∀ (R : Type u_1) [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {N : Type u_3}
  [inst_3 : AddCommGroup N] [inst_4 : Module R N] {ι : Type u_4} [inst_5 : Fintype ι] {m : ι → M} {n : ι → N},
  Function.Injective ⇑(LinearMap.rTensor N (Submodule.span R (Set.range m)).subtype) →
    (TensorProduct.VanishesTrivially R m n ↔ ∑ i, m i ⊗ₜ[R] n i = 0)