As @Ramiro Lafuente pointed out in a comment, there's a gap in this proof. 

You're apparently using the first edition of my *Smooth Manifolds* book. The problem is fixed in the second edition, because the closed subgroup theorem is proved before this theorem. 

Meanwhile, some years ago I posted a correction in my errata list for the first edition, which you can find [here][1].


  [1]: https://sites.math.washington.edu/~lee/Books/ISM/