AITA for denying lip fillers for my daughter?

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The original was posted on /r/amitheasshole by /u/ThrowAway45x65s762x on 2023-07-27 16:34:37+00:00.

Hello all, since days im in a huge fight with my daughter and her mom (her mom and I split up some years ago). For context, my daughter is turning 16 and she wants to get lip fillers (1ml).

I told her no and that she can do things like that when she is 18. So of course she went straight to her mom and she said YES. Surprise, surprise. Now her mom already went ahead and booked an appointment (turns out its very easy to do that). I told her to cancel it and that I do not want my daughter to idolize figures like Karshadians and start modifying her body. Her mom is claiming that I cant tell our daughter what to do with her body and that its a gift from HER and I cant tell her what to gift and what not to gift. Of course my daughter is siding 100% with her mom and is now staying at her place since days...

Am I really in the wrong here??