Are you just showing off your 6000 battled Vaporeon!? I don't think there's another way to see it but I'm impressed :).
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While your Pokemon is defending a gym, if you check your badges, it'll show a profile of your Pokemon with it's remaining CP/motivation and how many battles it's won.
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It looks like when you're going off into battle the number they show is how many times you've used that Pokémon to defeat another. As for battles won defensively, I have no idea. I'm wondering as well