by move/ltnW->.
rewrite leq_eqVlt; case/orP; first by move/eqP->.
by move/modp_small; rewrite rpq => h; move: nq0; rewrite h eqxx.
Qed.

Lemma root_bigmul : forall x (ps : seq {poly R}),
~~root (\big[*%R/1]_(p <- ps) p) x = all (fun p => ~~ root p x) ps.