Notation "\big [ op / idx ]_ ( i < n ) F" :=
  (\big[op/idx]_(i : ordinal n) F) : big_scope.Local Notation "x + y" := (plus x y).Notation "\max_ ( i 'in' A ) F" :=
 (\big[maxn/0%N]_(i in A) F%N) : nat_scope.

Lemma big_distrlr I J rI rJ (pI : pred I) (pJ : pred J) F G : (\big[+%M/0]_(i <- rI | pI i) F i) * (\big[+%M/0]_(j <- rJ | pJ j) G j) = \big[+%M/0]_(i <- rI | pI i) \big[+%M/0]_(j <- rJ | pJ j) (F i * G j).