Proving that the order of a group is greater than or equal to the product of orders of 2 subgroups. Let $H$ and $K$ be subgroups of a finite group $G$ such that $H \cap K = \{e\}$
Prove that $|G|\ge |H||K|$
What I think is the correct step is to consider the cosets $hK, h \in H$, and then using properties of cosets to prove that the product of orders can't be greater than $|G|$, but this is where I get stuck.

Hint: Consider $\phi: H \times K \to G$ given by $\phi(h,k)=hk$. Prove that $\phi$ is injective.
