no goals
R : Type u
inst✝ : CommRing R
I J : Ideal R
⊢ RingHom.ker (quotLeftToQuotSup I J) = map (Ideal.Quotient.mk I) J
simp only [mk_ker, sup_idem, sup_comm, quotLeftToQuotSup, Quotient.factor, ker_quotient_lift,
  map_eq_iff_sup_ker_eq_of_surjective (Ideal.Quotient.mk I) Quotient.mk_surjective, ← sup_assoc]