-- Module: Mathlib.NumberTheory.KummerDedekind
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
x : S
I : Ideal R
inst_3 : IsDomain R
inst_4 : IsIntegrallyClosed R
inst_5 : IsDedekindDomain S
inst_6 : Module.IsTorsionFree R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (F := R →+* S) (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R x
hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))
mem_norm_factors :
  Membership.mem (α := (R ⧸ I)[X]) (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
    ((normalize : (a : (R ⧸ I)[X]) → (R ⧸ I)[X]) (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) : (R ⧸ I)[X])
h :
  Eq (α := Multiset (Ideal S))
    (Multiset.map
      (fun
          (f :
            (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
              Type u_1)) ↦
        (↑(((Equiv.symm (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') :
                  Equiv.{u_1 + 1, u_2 + 1}
                    (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                      Type u_1)
                    (↑{J : Ideal S |
                          Membership.mem (γ := Multiset (Ideal S))
                            (normalizedFactors (Ideal.map (F := R →+* S) (algebraMap R S) I)) J} :
                      Type u_2)) :
                (a :
                    (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                      Type u_1)) →
                  (↑{J : Ideal S | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} : Type u_2))
              f) :
          Ideal S))
      (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
    {(↑(((Equiv.symm (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') :
                Equiv.{u_1 + 1, u_2 + 1}
                  (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                    Type u_1)
                  (↑{J : Ideal S |
                        Membership.mem (γ := Multiset (Ideal S))
                          (normalizedFactors (Ideal.map (F := R →+* S) (algebraMap R S) I)) J} :
                    Type u_2)) :
              (a :
                  (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                    Type u_1)) →
                (↑{J : Ideal S | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} : Type u_2))
            (Subtype.mk (p := fun (x_1 : (R ⧸ I)[X]) ↦
              Membership.mem (γ := Set (R ⧸ I)[X])
                {d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} x_1)
              ((normalize : (a : (R ⧸ I)[X]) → (R ⧸ I)[X]) (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)))
              mem_norm_factors)) :
        Ideal S)}
⊢ ∃ (y : Ideal S),
    Eq (α := Multiset (Ideal S))
      (Multiset.map
        (fun
            (f :
              (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                Type u_1)) ↦
          (↑(((Equiv.symm (normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') :
                    Equiv.{u_1 + 1, u_2 + 1}
                      (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                        Type u_1)
                      (↑{J : Ideal S |
                            Membership.mem (γ := Multiset (Ideal S))
                              (normalizedFactors (Ideal.map (F := R →+* S) (algebraMap R S) I)) J} :
                        Type u_2)) :
                  (a :
                      (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} :
                        Type u_1)) →
                    (↑{J : Ideal S | J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)} : Type u_2))
                f) :
            Ideal S))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
      {y}

Tactic:
use ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm
        ⟨normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors⟩ :
      Ideal S)

Premises used: Ideal.Quotient.mk, minpoly, Subtype.mk, KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk, Polynomial.map, normalize, Equiv.symm, Subtype.val, DFunLike.coe