In proposition 5.2 of Bump's Lie Groups, he states:
Let $G$ be a closed Lie subgroup of $GL(n, \mathbb{C})$, and let $X \in Mat_n (\mathbb{C})$. Then the path $t \to exp(tX)$ is tangent to the submanifold $G$ of $GL(n, \mathbb{C})$ at $t = 0$ if and only if it is contained in $G$ for all $t$.
The first condition is "the path $t \to exp(tX)$ is tangent to the submanifold $G$ of $GL(n, \mathbb{C})$ at $t = 0$" where the second just says that $exp(tX) \in G$ for all $t \in \mathbb{R}$.
What is the first condition really saying? It's been a while since I've seen any differential geometry... but this threw me off.
Just from my "knowledge" of Lie groups from conversations and so on I know that the first condition must reduce to saying something like $X$ is in the tangent space of $G$ at the identity... but I don't see how this is the case here.
I'd like an answer which would explain what this theorem means and how to use it to define a Lie group.