Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case a
A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range fun x_1 ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * (minpolyDiv K x).coeff ↑x_1) ≤
    span A (Set.range ((fun a ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * a) ∘ (minpolyDiv K x).coeff))

pp: case a
A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range ((fun a ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * a) ∘ (minpolyDiv K x).coeff)) ≤
    span A (Set.range fun x_1 ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * (minpolyDiv K x).coeff ↑x_1)

Tactic:
<;>

Goal state after:
case a
A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range fun x_1 ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * (minpolyDiv K x).coeff ↑x_1) ≤
    span A (Set.range ((fun a ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * a) ∘ (minpolyDiv K x).coeff))

case a
A : Type u_1
K : Type u_2
L : Type u
inst✝¹¹ : CommRing A
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra A K
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : IsDomain A
inst✝³ : IsFractionRing A K
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsIntegrallyClosed A
x : L
hx : K[x] = ⊤
hAx : IsIntegral A x
hKx : IsIntegral K x
pb : PowerBasis K L
pbgen : pb.gen = x
hnondeg : (traceForm K L).Nondegenerate
hpb :
  ⇑((traceForm K L).dualBasis hnondeg pb.basis) = fun x ↦
    (minpolyDiv K pb.gen).coeff ↑x / (aeval pb.gen) (derivative (minpoly K pb.gen))
this : Subalgebra.toSubmodule A[x] = span A (Set.range ⇑pb.basis)
⊢ span A (Set.range ((fun a ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * a) ∘ (minpolyDiv K x).coeff)) ≤
    span A (Set.range fun x_1 ↦ ((aeval x) (derivative (minpoly K x)))⁻¹ * (minpolyDiv K x).coeff ↑x_1)