-- 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))
have :
  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])
have₁ :
  ∀
    (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])),
    Irreducible.{u_2} (M := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I)
have₂ :
  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])
have₃ have₄ :
  ∀
    (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])),
    Irreducible.{u_2} (M := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I)
have₅ :
  ∀
    (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])),
    ∃ (y : Ideal S), Eq (α := Multiset (Ideal S)) (normalizedFactors (Ideal.map (F := R →+* S) (algebraMap R S) I)) {y}
have₆ :
  ∀
    (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])),
    ∃ (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}
h :
  ∀
    (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])),
    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)}
h₁ :
  ∀
    (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])),
    ∃ (a : (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} : Type u_1)),
      (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach = {a} ∧
        Eq.{u_2 + 1} (α := Ideal S)
          (↑(((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))
                a) :
            Ideal S)
          (↑(((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)
h₂ :
  ∀
    (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])),
    (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach =
        singleton (α :=
          (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} : Type u_1))
          (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 :
            { 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 }) ∧
      Eq.{u_2 + 1} (α := Ideal S)
        (↑(((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)
        (↑(((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)
h₃ :
  ∀
    (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])),
    (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach =
      singleton (α :=
        (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} : Type u_1))
        (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 :
          { 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 })
h₄ :
  ∀
    (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])),
    Eq (α := Multiset (R ⧸ I)[X])
      (Multiset.map
        (fun
            (a : { x_1 : (R ⧸ I)[X] // x_1 ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) }) ↦
          (↑a : (R ⧸ I)[X]))
        (normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))).attach)
      (Multiset.map
        (fun
            (a : { x_1 : (R ⧸ I)[X] // x_1 ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) }) ↦
          (↑a : (R ⧸ I)[X]))
        (singleton (α :=
          (↑{d : (R ⧸ I)[X] | d ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))} : Type u_1))
          (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 :
            { 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 })))
h₅ :
  ∀
    (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])),
    normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) =
      singleton (α := (R ⧸ I)[X])
        ((normalize : (a : (R ⧸ I)[X]) → (R ⧸ I)[X]) (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) : (R ⧸ I)[X])
⊢ Irreducible.{u_2} (M := Ideal S) (Ideal.map (F := R →+* S) (algebraMap R S) I)

Tactic:
classical
  have mem_norm_factors : normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) ∈
      normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) := by
    simp [normalizedFactors_irreducible hf]
  suffices ∃ y, normalizedFactors (I.map (algebraMap R S)) = {y} by
    obtain ⟨y, hy⟩ := this
    have h := prod_normalizedFactors (show I.map (algebraMap R S) ≠ 0 by
          rwa [← bot_eq_zero, Ne,
            map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective R S)])
    rw [associated_iff_eq, hy, Multiset.prod_singleton] at h
    rw [← h]
    exact
      irreducible_of_normalized_factor y
        (show y ∈ normalizedFactors (I.map (algebraMap R S)) by simp [hy])
  rw [normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map hI hI' hx hx']
  use ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm
        ⟨normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors⟩ :
      Ideal S)
  rw [Multiset.map_eq_singleton]
  use ⟨normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)), mem_norm_factors⟩
  refine ⟨?_, rfl⟩
  apply Multiset.map_injective Subtype.coe_injective
  rw [Multiset.attach_map_val, Multiset.map_singleton, Subtype.coe_mk]
  exact normalizedFactors_irreducible hf

Premises used: Subtype.coe_mk, Multiset.map_singleton, Multiset.attach_map_val, Subtype.coe_injective, Multiset.map_injective, KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map, Multiset.map_eq_singleton