Let $I$-identity operator, $\Pi_N$ is the orthogonal projection in $L_2$ onto subspace by the first $N$ eigenfunctions of the Stokes operator in $\Omega$, $\alpha_j$ denotes the increasing sequence of the eigenvalues for the Stokes operator, $c>0$ is the some constant not depending on $N$. We know that $\Pi_N v = 0$.

How to derive the following inequality using the Poincare inequality? Which form of inequality is used here? $|v|_{L_2}=|(I-\Pi_N)v|_{L_2}\le c\alpha_N^{-1/2}|v|_{H^1}$.