∀ {ι : Type u_1} {A : Type u_2} {σ : Type u_3} [inst : DecidableEq ι] [inst_1 : AddMonoid ι] [inst_2 : Semiring A]
  [inst_3 : SetLike σ A] [inst_4 : AddSubmonoidClass σ A] (𝒜 : ι → σ) [inst_5 : GradedRing 𝒜]
  (a b : DirectSum ι fun i => ↥(𝒜 i)),
  (DirectSum.coeRingHom 𝒜) (a * b) = (DirectSum.coeRingHom 𝒜) a * (DirectSum.coeRingHom 𝒜) b