-- Mathlib/RingTheory/DedekindDomain/Different.lean
lemma conductor_mul_differentIdeal [NoZeroSMulDivisors A B]
    (x : B) (hx : Algebra.adjoin K {algebraMap B L x} = ⊤) :
    (conductor A x) * differentIdeal A B = Ideal.span {aeval x (derivative (minpoly A x))}

Proof state:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
⊢ (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0

Tactic:
rwa [minpoly.isIntegrallyClosed_eq_field_fractions K L hAx, derivative_map,
  aeval_map_algebraMap, aeval_algebraMap_apply] at hne₁

Result:
no goals

Proof state:
case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
⊢ ↑(conductor A x) =
    FractionalIdeal.spanSingleton B⁰ ((algebraMap B L) ((aeval x) (derivative (minpoly A x)))) *
      FractionalIdeal.dual A K 1

case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
⊢ FractionalIdeal.dual A K 1 ≠ 0

Tactic:
swap

Result:
case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
⊢ FractionalIdeal.dual A K 1 ≠ 0

case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this : IsFractionRing B L
⊢ ↑(conductor A x) =
    FractionalIdeal.spanSingleton B⁰ ((algebraMap B L) ((aeval x) (derivative (minpoly A x)))) *
      FractionalIdeal.dual A K 1

Proof state:
case a.a.h
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this✝¹ : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
this✝ : (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0
hne₂ : ((aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))))⁻¹ ≠ 0
this : IsIntegral A ((algebraMap B L) x)
⊢ (∀ (z : B),
      ∀ y_1 ∈ Subalgebra.map (IsScalarTower.toAlgHom A B L) (adjoin A {x}),
        (Algebra.trace K L)
            (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * (y * ((algebraMap B L) z * y_1))) ∈
          1) ↔
    ∀ (a : B),
      (Algebra.trace K L) (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * (y * (algebraMap B L) a)) ∈
        (algebraMap A K).range

Tactic:
simp only [Subalgebra.mem_map, IsScalarTower.coe_toAlgHom', Submodule.one_eq_range,
  forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, ← _root_.map_mul]

Result:
case a.a.h
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this✝¹ : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
this✝ : (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0
hne₂ : ((aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))))⁻¹ ≠ 0
this : IsIntegral A ((algebraMap B L) x)
⊢ (∀ (z a : B),
      a ∈ adjoin A {x} →
        (Algebra.trace K L)
            (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * (y * (algebraMap B L) (z * a))) ∈
          LinearMap.range (Algebra.linearMap A K)) ↔
    ∀ (a : B),
      (Algebra.trace K L) (((algebraMap B L) ((aeval x) (derivative (minpoly A x))))⁻¹ * (y * (algebraMap B L) a)) ∈
        (algebraMap A K).range

Proof state:
case a.a.h.hI
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsDedekindDomain B
inst✝ : NoZeroSMulDivisors A B
x : B
hx : adjoin K {(algebraMap B L) x} = ⊤
hAx : IsIntegral A x
this✝ : IsFractionRing B L
y : L
hne₁ : (aeval ((algebraMap B L) x)) (derivative (minpoly K ((algebraMap B L) x))) ≠ 0
this : (algebraMap B L) ((aeval x) (derivative (minpoly A x))) ≠ 0
⊢ 1 ≠ 0

Tactic:
exact one_ne_zero

Result:
no goals