A simple (but not always accurate) rule of thumb for stringed instruments is that an active instrument needs power. Many bass pickups, for example, are actually powered by a battery pack hidden somewhere on the body. From what I can see, the Tremonti's pickups are just regular humbuckers so you don't have active pickups. You should be fine.

You don't actually have to use a DI box. Do you have a guitar multi-effects pedal with amp modeling? That's actually your DI box right there.

And just an an FYI, I realized that this post is almost 4 years old. I am using the exact same setup today and it serves me brilliantly. I supplement this with a Portastudio (http://tascam.com/product/dp-008/) for recording on a couch or impromptu jams and I am a happy camper.