have : comap ((Quotient.lift I f H).comp (Quotient.mk I)) ⊥ = ker f := rfl
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
ι : Type ?u.16994
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
⊢ ker (Quotient.lift I f H) = map (Quotient.mk I) (ker f)
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
ι : Type ?u.16994
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
this : comap ((Quotient.lift I f H).comp (Quotient.mk I)) ⊥ = ker f
⊢ ker (Quotient.lift I f H) = map (Quotient.mk I) (ker f)