Let $Y|X=x\sim N_p(Ax+B,\Sigma)$ and $X\sim N_p(\mu,\Psi)$ verify that $$Y\sim N_p(A\mu+B,A\Psi A^T+\Sigma)$$ where $A$ and $B$ are matrix of constant terms.
I know that $$E[Y]=E[E[Y|X]]=E[AX+B]=A\mu+B$$ $$Var(Y)=E[Var(Y|X)]+Var(E[Y|X])=\Sigma+Var(AX+B)=\Sigma+A\Psi A^T$$ the law of variance total and this conditional expectation property is also valid in the multivariate case?
If the conditional distribution is normal and $X$ too, I can assume that $Y$ is normal?
I know that I could try to work with the joint distribution, but it is a lot of work. Is there another way if the above is not true?