product ampleness

let X a smooth relative projective scheme over a normal basis S. Let $\{(Y_i,L_i)\}_{i=1,2}$ be schemes with relatively projective morphisms $Y_i\rightarrow X$ induced by ample line bundles $L_i$ on $Y_i$. Assume that $L_i^{d_i}$ is very ample for $d_i\geq r_i >0$ and that $r_1\geq r_2$. Is it true that $(L_1\otimes L_2)^d$ is very ample on $Y_1\times_X Y_2$ for $d\geq r_1$? If it helps we can also assume that $Y_i=Proj_A(\cal{E})$ for $\cal{E}$ a finitely generated $\cal{O}_X$-algebra.

2022-07-25 20:42:42
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