Let $(P,\pi,B,G)$ be a principle bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$.

Now I am searching for a good reference (with proofs) for the following facts:

1. The fundamental vector fields on $P$ span pointwise the vertical space - or equivalently they generate the $C^\infty(P)$-module of smooth sections of the vertical bundle.

2. Let $\gamma \colon TP \to \mathrm{Lie}(G)$ a connection one-form. The horizontal lifts of vector fields span pointwise the horizontal space - or equivalently they generate the $C^\infty(P)$-module of smooth sections of the horizontal bundle.