∀ {ι : Type u_4} {M : ι → Type u_5} [inst : (i : ι) → AddZeroClass (M i)] [inst_1 : DecidableEq ι] {I : Set ι}
  {S : (i : ι) → AddSubmonoid (M i)}, ⨆ i, AddSubmonoid.map (AddMonoidHom.single M i) (S i) ≤ AddSubmonoid.pi I S