Generalized method of moments: necessary condition for identification

Question

Consider a situation where we have a $K\times1$ vector of parameters $\theta$ and a set of restrictions$$E[\psi(X_{i};\theta)]=0$$where function $\psi$ is $M\times1$, with $M>K$, and $X_{i}$ denotes our observations. Given a positive definite symmetric matrix $C$, the true parameter $\theta^{*}$ minimizes the quadratic form$$Q(\theta)=E[\psi(X_{i};\theta)]'CE[\psi(X_{i};\theta)].$$Identification requires that $\theta^{*}$ uniquely minimizes $Q(\theta)$, that is$$\theta\neq\theta^{*}\Rightarrow Q(\theta)>Q(\theta^{*}).$$It is often stated that a necessary condition for identification is$$\text{rank}C^{\frac{1}{2}}E[\frac{\text{d}}{\text{d}\theta'}\psi(X_{i};\theta^{*})]=K.$$How can we prove that this rank condition follows form identification?