I am studying Lazarsfeld's proof of Mumford's Regularity Theorem. At some point, under the assumption of $\mathcal{F}$ being $m$-regular, he considers the exact sequence
\begin{equation} \cdots \to \bigwedge^2 V_{\mathbb{P}} \otimes \mathcal{F}(m-1) \to V_{\mathbb{P}} \otimes \mathcal{F}(m) \to \mathcal{F}(m+1) \to 0 \end{equation}
where $V_{\mathbb{P}} = V \otimes \mathcal{O}_{\mathbb{P}}$ where $\mathbb{P}$ is an $r$-dimensional projective space over $\mathbb{C}$ and $V$ is an $r+1$-dimensional complex vector space such that $\mathbb{P} = \mathbb{P}(V)$. He states that by the regularity of $\mathcal{F}$ one concludes the vanishing of
\begin{equation} H^i(\mathbb{P},\bigwedge^{i+1} V_{\mathbb{P}} \otimes \mathcal{F}(m-i)) \end{equation}
for every $i$. I tried with the definition of cohomology as derived functors, I tried in hoping that $V_{\mathbb{P}}$ could be flasque and then apply Kunneth formula, but it isn't flat or so it seems, and I also tried to look after some vanishing results that could help me here, but nothing seemed to be applying. Why is such statement true?