conditional expectation simplification 
On the right hand of the expression above, why could we move the condition  ' |X '   to the numerator without making any other changes?

The notation $\mathbb E[\cdot|X]$ is just a notation for the conditional expectation that associates to a random variable $Z$ the random variable $\mathbb E[Z|X]$ as the closest function of $X$, in the sense  of the least square solution
$$f^* = \arg \min_f \mathbb E[(Z-f(X))^2]$$
i.e. $f^*(X)=\mathbb E[Z|X]$. The notation $|X$ thus does not have a meaning by itself and cannot be "moved" around the expectation argument. In other words, $Y|X$ does not exist.
