A line in Hall's proof that the Lie algebra is the tangent space to the Lie group at the identity

Question

I am having trouble understanding the highlighted line below from Hall's book: Lie groups, Lie algebras and representations.

I suspect that this line follows from this theorem:

Here is my attempt to verify this line:

By Theorem 3.42, we know that in a small enough neighbourhood $0 \in U \subseteq \mathbb{C}$, there exists a function $\delta: U \rightarrow \mathfrak{g}$ such that $\gamma(t)=e^{\delta (t)}$ for all $t \in U$. We know $\gamma$ is smooth. Does that mean $\delta$ must also be smooth?