Also impressive, and correct
(but for "So Vx prop is slightly less than Vmp" that should read "So Vx prop is slightly less than Vmd"
So, PowderMonkey, with Keith's explanations on combining thrust curve and drag curve you know why Vx <= Vmd (with Vx = Vmd for jets),
and with his explanations on combining power available curve and power required curve, you have why Vmp < Vy.
And with my post you have why Vx < Vy and why Vmp < Vmd
Eventually you have
Vmp < Vx <= Vmd < Vy
The fact that Vmd is lower than Vy is obvious when Vx = Vmd (engine with almost constant thrust).
It stays in that relation as long as the effect of drag increase at speed lower than Vmd
is more important than the effect of engine thrust decrease as speed get higher than Vmp.
Luc