mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
hw : ¬IsUnramified k w
⊢ Even (Nat.card Gal(K/k))

Tactic:
by_cases H : Finite Gal(K/k)

Goal state after:
case pos
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
hw : ¬IsUnramified k w
H : Finite Gal(K/k)
⊢ Even (Nat.card Gal(K/k))

case neg
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
w : InfinitePlace K
inst✝ : IsGalois k K
hw : ¬IsUnramified k w
H : ¬Finite Gal(K/k)
⊢ Even (Nat.card Gal(K/k))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
c : index K
⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =
    ∑ x_1,
        fromBlocks 1 0 0
            ((blockDiagonal fun x ↦ 2⁻¹ • !![1, 1; -I, I]).submatrix ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm
              ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
            c (Sum.inl x_1) *
          x (↑x_1).embedding +
      ∑ x_1,
        (fromBlocks 1 0 0
              ((blockDiagonal fun x ↦ 2⁻¹ • !![1, 1; -I, I]).submatrix
                ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
              c (Sum.inr (x_1, 0)) *
            x (↑x_1).embedding +
          fromBlocks 1 0 0
              ((blockDiagonal fun x ↦ 2⁻¹ • !![1, 1; -I, I]).submatrix
                ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
              c (Sum.inr (x_1, 1)) *
            x (ComplexEmbedding.conjugate (↑x_1).embedding))

Tactic:
smul_of,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
c : index K
⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =
    ∑ x_1,
        fromBlocks 1 0 0
            ((blockDiagonal fun x ↦ of (2⁻¹ • ![![1, 1], ![-I, I]])).submatrix
              ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
            c (Sum.inl x_1) *
          x (↑x_1).embedding +
      ∑ x_1,
        (fromBlocks 1 0 0
              ((blockDiagonal fun x ↦ of (2⁻¹ • ![![1, 1], ![-I, I]])).submatrix
                ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
              c (Sum.inr (x_1, 0)) *
            x (↑x_1).embedding +
          fromBlocks 1 0 0
              ((blockDiagonal fun x ↦ of (2⁻¹ • ![![1, 1], ![-I, I]])).submatrix
                ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm ⇑(Equiv.prodComm (Fin 2) { w // w.IsComplex }).symm)
              c (Sum.inr (x_1, 1)) *
            x (ComplexEmbedding.conjugate (↑x_1).embedding))
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: case neg
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
v : InfinitePlace K
w : InfinitePlace L
left✝ : w.LiesOver v
hw : IsUnramified K w
h : ¬ComplexEmbedding.LiesOver w.embedding v.embedding
⊢ v.embeddingConjugateIte w ∈ unmixedEmbeddingsOver L v.embedding

Tactic:
simpa [embeddingConjugateIte_neg h] using
      ⟨⟨(LiesOver.embedding_comp_eq_or_conjugate_embedding_comp_eq w v).resolve_left
        (liesOver_iff.not.1 h)⟩, hw.isUnmixed_conjugate⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
v : InfinitePlace K
inst✝¹ : NumberField K
inst✝ : NumberField L
this : Algebra K ℂ := v.embedding.toAlgebra
σ : L →ₐ[K] ℂ
x✝ : σ ∈ ↑univ
⊢ σ.toRingHom ∈ ↑⋯.toFinset

Tactic:
by simpa using ⟨by aesop⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
v : InfinitePlace K
inst✝¹ : NumberField K
inst✝ : NumberField L
this : Algebra K ℂ := v.embedding.toAlgebra
ψ : L →+* ℂ
hψ : ComplexEmbedding.LiesOver ψ v.embedding
x✝ : K
⊢ (↑↑ψ).toFun ((algebraMap K L) x✝) = (algebraMap K ℂ) x✝

Tactic:
by simp [RingHom.algebraMap_toAlgebra, ← hψ.over]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ ↑1 ≤ √(1 + ↑B ^ 2)

Tactic:
rw [NNReal.coe_one, Real.le_sqrt' zero_lt_one, one_pow]

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case pos
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∈ s
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w

Tactic:
simp_rw [negAt_apply_isReal_and_mem _ hw, negAt, prodCongr_symm,
        prodCongr_apply, piCongrRight_symm_apply, if_pos hw, symm_neg,
        ContinuousLinearEquiv.neg_apply]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝⁶ : CommRing A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
inst✝ : IsGaloisGroup G A B
⊢ p.inertiaDegIn B = P.inertiaDeg A

Tactic:
by
  have h : ∃ P : Ideal B, P.IsPrime ∧ P.LiesOver p := ⟨P, hPp, hp⟩
  obtain ⟨_, _⟩ := h.choose_spec
  rw [inertiaDegIn, dif_pos h]
  exact inertiaDeg_eq_of_isGaloisGroup p h.choose P G

Goal state after:
Module: Mathlib.RingTheory.Ideal.Quotient.Index

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
s : Finset R
hs : span ↑s = I
inst✝ : Finite (R ⧸ I)
n : ℕ
this : ∀ (n : ℕ), Finite (R ⧸ I ^ n)
⊢ (Submodule.toAddSubgroup (I ^ n)).index ≤ (Submodule.toAddSubgroup I).index ^ ∑ i ∈ Finset.range n, s.card ^ i

Tactic:
induction n with
  | zero =>
    simp
  | succ n IH =>
    refine (Submodule.index_smul_le (I ^ n) s hs).trans ?_
    refine (Nat.mul_le_mul (Nat.pow_le_pow_left IH _) le_rfl).trans ?_
    rw [← pow_mul, ← pow_succ, geom_sum_succ, mul_comm]

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι R S
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by
      rw [Set.range_comp]
      exact ((quotient_span_eq_top_iff_span_eq_top _).mpr b.span_eq).ge

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ ((basisQuotient b).repr ((Ideal.Quotient.mk pS) x)) i = (Ideal.Quotient.mk p) ((b.repr x) i)

Tactic:
by
  refine congr_fun (g := Ideal.Quotient.mk p ∘ b.repr x) ?_ i
  apply (Finsupp.linearEquivFunOnFinite (R ⧸ p) _ _).symm.injective
  apply (basisQuotient b).repr.symm.injective
  simp only [Finsupp.linearEquivFunOnFinite_symm_coe, LinearEquiv.symm_apply_apply,
    Basis.repr_symm_apply]
  rw [Finsupp.linearCombination_eq_fintype_linearCombination_apply (R ⧸ p),
    Fintype.linearCombination_apply]
  simp only [Function.comp_apply, basisQuotient_apply,
    Ideal.Quotient.mk_smul_mk_quotient_map_quotient, ← Algebra.smul_def]
  rw [← map_sum, Basis.sum_repr b x]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case pos
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
hK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}
this✝ : CharZero ↥K
this : NumberField ↥K
w₀ : InfinitePlace ↥K
hw₀ : w₀.IsReal
⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N} ∪ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}

pp: case neg
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
hK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}
this✝ : CharZero ↥K
this : NumberField ↥K
w₀ : InfinitePlace ↥K
hw₀ : ¬w₀.IsReal
⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N} ∪ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}

Tactic:
· apply Set.mem_union_left
    exact ⟨⟨w₀, hw₀⟩, hK₁⟩

Goal state after:
case neg
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
hK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}
this✝ : CharZero ↥K
this : NumberField ↥K
w₀ : InfinitePlace ↥K
hw₀ : ¬w₀.IsReal
⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N} ∪ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
x : R
hP : ⊥.IsPrime
x_mem : x ∈ ⊥
hxP2 : x ∉ ⊥ ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ ⊥ → x ∉ Q
⊢ span {x} = ⊥

Tactic:
span_singleton_eq_bot,

Goal state after:
case pos
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
x : R
hP : ⊥.IsPrime
x_mem : x ∈ ⊥
hxP2 : x ∉ ⊥ ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ ⊥ → x ∉ Q
⊢ x = 0
Module: Mathlib.RingTheory.DedekindDomain.Instances

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : IsDomain R
inst✝⁴ : IsDomain S
inst✝³ : IsDomain T
inst✝² : Algebra R S
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : FaithfulSMul R S
⊢ IsTorsionFree R Sₚ

Tactic:
have := IsLocalization.AtPrime.faithfulSMul Rₚ R P
  exact IsTorsionFree.trans_faithfulSMul R Rₚ _

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
⊢ spanNorm R ⊤ = ⊤

Tactic:
by
  simp [← Ideal.span_singleton_one]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ↑p ∉ span {↑p ^ 2}

Tactic:
Ideal.mem_span_singleton

Goal state after:
case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ¬↑p ^ 2 ∣ ↑p
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_2
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Nat.Prime ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs

Tactic:
← Int.prime_iff_natAbs_prime

Goal state after:
case refine_2
p k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: case refine_2
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ Nat.Prime ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs

Tactic:
← Int.prime_iff_natAbs_prime

Goal state after:
case refine_2
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ (2 ^ (k + 1))
this : NumberField K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case pos.inr
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQzero : Q = 0
H₁ : z = 0
⊢ 0 ∈ R[B.gen]

Tactic:
exact Subalgebra.zero_mem _

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic

Goal state before:
pp: K : Type u
inst✝² : Field K
ζ : K
inst✝¹ : CharZero K
n : ℕ
inst✝ : IsCyclotomicExtension {n} ℚ K
hζ : IsPrimitiveRoot ζ n
h₁ : 2 < n
h₂ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n
⊢ have this := ⋯;
  (Algebra.norm ℤ) (hζ.toInteger - 1) = 1

Tactic:
by
  have : NumberField K := IsCyclotomicExtension.numberField {n} ℚ K
  have : NeZero n := NeZero.of_gt h₁
  dsimp only
  rw [norm_eq_iff ℤ (Sₘ := K) (Rₘ := ℚ) le_rfl, map_sub, map_one, map_one, RingOfIntegers.map_mk,
    sub_one_norm_eq_eval_cyclotomic hζ h₁ (cyclotomic.irreducible_rat (NeZero.pos _)),
    eval_one_cyclotomic_not_prime_pow h₂, Int.cast_one]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A ((traceMatrix K ⇑b).cramer (fun i ↦ (Algebra.trace K L) (x * a * b i)) i • b i)

Tactic:
Algebra.smul_def

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
⊢ IsIntegral A ((algebraMap K L) ((traceMatrix K ⇑b).cramer (fun i ↦ (Algebra.trace K L) (x * a * b i)) i) * b i)
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: case isTrue
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : ∀ a ∈ I, IsIntegral A (((traceForm K L) x) a)
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
i : ι
a✝ : i ∈ Finset.univ
j k : ι
h✝ : k = i
y : B
hy : (algebraMap B L) y = b j
⊢ y • a ∈ I

Tactic:
exact I.smul_mem _ (ha)

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: 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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
⊢ ↑(dual A K I) = (↑I)ᵛ

Tactic:
dual,

Goal state after:
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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
⊢ ↑(if hI : I = 0 then 0 else ⟨(↑I)ᵛ, ⋯⟩) = (↑I)ᵛ
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: 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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
⊢ dual A K 0 = 0

Tactic:
dual,

Goal state after:
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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
⊢ (if hI : 0 = 0 then 0 else ⟨(↑0)ᵛ, ⋯⟩) = 0
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: 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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
x : L
⊢ x ∈ dual A K I ↔ ∀ a ∈ I, ((traceForm K L) x) a ∈ (algebraMap A K).range

Tactic:
dual,

Goal state after:
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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
x : L
⊢ (x ∈ if hI : I = 0 then 0 else ⟨(↑I)ᵛ, ⋯⟩) ↔ ∀ a ∈ I, ((traceForm K L) x) a ∈ (algebraMap A K).range
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: 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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
b : B
hb : b ∈ B⁰
hb' : ∀ b_1 ∈ ↑I, IsInteger B (b • b_1)
⊢ (algebraMap B L) b ∈ dual A K I

Tactic:
mem_dual hI

Goal state after:
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 : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I : FractionalIdeal B⁰ L
hI : I ≠ 0
b : B
hb : b ∈ B⁰
hb' : ∀ b_1 ∈ ↑I, IsInteger B (b • b_1)
⊢ ∀ a ∈ I, ((traceForm K L) ((algebraMap B L) b)) a ∈ (algebraMap A K).range