Receipt required for Warranty service?

[ptf]

Hi, I recently purchased my first Rolex, a 16610 Submariner, from a grey market dealer who I trust (a reputable jeweler who is also a brother of a friend). I am very happy with it. I received all the packaging, plus the plastic warranty card with the original AD's name (not the name of the actual jeweler who sold it to me though) printed on it. The day I bought it, we filled out my name and the date of purchase.

If I need servicing however, is it enough to submit the watch with the warranty card? Or, would I need an original receipt from the AD (which obviously I don' t have)? In particular, how is the RSC in New York as far as those things?