Show that the general linear group $GL(n, \Bbb R)$ is a Lie group under matrix multiplication.
I'm reading an introduction to manifolds by Tu and found this problem there. The definition I have for a group to be Lie is that the multiplication map $\mu : GL(n, \Bbb R) \times GL(n, \Bbb R) \to GL(n, \Bbb R), \mu (x,y)=xy$ is smooth and $\iota : GL(n, \Bbb R) \to GL(n, \Bbb R), \iota(x)=x^{-1}$ is smooth.
Now $\mu$ is smooth if for any $p \in GL(n, \Bbb R) \times GL(n, \Bbb R)$ and $F(p) \in GL(n, \Bbb R)$ there exists charts $(U, \varphi)$ and $(V, \psi)$ such that $\psi \circ \mu \circ \varphi^{-1}$ is smooth from $\Bbb R^{2n^2} \to \Bbb R^{n^2}$. The chart maps are smooth, but I would need $\mu$ to be smooth in order to use the fact that compositions are smooth, but smoothness of $\mu$ is what I'm trying to show here.
How should I work with this definition? Since it's how we define smooth maps I would assume that smoothness of $\psi \circ \mu \circ \varphi^{-1}$ can be shown without smoothness of $\mu$?
The multiplication $\mu\colon\Bbb R^{n\times n}\times\Bbb R^{n\times n}\longrightarrow\Bbb R^{n\times n}$ is a smooth map (since each component is a polynomial function) and $GL(n,\Bbb R)$ is an open subset of of $\Bbb R^{n\times n}$. Therefore, $\mu|_{GL(n,\Bbb R)\times GL(n,\Bbb R)}$ is smooth.