We fix $p$ prime and $n$ a natural number. We let $K(n)$ be the $2(p^{n}-1)$-periodic Morava $K$-theory, i.e. $K(n)_*=\mathbb{F}_p[v_n^{\pm 1}]$ with $|v_n|=2(p^n-1)$. I distinctly recall that we should have $\pi_*(L_{K(n)}BP)\cong (v_n^{-1}BP_*)^{\wedge}_{I_n}$, yet I am unable to find an explicit reference in the literature to this fact. Do you have any idea where I can find the proof of such computation?

Also, I was wondering if we apply additional localizations with respect to these Morava $K$-theories this behavior continues. E.g. for $m<n$ do we have $\pi_*(L_{K(m)}L_{K(n)}BP)\cong (v_{m}^{-1}(\pi_*(L_{K(n)}BP)))^{\wedge}_{I_m}$ and so on?