Jacobian Criterion from $\mathbb R^n$ to $\mathbb P^n(\mathbb R)$

Question

$\newcommand\R{\mathbb{R}} \renewcommand\P{\mathbb{P}}$ $\newcommand\pd[2]{\frac{\partial #1}{\partial #2}}$ Let $f:\R^n\rightarrow \P^n(\R)$ be smooth, then why can one apply the inverse function theorem on $f$ at $\vec{x}\in\R^n$ iff the matrix (evaluated at $\vec{x}$) $$ \begin{pmatrix} \pd{f_0}{x_1} & \pd{f_0}{x_2} & \dots & \pd{f_0}{x_n} & f_0\\[2ex] \pd{f_1}{x_1} & \pd{f_1}{x_2} & \dots & \pd{f_1}{x_n} & f_1\\[2ex] \vdots & \vdots & \ddots & \vdots & \vdots \\[2ex] \pd{f_n}{x_1} & \pd{f_n}{x_2} & \dots & \pd{f_n}{x_n} & f_n\\[2ex] \end{pmatrix} $$ has a non-zero determinant? Here: coordinates in the affine space are numbered $1$ to $n$ and in the projective space from $0$ to $n$, and $f(\vec x)=[f_0(\vec x):\ldots:f_n(\vec x)]$ for real valued infinitely differentiable functions $f_i$ ($i=0,\dots,n$). I understand the case between two affine spaces and between two projective spaces (Jacobian criterion?) but I do not get this one.