Good. Well done. Now do it for the engine failure case
(I did. Trying to be a total smart-Alec. And when I got to BGS was told that such questions never appeared for ETP/CTP!)
If you're looking for another couple of proofs to do how about Vmp=0.76Vmd or Vrange=1.32Vmd? Subject to jet/prop considerations of course....
Enjoy!