Constructing a principal $G$-bundle

Question

Let $M$ be an $n$-dimensional smooth manifold, let $Fr(TM)$ be its frame bundle, and let $\{\theta^1,\ldots,\theta^r\}$ be a partial co-frame of $M$, i.e. each $\theta^i$ is a section of $T^\ast M$ such that $\{\theta^1(m),\ldots,\theta^r(m)\}$ is linearly independent for each $m\in M$. Define the subset \begin{equation} P_\theta:=\{\phi\in Fr(TM):\theta^i(\phi_j)=\delta^i_j:1\le i\le r,1\le j\le n\}$. \end{equation} It is clear that $P_\theta$ is invariant under the action of all matrices of type \begin{equation} \begin{pmatrix} 1&0 \\ A&B \end{pmatrix},A\in Mat(n-r,r), B\in GL_{n-r}. \end{equation} I have shown that the set of all matrices of this type (call it $G$) is a Lie subgroup of $GL_{n-r}$, so that the action of $GL_n$ on $P_\theta$ restricts to an action of $G$ on $P_\theta$. This action is clearly free, and I want to show now that $P_\theta$ is a principal $G$-bundle over $M$. To this end, I could show either that $P_\theta$ is itself a manifold (or perhaps even that its a submanifold of $P$) and that the action of $G$ on $P_\theta$ is proper, which then implies that $P_\theta$ is a principal $G$-bundle over $P_\theta/G\cong M$. I have been trying this, unsuccesfully. Can someone help, or give hints?