$2x = 2y \Rightarrow x = y$ with $x,y \in \mathbb{Z}$ How can I show, $2x = 2y \Rightarrow x = y$ with $x,y \in \mathbb{Z}$, when I can only use elements out of $\mathbb{Z}$? It is $\frac{1}{2} \notin \mathbb{Z}$, so I can't multiply both sides with $\frac{1}{2}$. How can I prove it?

$$\begin{align}
2x&=2y
\\ \Rightarrow2x-2y&=0
\\ \Rightarrow 2(x-y)&=0
\\ \Rightarrow 2=0&\text{ or }x-y=0
\\ \Rightarrow x&=y.
\end{align}$$
EDIT: This (or rather I) implicitly uses the field axioms of $\mathbb{Q}$. We need this to fix it.
EDIT 2 How about this:
$$\begin{align}
2x&=2y
\\\Rightarrow x+x&=y+y
\\\Rightarrow x-y&=-(x-y)
\end{align}
$$
The only number that is its own inverse is zero. Hence $x=y$.
