I am learning about the tetrad basis for manifolds from this lecture notes. On pg 52, the spin connections ${{w_\mu}^a}_b$ are defined as $${{w_\mu}^a}_b=e^a_\nu e^\lambda_b\Gamma^\nu_{\mu\lambda}-e^\lambda_b\partial_\mu e^a_\lambda,$$
where the coordinate basis vectors $\hat{e}_\mu$ and tetrad basis vectors $\hat{e}_a$ are related by $$\hat{e}_\mu=e^a_\mu\hat{e}_a,$$ $$\hat{e}_a=e^\mu_a\hat{e}_\mu.$$
The author claimed that the first equation is equivalent to $$\nabla_\mu e^a_\nu=0.$$ How can this be shown to be true?