awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : IsDomain A
inst_14 : Algebra.IsSeparable K L
⊢ IsLocalization (algebraMapSubmonoid.{u_1, u_4} (R := A) C A⁰) L

Tactic:
IsIntegralClosure.isLocalization A K L C

Premises used: IsIntegralClosure.isLocalization
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
rw :
  Set.range.{u_3, u_5 + 1} (ι := ι) (⇑b : (a : ι) → L) ⊆
    (↑(LinearMap.BilinForm.dualSubmodule.{u_1, u_2, u_3} (R := A) (S := K) (M := L)
            (LinearMap.BilinForm.flip (traceForm K L))
            (LinearMap.range
              ((↑A :
                  {S M : Type u_4} →
                    {M₂ : Type u_3} →
                      [inst : Semiring A] →
                        [inst_18 : Semiring S] →
                          [inst_19 : AddCommMonoid M] →
                            [inst_20 : AddCommMonoid M₂] →
                              [inst_21 : Module A M] →
                                [inst_22 : Module A M₂] →
                                  [inst_23 : Module S M] →
                                    [inst_24 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
                (Algebra.linearMap C L))) :
          Submodule A L) :
      Set L)
⊢ LE.le (α := Submodule A L)
    (LinearMap.range
      ((↑A :
          {S M : Type u_4} →
            {M₂ : Type u_3} →
              [inst : Semiring A] →
                [inst_18 : Semiring S] →
                  [inst_19 : AddCommMonoid M] →
                    [inst_20 : AddCommMonoid M₂] →
                      [inst_21 : Module A M] →
                        [inst_22 : Module A M₂] →
                          [inst_23 : Module S M] →
                            [inst_24 : Module S M₂] →
                              [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
        (Algebra.linearMap C L)))
    (Submodule.span A
      (Set.range.{u_3, u_5 + 1} (ι := ι)
        (⇑(LinearMap.BilinForm.dualBasis.{u_3, u_2, u_5} (V := L) (K := K) (ι := ι) (traceForm K L) ⋯ b : Basis ι K L) :
          (a : ι) → L)))

Tactic:
rw [← LinearMap.BilinForm.dualSubmodule_span_of_basis,
    ← LinearMap.BilinForm.le_flip_dualSubmodule, Submodule.span_le]

Premises used: LinearMap.BilinForm.dualSubmodule_span_of_basis, Submodule.span_le, LinearMap.BilinForm.le_flip_dualSubmodule
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
i : ι
y : C
simp :
  Membership.mem.{u_2, u_2} (α := K) (γ := Submodule A K) 1
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)
⊢ Membership.mem.{u_2, u_2} (α := K) (γ := Submodule A K) 1
    ((((LinearMap.BilinForm.flip (traceForm K L) : (a : L) → L →ₗ[K] K) ((b : (a : ι) → L) i) : L →ₗ[K] K) :
          (a : L) → K)
        (((↑A :
                  {S M : Type u_4} →
                    {M₂ : Type u_3} →
                      [inst : Semiring A] →
                        [inst_18 : Semiring S] →
                          [inst_19 : AddCommMonoid M] →
                            [inst_20 : AddCommMonoid M₂] →
                              [inst_21 : Module A M] →
                                [inst_22 : Module A M₂] →
                                  [inst_23 : Module S M] →
                                    [inst_24 : Module S M₂] →
                                      [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
                (Algebra.linearMap C L) :
              (a : C) → L)
            y :
          L) :
      K)

Tactic:
simp only [LinearMap.coe_restrictScalars, linearMap_apply, LinearMap.BilinForm.flip_apply,
    traceForm_apply]

Premises used: Algebra.traceForm_apply, LinearMap.coe_restrictScalars, Algebra.linearMap_apply, LinearMap.BilinForm.flip_apply
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
i : ι
y : C
⊢ Membership.mem.{u_2, u_2} (α := K) (γ := Submodule A K) 1
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)

Tactic:
Submodule.mem_one.mpr <| IsIntegrallyClosed.isIntegral_iff.mp ?_

Premises used: Iff.mpr, IsIntegrallyClosed.isIntegral_iff, Set.Mem, Submodule.mem_one, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
i : ι
y : C
refine :
  IsIntegral.{u_1, u_2} A (A := K)
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)
⊢ Membership.mem.{u_2, u_2} (α := K) (γ := Submodule A K) 1
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)

Tactic:
refine Submodule.mem_one.mpr <| IsIntegrallyClosed.isIntegral_iff.mp ?_

Premises used: Iff.mpr, IsIntegrallyClosed.isIntegral_iff, Submodule.mem_one, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
i : ι
y : C
⊢ IsIntegral.{u_1, u_2} A (A := K)
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)

Tactic:
isIntegral_trace ((IsIntegralClosure.isIntegral A L y).algebraMap.mul (hb_int i))

Premises used: IsIntegral.mul, Algebra.isIntegral_trace, IsIntegralClosure.isIntegral, IsIntegral.algebraMap
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
C : Type u_4
inst_5 : CommRing C
inst_6 : Algebra K L
inst_7 : Algebra A L
inst_8 : IsScalarTower A K L
inst_9 : Algebra C L
inst_10 : IsIntegralClosure C A L
inst_11 : Algebra A C
inst_12 : IsScalarTower A C L
inst_13 : FiniteDimensional K L
inst_14 : Algebra.IsSeparable K L
ι : Type u_5
inst_15 : Finite ι
inst_16 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_17 : IsIntegrallyClosed A
i : ι
y : C
⊢ IsIntegral.{u_1, u_2} A (A := K)
    ((trace K L : (a : L) → K)
        (HMul.hMul (α := L) (β := L) ((algebraMap C L : (a : C) → L) y : L) ((b : (a : ι) → L) i : L)) :
      K)

Tactic:
exact isIntegral_trace ((IsIntegralClosure.isIntegral A L y).algebraMap.mul (hb_int i))

Premises used: IsIntegral.mul, Algebra.isIntegral_trace, IsIntegralClosure.isIntegral, IsIntegral.algebraMap
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : Algebra.IsSeparable K L
ι : Type u_5
inst_10 : Finite ι
inst_11 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_12 : IsIntegrallyClosed A
⊢ LE.le (α := Submodule A L)
    ((Subalgebra.toSubmodule : (a : Subalgebra A L) → Submodule A L) (integralClosure A L) : Submodule A L)
    (Submodule.span A
        (Set.range.{u_3, u_5 + 1} (ι := ι)
          (⇑(LinearMap.BilinForm.dualBasis.{u_3, u_2, u_5} (V := L) (K := K) (ι := ι) (traceForm K L) ⋯ b :
                Basis ι K L) :
            (a : ι) → L)) :
      Submodule A L)

Tactic:
le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)

Premises used: Membership.mem, Subtype, IsIntegralClosure.range_le_span_dualBasis, integralClosure, le_trans
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : Algebra.IsSeparable K L
ι : Type u_5
inst_10 : Finite ι
inst_11 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_12 : IsIntegrallyClosed A
refine :
  LE.le (α := Submodule A L)
    ((Subalgebra.toSubmodule : (a : Subalgebra A L) → Submodule A L) (integralClosure A L) : Submodule A L)
    (LinearMap.range (M := ↥(integralClosure A L))
        ((↑A :
            {S M M₂ : Type u_3} →
              [inst : Semiring A] →
                [inst_13 : Semiring S] →
                  [inst_14 : AddCommMonoid M] →
                    [inst_15 : AddCommMonoid M₂] →
                      [inst_16 : Module A M] →
                        [inst_17 : Module A M₂] →
                          [inst_18 : Module S M] →
                            [inst_19 : Module S M₂] →
                              [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
          (Algebra.linearMap (↥(integralClosure A L)) L)) :
      Submodule A L)
⊢ LE.le (α := Submodule A L)
    ((Subalgebra.toSubmodule : (a : Subalgebra A L) → Submodule A L) (integralClosure A L) : Submodule A L)
    (Submodule.span A
        (Set.range.{u_3, u_5 + 1} (ι := ι)
          (⇑(LinearMap.BilinForm.dualBasis.{u_3, u_2, u_5} (V := L) (K := K) (ι := ι) (traceForm K L) ⋯ b :
                Basis ι K L) :
            (a : ι) → L)) :
      Submodule A L)

Tactic:
refine le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)

Premises used: Membership.mem, Subtype, IsIntegralClosure.range_le_span_dualBasis, integralClosure, le_trans
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : Algebra.IsSeparable K L
ι : Type u_5
inst_10 : Finite ι
inst_11 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_12 : IsIntegrallyClosed A
x : L
hx :
  Membership.mem (γ := Submodule A L)
    ((Subalgebra.toSubmodule : (a : Subalgebra A L) → Submodule A L) (integralClosure A L) : Submodule A L) x
⊢ Membership.mem (γ := Submodule A L)
    (LinearMap.range (M := ↥(integralClosure A L))
      ((↑A :
          {S M M₂ : Type u_3} →
            [inst : Semiring A] →
              [inst_13 : Semiring S] →
                [inst_14 : AddCommMonoid M] →
                  [inst_15 : AddCommMonoid M₂] →
                    [inst_16 : Module A M] →
                      [inst_17 : Module A M₂] →
                        [inst_18 : Module S M] →
                          [inst_19 : Module S M₂] →
                            [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
        (Algebra.linearMap (↥(integralClosure A L)) L)))
    x

Tactic:
⟨⟨x, hx⟩, rfl⟩

Premises used: Set.Mem, setOf, Subtype.mk, Exists.intro, Set.range, rfl
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : Algebra.IsSeparable K L
ι : Type u_5
inst_10 : Finite ι
inst_11 : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral.{u_1, u_3} A (A := L) ((b : (a : ι) → L) i : L)
inst_12 : IsIntegrallyClosed A
x : L
hx :
  Membership.mem (γ := Submodule A L)
    ((Subalgebra.toSubmodule : (a : Subalgebra A L) → Submodule A L) (integralClosure A L) : Submodule A L) x
⊢ Membership.mem (γ := Submodule A L)
    (LinearMap.range (M := ↥(integralClosure A L))
      ((↑A :
          {S M M₂ : Type u_3} →
            [inst : Semiring A] →
              [inst_13 : Semiring S] →
                [inst_14 : AddCommMonoid M] →
                  [inst_15 : AddCommMonoid M₂] →
                    [inst_16 : Module A M] →
                      [inst_17 : Module A M₂] →
                        [inst_18 : Module S M] →
                          [inst_19 : Module S M₂] →
                            [LinearMap.CompatibleSMul M M₂ A S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[A] M₂)
        (Algebra.linearMap (↥(integralClosure A L)) L)))
    x

Tactic:
exact ⟨⟨x, hx⟩, rfl⟩
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
s : Finset L
⊢ ∃ (y : A), y ≠ 0 ∧ ∀ x ∈ s, IsIntegral A (y • x)

Tactic:
have := IsLocalization.isAlgebraic K (nonZeroDivisors A)
  have := Algebra.IsAlgebraic.trans A K L
  Algebra.IsAlgebraic.exists_integral_multiples ..

Premises used: Algebra.IsAlgebraic.exists_integral_multiples, nonZeroDivisors, Algebra.IsAlgebraic.trans, IsLocalization.isAlgebraic
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
obtain :
  ∀ (y : A) (hy : y ≠ 0)
    (his' :
      ∀ (x : L),
        Membership.mem (γ := Finset L)
            (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
              (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
            x →
          IsIntegral A (y • x)),
    ∃ (s : Finset L) (b : Basis (↥s) K L), ∀ (x : ↥s), IsIntegral A (A := L) ((b : (a : ↥s) → L) x : L)
⊢ ∃ (s : Finset L) (b : Basis (↥s) K L), ∀ (x : ↥s), IsIntegral A (A := L) ((b : (a : ↥s) → L) x : L)

Tactic:
obtain ⟨y, hy, his'⟩ := exists_integral_multiples A K (Finset.univ.image bs')

Premises used: Finset.univ, exists_integral_multiples, DFunLike.coe, Finset.image
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
⊢ Ne.{u_3 + 1} (α := L) ((algebraMap A L : (a : A) → L) y : L) (0 : L)

Tactic:
mt ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_ _) hy

Premises used: Algebra.algebraMap, mt, injective_iff_map_eq_zero, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
refine : Function.Injective.{u_1 + 1, u_3 + 1} (α := A) (β := L) (⇑(algebraMap A L) : (a : A) → L)
⊢ Ne.{u_3 + 1} (α := L) ((algebraMap A L : (a : A) → L) y : L) (0 : L)

Tactic:
refine mt ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_ _) hy

Premises used: Algebra.algebraMap, mt, injective_iff_map_eq_zero, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
rw :
  Function.Injective.{u_1 + 1, u_3 + 1} (α := A) (β := L)
    (⇑(@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap K L) (algebraMap A K)) : (a : A) → L)
⊢ Function.Injective.{u_1 + 1, u_3 + 1} (α := A) (β := L) (⇑(algebraMap A L) : (a : A) → L)

Tactic:
rw [IsScalarTower.algebraMap_eq A K L]

Premises used: IsScalarTower.algebraMap_eq
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
⊢ Function.Injective.{u_1 + 1, u_3 + 1} (α := A) (β := L)
    (⇑(@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap K L) (algebraMap A K)) : (a : A) → L)

Tactic:
(algebraMap K L).injective.comp (IsFractionRing.injective A K)

Premises used: Algebra.algebraMap, IsFractionRing.injective, RingHom.injective, Function.Injective.comp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
⊢ Function.Injective.{u_1 + 1, u_3 + 1} (α := A) (β := L)
    (⇑(@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap K L) (algebraMap A K)) : (a : A) → L)

Tactic:
exact (algebraMap K L).injective.comp (IsFractionRing.injective A K)

Premises used: Algebra.algebraMap, IsFractionRing.injective, RingHom.injective, Function.Injective.comp
-- Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure
Goal:
A : Type u_1
K : Type u_2
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
L : Type u_3
inst_4 : Field L
inst_5 : Algebra K L
inst_6 : Algebra A L
inst_7 : IsScalarTower A K L
inst_8 : FiniteDimensional K L
inst_9 : IsDomain A
this : DecidableEq L := Classical.decEq L
s' : Finset L := IsNoetherian.finsetBasisIndex K L
bs' : Basis (↥(IsNoetherian.finsetBasisIndex K L)) K L := IsNoetherian.finsetBasis K L
y : A
hy : y ≠ 0
his' :
  ∀ (x : L),
    Membership.mem (γ := Finset L)
        (Finset.image (α := ↥(IsNoetherian.finsetBasisIndex K L))
          (⇑bs' : (a : ↥(IsNoetherian.finsetBasisIndex K L)) → L) Finset.univ)
        x →
      IsIntegral A (y • x)
⊢ Ne.{u_3 + 1} (α := L) ((algebraMap A L : (a : A) → L) y : L) (0 : L)

Tactic:
have hy' : algebraMap A L y ≠ 0 := by
    refine mt ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_ _) hy
    rw [IsScalarTower.algebraMap_eq A K L]
    exact (algebraMap K L).injective.comp (IsFractionRing.injective A K)