Advice Please

So ladies I need your advice good or bad please. Here's the issue, my boyfriend and I are in an open relationship and have been together for nearly 5 years. My issue is that he has been lying about small things and I think he is cheating but I can't get 100% solid proof, I can't just pick up and leave because we have a small daughter but I need to know if he is or not, how can I find solid proof that he either is or isn't cheating? Examples of the lying: he went fishing with a male friend and told me it was just them but there was a 3rd fishing pole and female sunglasses in his car, I was cleaning and noticed a condom from our box was missing couldn't find it anywhere and when I asked him he apparently didn't take it, but I've just noticed another condom gone and when I asked he said it was in his wallet in case he needed it but he lied about that because I was in his wallet the day before getting my bank card back plus a whole lot of other little lies. The thing is I also don't see why he would cheat because we're in an open relationship he can have sex with whoever he wants so why the lying? Please help how do I find out the truth?

Edit: we have rules in the open relationship about telling each other who we see and when and that is why I think he is cheating because he is lying