Daughter wants to move in with her mom

You obviously care about your daughter. You are an intelligent person and the mother is not. The only reason your daughter wants to live with the mother is so she doesn't have to do what she thinks is a waste of time, (school, studying, working at a savings account, earning respect from mentors, and many more). She is lazy like every other 13 year old without direction. She is unaware of life's hardships. She sees her mother living the "I don't give a ****" lifestyle and wants the same because its easy. You are not going to change her mind.
This is all normal 13 year old brains at work. You have 2 choices, first, you can let her go and find out the hard way, or, you can file for full custody before she does. What could help your cause, and your daughters cause, if a phone call to child family protection services about the wife's current parenting procedures and have them do an inspection of her life, for the protection of ALL the children. Your daughter will most likely fight you on staying with you and it will be hard for both of you but I know for sure that she will thank you some day for saving her. Stop worrying about her hating you now, once the mother's life crashes, and it will, your daughter, hopefully will see who was right.