Does the FIDE 75-move rule apply after checkmate or resignation? 75 move rule says


  9.6 If one or both of the following occur(s) then the game is drawn: (...) any series of at least 75 moves have been made by each player without the movement of any pawn and without any capture.


This rule can be read to mean that the game ends after 75 moves without capture or pawn move and anything that occurs after that doesn't matter. Let's say I defend a pawnless endgame and over 75 moves pass without capture. I don't realize it and a few moves later I'm about to be checkmated and resign. The scoresheet is signed as a loss. Then I count the moves and see that 75 moves were exceeded. Am I right that I can claim that the game had ended in a draw before I resigned and my resignation has no effect? I think FIDE rules could be clearer on this point, but is this interpretation valid?

Bonus question: if the game is in rapid with no scoresheets, but with video recording, such as at top level tournments, could a player who resigned make the same kind of claim and request counting moves from DGT or video to ascertain that 75 moves have passed?
The FIDE Laws of Chess (I'm giving a link to the version in the Arbiter's Handbook because FIDE have Munged their own site) do give a definitive answer but it takes some searching to find. The key appears in the section on the Chess clock!


  6.2.1 During the game each player, having made his move on the chessboard, shall stop his own clock and start his opponent‟s clock
  (that is to say, he shall press his clock). This “completes” the move.
  A move is also completed if:
  6.2.1.1 the move ends the game (see Articles 5.1.1, 5.2.1, 5.2.2, 9.6.1 and 9.6.2),


We turn to 9.6.1 and 9.6.2 -


  9.6 If one or both of the following occur(s) then the game is drawn:
  9.6.1 the same position has appeared, as in 9.2.2 at least five times.    
  
  9.6.2 any series of at least 75 moves have been made by each player without the movement of any pawn and without any capture. If the last
  move resulted in checkmate, that shall take precedence.


Checkmate or resignation occurred after the 75 moves hence the game had already ended and provided the rules of the competition allow for appeals / claims then your claim of a draw is valid. Note that if you wait too long the decision is not going to be reversed. For instance if the results have already been sent to FIDE for rating then it is way too late. If you notice 10 minutes after the game then you should still be able to appeal and claim the draw.

Note that had you been paying attention then you would have claimed the draw after 50 moves. After 75 moves it is the arbiter who is supposed to be paying attention and you should not have to take any action. It is his responsibility to pay attention and step in to declare the game drawn. The only reason you have to take action and appeal in this case is to correct the mistake of the arbiter.