I'm trying to understand the slice-theorem for proper Lie-group actions. Having a smooth manifold $M$ and a Liegroup $G$ acting on $M$ in a proper way, we have the slice theorem, saying that at each point $x \in M$ we find a slice and a tube.

So choosing $x \in M$, denote by $H = G_x$ the isotropy-group of $G$ at $x$. Using the tube-theorem we find a $G$-invariant neighborhood $U \subset M$ of the orbit $G \cdot x$, a vectorspace $V \cong T_xM / T_x(G \cdot x)$ where $H$ acts linearly on $V$, a $H$-invariant vector-subspace $D \subset V$ and a $G$-equivariant diffeomorphism $$\phi \colon G\times_H D \to U$$ such that $\phi([e,0])=x, \ \phi([G,0])=G \cdot x,\ \phi([e,D])=S$ where $S$ is a slice of $G \cdot x$.

From now on I'm considering the case, that $M$ is connected and we choose $x \in M$ such that the $G$-orbit is a principal orbit.

Now I want to construct $k = \dim M -\dim G \cdot x$ functions $f_1, \dots, f_k$ on $U$, such that they are $G$-invariant, with $d_xf_1 \wedge \dots \wedge d_xf_k \neq 0$ and such that $G \cdot x = \{y \in U \ |\ f_1(y) = \dots = f_k(y)=0\}$.

Using the slice theorem I have to find functions $\tilde{f}_1, \dots, \tilde{f}_k \in C^\infty(D)$ which are $H$-invariant, and satisfy $\tilde{f}_j(0)=0$ then I could take the extension \begin{align} \tilde{F}_j \colon G \times_H V \to \mathbb{R}, \quad F_j([g,v])= \tilde{f}_j(v) \end{align}

and the functions $$F_j = \tilde{F}_j \circ \phi^{-1} \colon U \to \mathbb{R}$$ are $G$-invariant and satisfy $G \cdot x = \{y\in U \ | \ F_1(y)=\dots = F_k(y)=0\}$.

But I'm not sure how to construct these functions $\tilde{f}_1,\dots, \tilde{f}_k$ on $D\subset V$.

Edit: What I forgot to post: Since $x$ is a regular point, i.e. the dimension of $G \cdot x$ is maximal, we find a neighborhood $W \subset U \subset M$ of $x$, such that $\dim G \cdot y = \dim G \cdot x$ for all $y \in W$. Since now $T_yM = T_yS + T_y (G\cdot y)$ and $T_yS \cap T_y (G \cdot y)=T_y (H \cdot y)$ we have \begin{align} \dim M &= \dim G \cdot y + \dim S - \dim H \cdot y \\&= \dim G \cdot x +\dim S - \dim H \cdot y \\&= \dim M -\dim H \cdot y \end{align} we have $\dim H \cdot y =0$ for all $y \in S \cap W$. From that I concluded, that $H$ acts as the identity on $D$. But is this true?