Which GPU was it? As of 2 months ago, they still had a program in place to replace 6970m GPUs due to a known issue with them. Mine kept shutting itself off, presumably due to heat. When I called in about this particular issue, they knew about it already and immediately setup an on-site appointment for me to get it replaced along with some other work. I didn't have to prove anything to anyone.
Also, if it is more convent for you and you live within 50 miles of an Apple authorized and trained service tech company (Not Apple but there are such repair techs they use all over North America at least), you are entitled to on-site service for your iMac with AppleCare. I had 3 on-site service calls and I live in a rather rural location but it was close enough to civilization for a tech to be available to come out to me.
I've never dealt with an Apple Store tech although I've read enough stories to wonder if you aren't better off dealing with AppleCare over the phone and asking for on-site. I had very good experience that way even if the machine kept having problems. The actual repairs were done properly by pros.
Also, if they need to fix your machine too many times while it is still under warranty, I think it becomes reasonable to ask for a replacement. I did and they granted the request with a brand new iMac I should be getting by the end of next week. At some point, they get tired of throwing money at it probably as much as you become tired of being inconvenienced with issues and repairs. I think their general rule of thumb is if the same problem needs fixing 3 times it becomes time to consider the option of replacement. In my case, we were at the point of me needing LCD number 3 and they aren't cheap. However, if I had enough miscellaneous troubles, I'd still ask for that because I expect a certain level of quality from Apple which I am willing to pay for but also expect to get.