Double-Up would not proc it, since Mystic shot doesn't proc Rylai's either. Rylai's only applies to spells that deal Magic Damage.

Everybody going AP on miss fortune needs a swift kick to the nuts

I think the game code may treat Mystic Shot differently, though. I'm pretty certain I've seen Spell Vamp triggering on Spear Shot, and I'm under the impression that Spell Vamp triggers on pretty much everything Rylai's does.

I'll actually try this out later with Pantheon. I feel it needs tested on someone with a physical-damage effect which does not proc onhits.

EDIT: Talking about Double Up.

It doesn't. It's a buff, not an actual targetted spell. Phage and Hammer, however, will proc.

I just tested it. As you can see, the golem is debuffed with a Slow effect, I have a Rylai's, and I just hit it with Spearshot (which is on CD now).

If it's just applying a ghost debuff for some idiotic reason, then that's one thing, but I AM applying slow debuffs on use of skills with purely physical damage components. I'm fairly certain at this point that the only physical damage spells which won't work with Rylai's or spell vamp are those that specifically enable on-hit effects (Parrley and Mystic Shot) and that pure-physical damage spells will proc it.

EDIT: Wow, low-res pic. Sorry. Point still stands though.