Help with a certain proof For all $x,y \in \mathbb{R} - \{0\}$, $(xy)^{-1}=x^{-1}y^{-1}$.


I was wondering how I could solve this.
$\mathbb R$ is a field, so nonzero elements are invertible, then
$$xy\cdot x^{-1}y^{-1}=xy\cdot y^{-1}x^{-1}=x\cdot1\cdot x^{-1}=x\cdot x^{-1}=1$$
So $(xy)^{-1}=x^{-1}y^{-1}$.
If $xy\neq 0$, we can write $(xy)^{-1}=\frac{1}{xy}$ . So we can write $\frac{1}{xy}=\frac{1}{(x^1 y^1 )}= x^{-1} y^{-1}$