In (Bellucci, 2010; pg89) it is said (my wording):
A pure bipartite state is separable if and only if the two reduced density matrices are pure.
Proving the "only if" part is easy but I am struggling to get anywhere with the "if" part. I.e. how do we prove that if $\rho_A$ and $\rho_B$ are pure then $\rho_{AB}$ is separable? (Where here $\rho_A$ and $\rho_B$ are the corresponding reduced density matrices of $\rho_{AB}$.)
Intuitively it is obvious that this should be the case - but I am looking for a mathematical proof.