Let $M\subset\mathbb{R}^n$ be a smooth submanifold of dimension $m<n$, and a tubular neighborhood $\mathcal{V}=E(\{(x,v)\in NM:|v|<\delta(x)\})$ (diffeomorphic image of the smooth map $E:NM\to\mathbb{R}^n$ defined by $E(x,v)=x+v$) with $\delta:M\to]0,\infty[$ continuous function.
Fixed a point $p$, we can consider a local parametrization $\varphi:\mathbb{R}^m\to U_p\subset M$ with $\varphi(0)=p$ such that $T_pM=\varphi'(0)(\mathbb{R}^m)$.
From this, it can be shown that there exists a nbd $B_r(0)\subset\mathbb{R}^m$ such that the part $p+\varphi'(0)(B_r(0))$ of the affine tangent space $p+T_pM$ is contained in the tubular nbd $\mathcal{V}$, and there exists other nbd $V_p\subset U_p$ such that the affine normal spaces $x+N_xM$ intersects $p+\varphi'(0)(B_r(0))$ in a unique point $p_x$ for all $x\in V_p$ (show the image below in the case $n=3$ and $m=1$).
My question:
Taking into account that the normal bundle $NM$ is a smooth submanifold of $\mathbb{R}^{2n}$, it can be proven that the map $x\mapsto p_x$ (well defined in $V_p$) is smooth too?
