Let $\omega_p:T_pP\longrightarrow T_eG$ be any Lie-algebra valued one-form on a principal bundle $(P,\pi,M)$ satisfying $$ (a)\;\;\omega_p(X^{A}_p)=A $$ $$ (b)\;\;((\triangleleft g)^*\omega)_p(X_p)=(Ad_{g^{-1}})_*(\omega_p(X_p)) $$ where $X^{A}$ is the fundamental vector field induced by the Lie algebra element $A\in T_eG$.
I want to prove that any one-form satisfying these properties induces a horizontal subspace, ie, is in particular a connection one-form. Can anyone push me towards the right direction to start? I'm not even sure what I'm supposed to be driving at, what would a solution to this sort of thing look like?
Am I trying to prove that we are forced to define $\omega_p$ in such a way that it involves horizontal subspaces if we want it to satisfy the above properties, or is that barking up the wrong tree?