Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case neg.hi.convert_2
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
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
f : ℕ → L
hf :
  ∀ (i : ℕ),
    (Polynomial.map (algebraMap R L) (minpoly R B.gen)).natDegree ≤ i →
      f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
hf₁ :
  ∀ k ∈ (range (Q.natDegree - j)).erase 0,
    Q.coeff (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (P.natDegree - (j + 2)) =
      (algebraMap R L) p * Q.coeff (j + 1 + k) • f (k + P.natDegree - 1)
hintsum :
  IsIntegral R
    (z * B.gen ^ (P.natDegree - (j + 2)) -
      (∑ x ∈ (range (Q.natDegree - j)).erase 0, Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
        ∑ x ∈ range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2))))
r : R
hQ :
  (algebraMap R K)
      (Q.coeff (j + 1) ^ B.dim * ((-1) ^ B.dim * (minpoly R B.gen).coeff 0) ^ (j + 1 + (P.natDegree - (j + 2)))) =
    (algebraMap R K) (p ^ B.dim * r)
hr :
  (algebraMap R K) r =
    (Algebra.norm K)
      (z * B.gen ^ (P.natDegree - (j + 2)) -
        (∑ x ∈ (range (Q.natDegree - j)).erase 0, Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
          ∑ x ∈ range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2))))
hppdiv :
  p ^ B.dim ∣
    (↑(-1) ^ B.dim) ^ (j + 1 + (P.natDegree - (j + 2))) *
        (minpoly R B.gen).coeff 0 ^ (j + 1 + (P.natDegree - (j + 2))) *
      Q.coeff (j + 1) ^ B.dim
⊢ p ^ n.succ ∣ Q.coeff j.succ ^ n.succ * (minpoly R B.gen).coeff 0 ^ (j.succ + (P.natDegree - (j + 2)))

Tactic:
←
      Units.val_pow_eq_pow_val,

Goal state after:
case neg.hi.convert_2
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
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ k ∈ range (j + 1), Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
f : ℕ → L
hf :
  ∀ (i : ℕ),
    (Polynomial.map (algebraMap R L) (minpoly R B.gen)).natDegree ≤ i →
      f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
hf₁ :
  ∀ k ∈ (range (Q.natDegree - j)).erase 0,
    Q.coeff (j + 1 + k) • B.gen ^ (j + 1 + k) * B.gen ^ (P.natDegree - (j + 2)) =
      (algebraMap R L) p * Q.coeff (j + 1 + k) • f (k + P.natDegree - 1)
hintsum :
  IsIntegral R
    (z * B.gen ^ (P.natDegree - (j + 2)) -
      (∑ x ∈ (range (Q.natDegree - j)).erase 0, Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
        ∑ x ∈ range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2))))
r : R
hQ :
  (algebraMap R K)
      (Q.coeff (j + 1) ^ B.dim * ((-1) ^ B.dim * (minpoly R B.gen).coeff 0) ^ (j + 1 + (P.natDegree - (j + 2)))) =
    (algebraMap R K) (p ^ B.dim * r)
hr :
  (algebraMap R K) r =
    (Algebra.norm K)
      (z * B.gen ^ (P.natDegree - (j + 2)) -
        (∑ x ∈ (range (Q.natDegree - j)).erase 0, Q.coeff (j + 1 + x) • f (x + P.natDegree - 1) +
          ∑ x ∈ range (j + 1), g x • B.gen ^ x * B.gen ^ (P.natDegree - (j + 2))))
hppdiv :
  p ^ B.dim ∣
    ↑((-1) ^ B.dim) ^ (j + 1 + (P.natDegree - (j + 2))) *
        (minpoly R B.gen).coeff 0 ^ (j + 1 + (P.natDegree - (j + 2))) *
      Q.coeff (j + 1) ^ B.dim
⊢ p ^ n.succ ∣ Q.coeff j.succ ^ n.succ * (minpoly R B.gen).coeff 0 ^ (j.succ + (P.natDegree - (j + 2)))