mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
⊢ Polynomial.map (↑hx.aevalEquiv) Polynomial.X = Polynomial.X

Tactic:
Polynomial.map_X

Goal state after:
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
⊢ Polynomial.X = Polynomial.X
Module: Mathlib.RingTheory.AlgebraicIndependent.Basic

Goal state before:
pp: case refine_2.none
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
a : A
⊢ (Polynomial.aeval a) (hx.mvPolynomialOptionEquivPolynomialAdjoin (X none)) = (aeval fun o ↦ o.elim a x) (X none)

Tactic:
hx.mvPolynomialOptionEquivPolynomialAdjoin_X_none,

Goal state after:
case refine_2.none
ι : Type u
R : Type u_2
A : Type v
x : ι → A
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
hx : AlgebraicIndependent R x
a : A
⊢ (Polynomial.aeval a) Polynomial.X = (aeval fun o ↦ o.elim a x) (X none)
Module: Mathlib.GroupTheory.GroupAction.FixedPoints

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
g : G
a : α
h : a ∈ fixedBy α g
j : ℤ
⊢ ↑1 ∣ j

Tactic:
Int.natCast_one

Goal state after:
α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
g : G
a : α
h : a ∈ fixedBy α g
j : ℤ
⊢ 1 ∣ j
Module: Mathlib.GroupTheory.GroupAction.FixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
a : α
a_in_s : a ∈ s
b : α
b_in_orbit : b ∈ orbit (↥(fixingSubgroup M sᶜ)) a
g : M
g_fixing : g ∈ fixingSubgroup M sᶜ
g_eq : ⟨g, g_fixing⟩ • a = b
⊢ b ∈ s

Tactic:
rw [Submonoid.mk_smul] at g_eq

Goal state after:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
a : α
a_in_s : a ∈ s
b : α
b_in_orbit : b ∈ orbit (↥(fixingSubgroup M sᶜ)) a
g : M
g_fixing : g ∈ fixingSubgroup M sᶜ
g_eq : g • a = b
⊢ b ∈ s
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u
inst✝ : Field F
⊢ natSepDegree 0 = 0

Tactic:
by
  rw [← C_0, natSepDegree_C]

Goal state after:
Module: Mathlib.FieldTheory.SeparableDegree

Goal state before:
pp: F : Type u_1
inst✝ : Field F
h : ∀ (f : F[X]), f.natSepDegree = 1 → f.Splits
p : ℕ
h✝ : ExpChar F p
x y : F
hy : eval y (X ^ p - C x) = 0
⊢ (frobenius F p) y = x

Tactic:
by rwa [eval_sub, eval_X_pow, eval_C, sub_eq_zero] at hy

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
⊢ p ∈ RingHom.ker (Polynomial.aeval s).toRingHom ↔ p ∈ Ideal.span {minpoly R s}

Tactic:
RingHom.mem_ker,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
hs : IsIntegral R s
p : R[X]
⊢ (Polynomial.aeval s).toRingHom p = 0 ↔ p ∈ Ideal.span {minpoly R s}
Module: Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed

Goal state before:
pp: case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp0 : p ≠ 0
hp : (Polynomial.aeval s) p = 0
hs : IsIntegral R s
⊢ (minpoly R s).degree ≤ p.degree

Tactic:
rw [degree_eq_natDegree (minpoly.ne_zero hs), degree_eq_natDegree hp0]

Goal state after:
case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : IsDomain R
inst✝³ : Algebra R S
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDomain S
inst✝ : IsTorsionFree R S
s : S
p : R[X]
hp0 : p ≠ 0
hp : (Polynomial.aeval s) p = 0
hs : IsIntegral R s
⊢ ↑(minpoly R s).natDegree ≤ ↑p.natDegree
Module: Mathlib.RingTheory.DedekindDomain.IntegralClosure

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝¹² : CommRing A
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra A K
inst✝⁹ : IsFractionRing A K
L : Type u_3
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra A L
inst✝⁵ : IsScalarTower A K L
inst✝⁴ : FiniteDimensional K L
inst✝³ : Algebra.IsSeparable K L
ι : Type u_5
inst✝² : Finite ι
inst✝¹ : DecidableEq ι
b : Basis ι K L
hb_int : ∀ (i : ι), IsIntegral A (b i)
inst✝ : IsIntegrallyClosed A
⊢ Subalgebra.toSubmodule (integralClosure A L) ≤ Submodule.span A (Set.range ⇑((traceForm K L).dualBasis ⋯ b))

Tactic:
refine le_trans ?_ (IsIntegralClosure.range_le_span_dualBasis (integralClosure A L) b hb_int)
  intro x hx
  exact ⟨⟨x, hx⟩, rfl⟩

Goal state after:
Module: Mathlib.FieldTheory.PurelyInseparable.Basic

Goal state before:
pp: case h
k : Type u_1
K : Type u_2
R : Type u_3
inst✝⁶ : Field k
inst✝⁵ : Field K
inst✝⁴ : Algebra k K
inst✝³ : CommRing R
inst✝² : Algebra k R
inst✝¹ : IsPurelyInseparable k K
q : ℕ
inst✝ : ExpChar k q
a✝ : Nontrivial (R ⊗[k] K)
hq : Nat.Prime q
x : R
y : K
n : ℕ
a : k
ha : (algebraMap k K) a = y ^ q ^ n
⊢ x ⊗ₜ[k] y ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range

Tactic:
have : (x ^ q ^ n) ⊗ₜ[k] (y ^ q ^ n) =
        (x ^ q ^ n) ⊗ₜ[k] (1 : K) * (1 : R) ⊗ₜ[k] (y ^ q ^ n) := by
      rw [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]

Goal state after:
case h
k : Type u_1
K : Type u_2
R : Type u_3
inst✝⁶ : Field k
inst✝⁵ : Field K
inst✝⁴ : Algebra k K
inst✝³ : CommRing R
inst✝² : Algebra k R
inst✝¹ : IsPurelyInseparable k K
q : ℕ
inst✝ : ExpChar k q
a✝ : Nontrivial (R ⊗[k] K)
hq : Nat.Prime q
x : R
y : K
n : ℕ
a : k
ha : (algebraMap k K) a = y ^ q ^ n
this : (x ^ q ^ n) ⊗ₜ[k] (y ^ q ^ n) = (x ^ q ^ n) ⊗ₜ[k] 1 * 1 ⊗ₜ[k] (y ^ q ^ n)
⊢ x ⊗ₜ[k] y ^ q ^ n ∈ (algebraMap R (R ⊗[k] K)).range
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: B : Type u_2
G : Type u_3
inst✝³ : CommRing B
inst✝² : Group G
inst✝¹ : MulSemiringAction G B
inst✝ : Fintype G
b : B
n : ℕ
g : G
⊢ g • (charpoly G b).coeff n = (charpoly G b).coeff n

Tactic:
← coeff_smul,

Goal state after:
B : Type u_2
G : Type u_3
inst✝³ : CommRing B
inst✝² : Group G
inst✝¹ : MulSemiringAction G B
inst✝ : Fintype G
b : B
n : ℕ
g : G
⊢ (g • charpoly G b).coeff n = (charpoly G b).coeff n
Module: Mathlib.RingTheory.IsGaloisGroup.Basic

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
H : Subgroup G
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring C
inst✝⁶ : Algebra A C
inst✝⁵ : MulSemiringAction G C
N : Subgroup G
inst✝⁴ : CommSemiring B
inst✝³ : Algebra B C
inst✝² : FaithfulSMul B C
inst✝¹ : IsGaloisGroup (↥N) B C
inst✝ : N.Normal
this : MulSemiringAction G B := mulSemiringActionOfNormal G B C N
q : G ⧸ N
x : B
g₁ g₂ : G
h : (QuotientGroup.leftRel N) g₁ g₂
⊢ g₁ • x = g₂ • x

Tactic:
apply FaithfulSMul.algebraMap_injective B C

Goal state after:
G : Type u_1
inst✝⁹ : Group G
H : Subgroup G
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring C
inst✝⁶ : Algebra A C
inst✝⁵ : MulSemiringAction G C
N : Subgroup G
inst✝⁴ : CommSemiring B
inst✝³ : Algebra B C
inst✝² : FaithfulSMul B C
inst✝¹ : IsGaloisGroup (↥N) B C
inst✝ : N.Normal
this : MulSemiringAction G B := ⋯
q : G ⧸ N
x : B
g₁ g₂ : G
h : (QuotientGroup.leftRel N) g₁ g₂
⊢ (algebraMap B C) (g₁ • x) = (algebraMap B C) (g₂ • x)
Module: Mathlib.RingTheory.IsGaloisGroup.Basic

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
H : Subgroup G
A : Type u_2
B : Type u_3
C : Type u_4
inst✝⁸ : CommSemiring A
inst✝⁷ : Semiring C
inst✝⁶ : Algebra A C
inst✝⁵ : MulSemiringAction G C
N : Subgroup G
inst✝⁴ : CommSemiring B
inst✝³ : Algebra B C
inst✝² : FaithfulSMul B C
inst✝¹ : IsGaloisGroup (↥N) B C
inst✝ : N.Normal
this : MulSemiringAction G B := ⋯
q : G ⧸ N
x : B
g₁ g₂ : G
h : (QuotientGroup.leftRel N) g₁ g₂
⊢ (algebraMap B C) (g₁ • x) = (algebraMap B C) (g₂ • x)

Tactic:
rw [algebraMap.smul', algebraMap.smul', smul_eq_iff_eq_inv_smul, ← smul_assoc, smul_eq_mul,
        Subgroup.smul_algebraMap C (by rwa [← QuotientGroup.leftRel_apply])]

Goal state after:
Module: Mathlib.RingTheory.Trace.Basic

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
E : Type u_7
inst✝⁴ : Field E
inst✝³ : Algebra K E
inst✝² : IsAlgClosed E
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
x : L
⊢ (algebraMap K E) ((Algebra.trace K L) x) = ∑ σ, σ x

Tactic:
have hx := Algebra.IsSeparable.isIntegral K x
  let pb := adjoin.powerBasis hx
  rw [trace_eq_trace_adjoin K x, Algebra.smul_def, map_mul, ← adjoin.powerBasis_gen hx,
    trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits _), ← Algebra.smul_def,
    algebraMap_smul]
  · exact (sum_embeddings_eq_finrank_mul L E pb).symm
  · have := Algebra.isSeparable_tower_bot_of_isSeparable K K⟮x⟯ L
    exact Algebra.IsSeparable.isSeparable K _

Goal state after:
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: B : Type u_2
inst✝⁴ : CommRing B
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
Q : Ideal B
inst✝ : Q.IsPrime
val✝ : Fintype G
P : Ideal B := {g | g • Q ≠ Q}.inf fun g ↦ g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝¹ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr✝ : eval b r ∈ Q
a : B := f.coeff j
ha : ∀ (g : G), g • a = a
g : G
hg : g • Q ≠ Q
hr : r = ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
⊢ eval b (∑ k ∈ Finset.range k, (monomial (k + 1)) (f.coeff (k + 1 + j)) + (monomial 0) (f.coeff (0 + j))) - a ∈ g • Q

Tactic:
Polynomial.eval_add,

Goal state after:
B : Type u_2
inst✝⁴ : CommRing B
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
Q : Ideal B
inst✝ : Q.IsPrime
val✝ : Fintype G
P : Ideal B := {g | g • Q ≠ Q}.inf fun g ↦ g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝¹ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr✝ : eval b r ∈ Q
a : B := f.coeff j
ha : ∀ (g : G), g • a = a
g : G
hg : g • Q ≠ Q
hr : r = ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
⊢ eval b (∑ k ∈ Finset.range k, (monomial (k + 1)) (f.coeff (k + 1 + j))) + eval b ((monomial 0) (f.coeff (0 + j))) -
      a ∈
    g • Q
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: B : Type u_2
inst✝⁴ : CommRing B
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
Q : Ideal B
inst✝ : Q.IsPrime
val✝ : Fintype G
P : Ideal B := {g | g • Q ≠ Q}.inf fun g ↦ g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝¹ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr✝ : eval b r ∈ Q
a : B := f.coeff j
ha : ∀ (g : G), g • a = a
g : G
hg : g • Q ≠ Q
hr : r = ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
⊢ eval b (∑ k ∈ Finset.range k, (monomial (k + 1)) (f.coeff (k + 1 + j))) + f.coeff j - a ∈ g • Q

Tactic:
add_sub_cancel_right

Goal state after:
B : Type u_2
inst✝⁴ : CommRing B
G : Type u_3
inst✝³ : Group G
inst✝² : Finite G
inst✝¹ : MulSemiringAction G B
Q : Ideal B
inst✝ : Q.IsPrime
val✝ : Fintype G
P : Ideal B := {g | g • Q ≠ Q}.inf fun g ↦ g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝¹ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr✝ : eval b r ∈ Q
a : B := f.coeff j
ha : ∀ (g : G), g • a = a
g : G
hg : g • Q ≠ Q
hr : r = ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
⊢ eval b (∑ k ∈ Finset.range k, (monomial (k + 1)) (f.coeff (k + 1 + j))) ∈ g • Q
Module: Mathlib.RingTheory.Trace.Basic

Goal state before:
pp: case neg.succ.hnc
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : ¬Algebra.IsSeparable K L
p : ℕ
hp : ExpChar K p
this✝ : p ≠ 0
x : L
h₀ : FiniteDimensional K L
hx : ¬IsSeparable K x
g : K[X]
hg₁ : g.Separable
n : ℕ
hg₂ : (expand K (p ^ (n + 1))) g = minpoly K x
h : p ^ (n + 1) ∣ g.natDegree * p ^ (n + 1) - 1
this : p ^ (n + 1) ∣ g.natDegree * p ^ (n + 1) - (g.natDegree * p ^ (n + 1) - 1)
⊢ 1 ≤ g.natDegree * p ^ (n + 1)

Tactic:
rw [Nat.one_le_iff_ne_zero]
          have : g.natDegree ≠ 0 := fun e ↦ by
            have := congr(natDegree $hg₂)
            rw [natDegree_expand, e, zero_mul] at this
            exact (minpoly.natDegree_pos (IsIntegral.isIntegral x)).ne this
          positivity

Goal state after:
Module: Mathlib.FieldTheory.Galois.IsGaloisGroup

Goal state before:
pp: case inl
G : Type u_1
K : Type u_3
L : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : MulSemiringAction G L
inst✝ : IsGaloisGroup G K L
val✝ : Fintype G
⊢ Nat.card G = finrank K L

Tactic:
have : FaithfulSMul G L := faithful K
    rw [← IntermediateField.finrank_bot', ← fixedPoints_eq_bot G, Nat.card_eq_fintype_card]
    exact (FixedPoints.finrank_eq_card G L).symm

Goal state after:
Module: Mathlib.FieldTheory.Galois.IsGaloisGroup

Goal state before:
pp: G : Type u_1
K : Type u_3
L : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : MulSemiringAction G L
hGKL : IsGaloisGroup G K L
inst✝ : Finite G
H : (Subgroup G)ᵒᵈ
⊢ (intermediateFieldEquivSubgroup G K L).symm H = FixedPoints.intermediateField ↥(OrderDual.ofDual H)

Tactic:
obtain ⟨H, rfl⟩ := OrderDual.toDual.surjective H

Goal state after:
G : Type u_1
K : Type u_3
L : Type u_4
inst✝⁵ : Group G
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : MulSemiringAction G L
hGKL : IsGaloisGroup G K L
inst✝ : Finite G
H : Subgroup G
⊢ (intermediateFieldEquivSubgroup G K L).symm (OrderDual.toDual H) =
    FixedPoints.intermediateField ↥(OrderDual.ofDual (OrderDual.toDual H))
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: case intro
A : Type u_1
B : Type u_2
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
G : Type u_3
inst✝¹⁵ : Group G
inst✝¹⁴ : Finite G
inst✝¹³ : MulSemiringAction G B
inst✝¹² : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹¹ : Q.IsPrime
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : Algebra.IsInvariant A B G
inst✝ : Module.IsTorsionFree (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
b₀ : B
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((Ideal.Quotient.stabilizerHom Q P G) g) ((Ideal.Quotient.mk Q) b₀) = (Ideal.Quotient.mk Q) b₀
a b : B
ha1 : ∀ (g : G), g • a = a
ha2 : a ∉ Q
hb : ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a * b₀ else 0)
M : A[X]
key : map ((algebraMap B (B ⧸ Q)).comp (algebraMap A B)) M = map (algebraMap B (B ⧸ Q)) (MulSemiringAction.charpoly G b)
⊢ f ((algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)) = (algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)

Tactic:
← algebraMap_eq,

Goal state after:
case intro
A : Type u_1
B : Type u_2
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
G : Type u_3
inst✝¹⁵ : Group G
inst✝¹⁴ : Finite G
inst✝¹³ : MulSemiringAction G B
inst✝¹² : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹¹ : Q.IsPrime
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : Algebra.IsInvariant A B G
inst✝ : Module.IsTorsionFree (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
b₀ : B
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((Ideal.Quotient.stabilizerHom Q P G) g) ((Ideal.Quotient.mk Q) b₀) = (Ideal.Quotient.mk Q) b₀
a b : B
ha1 : ∀ (g : G), g • a = a
ha2 : a ∉ Q
hb : ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a * b₀ else 0)
M : A[X]
key : map (algebraMap A (B ⧸ Q)) M = map (algebraMap B (B ⧸ Q)) (MulSemiringAction.charpoly G b)
⊢ f ((algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)) = (algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: case intro
A : Type u_1
B : Type u_2
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
G : Type u_3
inst✝¹⁵ : Group G
inst✝¹⁴ : Finite G
inst✝¹³ : MulSemiringAction G B
inst✝¹² : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹¹ : Q.IsPrime
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : Algebra.IsInvariant A B G
inst✝ : Module.IsTorsionFree (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
b₀ : B
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((Ideal.Quotient.stabilizerHom Q P G) g) ((Ideal.Quotient.mk Q) b₀) = (Ideal.Quotient.mk Q) b₀
a b : B
ha1 : ∀ (g : G), g • a = a
ha2 : a ∉ Q
hb : ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a * b₀ else 0)
M : A[X]
key₀ :
  ∀ (g : G),
    map (algebraMap B (B ⧸ Q)) (X - C (g • b)) = if g • Q = Q then X - C ((algebraMap B (B ⧸ Q)) (a * b₀)) else X
key :
  map ((algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))) (map (algebraMap A (A ⧸ P)) M) =
    map (algebraMap (B ⧸ Q) L)
      ((X - C ((algebraMap B (B ⧸ Q)) (a * b₀))) ^ {x | x • Q = Q}.card * X ^ {x | ¬x • Q = Q}.card)
⊢ f ((algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)) = (algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)

Tactic:
← algebraMap_eq,

Goal state after:
case intro
A : Type u_1
B : Type u_2
inst✝¹⁸ : CommRing A
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Algebra A B
G : Type u_3
inst✝¹⁵ : Group G
inst✝¹⁴ : Finite G
inst✝¹³ : MulSemiringAction G B
inst✝¹² : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹¹ : Q.IsPrime
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : Algebra.IsInvariant A B G
inst✝ : Module.IsTorsionFree (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
b₀ : B
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((Ideal.Quotient.stabilizerHom Q P G) g) ((Ideal.Quotient.mk Q) b₀) = (Ideal.Quotient.mk Q) b₀
a b : B
ha1 : ∀ (g : G), g • a = a
ha2 : a ∉ Q
hb : ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a * b₀ else 0)
M : A[X]
key₀ :
  ∀ (g : G),
    map (algebraMap B (B ⧸ Q)) (X - C (g • b)) = if g • Q = Q then X - C ((algebraMap B (B ⧸ Q)) (a * b₀)) else X
key :
  map (algebraMap (A ⧸ P) L) (map (algebraMap A (A ⧸ P)) M) =
    map (algebraMap (B ⧸ Q) L)
      ((X - C ((algebraMap B (B ⧸ Q)) (a * b₀))) ^ {x | x • Q = Q}.card * X ^ {x | ¬x • Q = Q}.card)
⊢ f ((algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)) = (algebraMap (B ⧸ Q) L) ((algebraMap B (B ⧸ Q)) b₀)
Module: Mathlib.RingTheory.ZMod

Goal state before:
pp: n : ℕ
x✝ : ℤ
⊢ x✝ ∈ RingHom.ker (Int.castRingHom (ZMod n)) ↔ ↑n ∣ x✝

Tactic:
RingHom.mem_ker,

Goal state after:
n : ℕ
x✝ : ℤ
⊢ (Int.castRingHom (ZMod n)) x✝ = 0 ↔ ↑n ∣ x✝
Module: Mathlib.LinearAlgebra.Quotient.Pi

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
Ms : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (Ms i)
inst✝⁴ : (i : ι) → Module R (Ms i)
N : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : (i : ι) → Submodule R (Ms i)
q : Submodule R N
f : (i : ι) → Ms i →ₗ[R] N
hf : ∀ (i : ι), p i ≤ comap (f i) q
i : ι
x : Ms i ⧸ p i
⊢ ((p i).mapQ q (f i) ⋯) (Pi.single i x i) = ((p i).mapQ q (f i) ⋯) x

pp: case h₀
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
Ms : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (Ms i)
inst✝⁴ : (i : ι) → Module R (Ms i)
N : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : (i : ι) → Submodule R (Ms i)
q : Submodule R N
f : (i : ι) → Ms i →ₗ[R] N
hf : ∀ (i : ι), p i ≤ comap (f i) q
i : ι
x : Ms i ⧸ p i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ((p b).mapQ q (f b) ⋯) (Pi.single i x b) = 0

pp: case h₁
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
Ms : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (Ms i)
inst✝⁴ : (i : ι) → Module R (Ms i)
N : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : (i : ι) → Submodule R (Ms i)
q : Submodule R N
f : (i : ι) → Ms i →ₗ[R] N
hf : ∀ (i : ι), p i ≤ comap (f i) q
i : ι
x : Ms i ⧸ p i
⊢ i ∉ Finset.univ → ((p i).mapQ q (f i) ⋯) (Pi.single i x i) = 0

Tactic:
· rw [Pi.single_eq_same]

Goal state after:
case h₀
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
Ms : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (Ms i)
inst✝⁴ : (i : ι) → Module R (Ms i)
N : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : (i : ι) → Submodule R (Ms i)
q : Submodule R N
f : (i : ι) → Ms i →ₗ[R] N
hf : ∀ (i : ι), p i ≤ comap (f i) q
i : ι
x : Ms i ⧸ p i
⊢ ∀ b ∈ Finset.univ, b ≠ i → ((p b).mapQ q (f b) ⋯) (Pi.single i x b) = 0

case h₁
ι : Type u_1
R : Type u_2
inst✝⁶ : CommRing R
Ms : ι → Type u_3
inst✝⁵ : (i : ι) → AddCommGroup (Ms i)
inst✝⁴ : (i : ι) → Module R (Ms i)
N : Type u_4
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
p : (i : ι) → Submodule R (Ms i)
q : Submodule R N
f : (i : ι) → Ms i →ₗ[R] N
hf : ∀ (i : ι), p i ≤ comap (f i) q
i : ι
x : Ms i ⧸ p i
⊢ i ∉ Finset.univ → ((p i).mapQ q (f i) ⋯) (Pi.single i x i) = 0