The area of differential manifold is foreign to me. I want to find a convenient way to prove that a set is a manifold.

Here is a smooth map $f:\mathbb{R}^n\rightarrow \mathbb{R^m}$, where $m<n$. Now we know that $rank(\frac{df}{dx})$ reaches its maximum $r<n$, i.e., $r=\max\{rank(\frac{df}{dx}):x\in\mathbb{R}^n\}$, at each point $x\in f^{-1}(0)$. 

Can I conclude that $f^{-1}(0)$ is a manifold of dimension $n-r$ ? How to prove it? If there is some theorem to employ, please try to give me a reference or textbook. 

Thank you very much!