Question on a Nullstellensatz proof

Question

I'm reading a proof of Nullstellensatz and there is a part I don't quite get. The proof starts by choosing an $f$ inside the ideal $I(V(J))$ of $k[x_1,...,x_n]$ where $k$ is an (algebraically closed) field and $J$ is also an ideal of the same polynomial ring. We then introduce a new variable $x_0$ and the new polynomial ring $k[x_0,x_1,...,x_n]$ and an ideal $K$ generated by the elements of $J$ and the element $1-x_0f$, inside the new ring. After that the proof states that $f$ is invertible in the new ring and that is the part I don't understand. Why is it invertible? Can somebody please explain this to me?

Answer 1

The proof surely states that $f$ is invertible in $k[x_0,\ldots,x_n]/K$. This is the point of introducing the auxiliary variable $x_0$. By modding out by $K$, you are setting $x_0=\frac{1}{f}$.