I want to show that the product of smooth quasi-projective varieties is smooth.
Call my varieties $A$ and $B$. Since both are smooth, at every point the dimension of the tangent space equals the dimension of the variety. Will the dimension of $A \times B$ be the sum of the dimensions of $A$ and $B$? If so, how can I show that the same thing happens to the dimension of the tangent space at any point?