`{Funext} {A B : Ring} (f : A $-> B) : A / ideal_kernel f ≅ rng_image f. Proof. snrapply Build_RingIsomorphism''. 1: rapply abgroup_first_iso. split. { srapply QuotientRing_ind2_hprop; intros x y. srapply path_sigma_hprop. exact (rng_homo_mult _ _ _). } srapply path_sigma_hprop. exact (rng_homo_one _). Defined.
Require Import WildCat.Core WildCat.Equiv. Require Import Algebra.Congruence. Require Import Algebra.AbGroups. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Import Ideal.Notation.