I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have a Lie group $G$, why do we have the requirement that $G \times G \to G$, $(x,y) \mapsto xy^{-1}$ is a smooth continuous mapping between the product manifold and the manifold $G$.
What is it that makes $G$ a topological manifold in the first place? How does it inherit differentiable structure?
When you ask that $G \times G \to G$ is a smooth map, it means "with respect to the smooth structure you put on $G$ and the product differentiable structure on $G \times G$", so that asking this map to be smooth makes sense since it becomes a map between smooth manifolds and you can ask yourself this question. So somewhere you already assumed there was a smooth structure on $G$ and this property you ask of multiplication is extra.
Hope that helps,