The product of two projective varieties over $X,Y$ is the fibered product $X\times_{\mathbb{Z}} Y$. I want to show that the projections $X\times_{\mathbb{Z}} Y \to X$ and $X\times_{\mathbb{Z}} Y \to Y$ are smooth if $X,Y$ are smooth.This somehow eludes me. Can anyone help?
EDIT: First, I mean smooth over a field $k$. Second,for $S$-Objects $X\to S$ and $Y\to S$, the product is the object $X\times_{S} Y$. Hence for smooth $k$-schemes the product is $X\times_{k} Y$ and not $X\times_{\mathbb{Z}} Y$. So the result follows via base change as pointed out in the answer below.