How do you know if you beat Batman Arkham City

In the menu after you've chosen your profile, there's a box that says "Game stats" or something of that nature. It breaks down the game into things such as the Riddler trophies, challenge maps, Catwoman story, and the plain ol' story. It should say 100% in all of them, but if you're just concerned about the main storyline, then there are two ways of knowing. At the end of the game, after you've beaten the storyline, the credits go. After the credits, you're standing next to Catwoman and she says something about her stuff being stolen, and she intends to get it all back. However, if these didn't happen, you're probably not done with the story. You can, however, check your progress in the "game stats" at the beginning. Hope this helps!