∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) (S : Type u_2) [inst_1 : CommSemiring S]
  [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] (I : Ideal R) (x : R) (s : ↥M),
  (IsLocalization.mk' S x s ∈ Ideal.map (algebraMap R S) I) = ∃ s ∈ M, s * x ∈ I