\
x^{-1} (xy) = x^{-1} (xz) & & \text{Multiply on left by $x^{-1}$} \\
\left(x^{-1} x\right)y = \left(x^{-1} x\right)z & & \text{Associativity} \\
1y = 1z & & \text{Inverse properties} \\
y = z
\end{align*}
Example 2: Without assuming arithmetic, and mirroring Rudin.
\begin{align*}
y & = 1 \cdot y & & \text{Multiplicative identity} \\
& = \left(x \cdot \frac{1}{x}\right) y & & \text{Mult inverse axiom with $x \neq 0$} \\
& = \left(\frac{1}{x} \cdot x\right)y & & \text{Commutativity of multiplication} \\
& = \frac{1}{x} \left(x \cdot y\right) & & \text{Associativity of multiplication} \\
& = \frac{1}{x} \left(xz\right) & & \text{Assumption that $xy = xz$} \\
& = \left(\frac{1}{x} \cdot x\right) z & & \text{Associativity of multiplication} \\
& = 1z & & \text{Inverse properties} \\
& = z
\end{align*}
Thanks in advance.
Best Answer
The two proofs are essentially the same and the first doesn't use arithmetic, but rather field axioms. I wouldn't use $\frac{1}{x}$, but that's more cosmetic than substantial.
More substantial is that you don't need to appeal to commutativity: \begin{align} y &=1y &&\text{(multiplicative identity)} \\ &=(x^{-1}x)y &&\text{($x\ne0$ has an inverse)} \\ &=x^{-1}(xy) &&\text{(associativity)} \\ &=x^{-1}(xz) &&\text{(hypothesis)} \\ &=(x^{-1}x)z &&\text{(associativity)}\\ &=1z &&\text{(property of the inverse)} \\ &=z &&\text{(multiplicative identity)} \end{align}
On the other hand, the other proof seems shorter \begin{align} & xy=xz &&\text{(hypothesis)} \\ & x^{-1}(xy)=x^{-1}(xz) && \text{($x\ne0$ has an inverse)} \\ & (x^{-1}x)y=(x^{-1}x)z && \text{(associativity)} \\ & 1y=1z && \text{(property of the inverse)} \\ & y=z && \text{(multiplicative identity)} \end{align} and less “rabbit out of a top hat”.