∀ {R : Type u_1} [inst : CommSemiring R] {A : Type u_2} {B : Type u_3} [inst_1 : Semiring A] [inst_2 : Semiring B]
  [inst_3 : Algebra R A] [inst_4 : Algebra R B] (f : A ≃ₐ[R] B) (x y : Submodule R A),
  (↑↑(Submodule.mapAlgHom ↑f).toRingHom).toFun (x + y) =
    (↑↑(Submodule.mapAlgHom ↑f).toRingHom).toFun x + (↑↑(Submodule.mapAlgHom ↑f).toRingHom).toFun y