Double Doppler for Free

The last post showed the relativistic Doppler effect measured in a stationary frame from bouncing signals off an object in a moving frame. How about two moving frames? Geometry Expressions does a really good job, but, this time, the results are sufficiently hairy that we must help out a bit. The Minkowski sketch is simple enough -- one frame has speed u and the other has speed v.

I chopped off this very hairy expression for the pulse duration in the v frame:

We all know that simplifying algebraic expressions is a tough problem, and sometimes doesn't give the results we want. So, I used GX's "copy to Mathematica form", and, after a FullSimplify in MMA, got this:

This still wasn't good enough for me, so, by hand, I figured this:

But, symbolic checking of equality is problematic because of the absolute-value functions, resulting from the fact that we didn't constrain u and v adequately to be less than one = c. I'm lazy, so I checked equality numerically: