Let $f:\mathbb{R}^{m+n}\to \mathbb{R}^m$ be a smooth map with $f(0)=0$. Assume that $0$ is a regular value that is for every $p\in f^{-1}\{0\} $ we have $Df(p):T_p\mathbb{R}^{m+n}\to T_0\mathbb{R}^m$ is surjective. Show that $f^{-1}\{0\}$ is a submanifold of dimension $n$.
I found the problem here. But as I want to do using the inverse and implicit function theorems. So I wanted to show that $\forall\ p\in f^{-1}(0),\ \exists U$ neighborhood of $p$ such that $U$ is homeomorphic to some ball in $R^n$ to conclude that it is a manifold of dimension $n$.
My attempt
Take any $p\in f^{-1}(0)\implies f(p)=0. $ We write $p=(p_x,p_y)$ with $p_x\in \mathbb{R}^n$ and $p_y\in \mathbb{R}^m$. Since $Df(p)$ is surjective, by some permutation we can assume that $\frac{\partial f}{\partial y}(p)$ is invertible and hence using implicit function theorem there exist neighborhoods $V$ and $W$ of $p_x$ and $p_y$ respectively and a smooth map $\phi: V\to W$ such that $f(x,\phi(x))=0,$ for $x\in V$. After that I stuck.
The map \begin{align*} \omega: \mathbb{R}^n & \to \mathbb{R}^{n+m} \\ x &\mapsto (x,\phi(x))\end{align*} is a chart on $f^{-1}(0)$. Let me know if this is unclear in any way!