Is my math off?? how do auto trade in's work?

I have 2 vehicals I want to trade in valued at 17,500.(via the dealer) I will be carying over 5000 dollars of negetive equity to the loan amount. My sticker price on a new vehical Is 37,000. I was given 10,000 in rebates. So the new sticker price is 27,000 before my trade in value and negitive equity is even thrown in the mix. Subtract my trade in value from the new sticker price and the total is 10,000. Add in the negitive equity back up to 15,000, which should be final sales price correct? How are the dealerships telling me that my monthly payments will be $580 - $640 a month on 72 months. On 15,000? At 72 months it would cost a 3rd of that monthly. Is there some sort of trade in secret I am not aware off?? With 10k in rebates and 17,500 off of sticker your looking at 27,500 in my favor off sticker price. What's the catch?