Consider a matrix $$ B\equiv \begin{pmatrix} D'D & D'F \\ 0_{d\times n} & P'F \end{pmatrix} $$ where
(1): $D'D$ is an $n\times n$ diagonal matrix with strictly positive elements in the diagonal
(2): $0_{d\times n}$ is a $d\times n$ matrix of zeros
(3): $D'F$ is an $n\times J$ matrix
(4): $P'F$ is a $d\times J$ matrix
Question: is it true that $B$ has column-rank $n+J$ if and only if $P'F$ has column-rank $J$? Which properties from (1)-(4) are you using to enstablish such a result? In particular, is (2) key to claim this result?