The Flanders theorem over division rings

Let D be a division ring and F be a subfield of the center of D over which D has finite dimension d. Let n, p, r be positive integers and V be an affine subspace of the F-vector space Mn,p(D) in which every matrix has rank less than or equal to r. Using a new method, we prove that dimF⁡V≤max⁡(n,p)rd and we characterize the spaces for which equality holds. This extends a famous theorem of Flanders which was known only for fields.