This is true for ```Add```, see http://www.ams.org/journals/proc/1970-025-03/S0002-9939-1970-0258803-3/home.html.
Unfortunately for ```Mul``` it isn't. A counterexample would be ```ex1 = 2**(1/3)``` and ```ex2 = exp(2/3 * I * pi)```,
then ```mp1 = x**3 - 2``` and ```mp2 = x**2 + x + 1```. Since ```ex1 * ex2``` is still a third root of 2 the minimal polynomial is again ```x**3 - 2```, but the resultant has degree 6. See the comments here: http://mathoverflow.net/questions/26832/degree-of-sum-of-algebraic-numbers/26837#26837.