-- Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
    (hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} {Q : R[X]} (hQ : aeval B.gen Q = p • z)
    (hzint : IsIntegral R z) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) : p ∣ Q.coeff 0

Proof state:
case h.e'_4
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this✝ : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
this : p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0
⊢ Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n =
    Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) * ↑((-1) ^ (n.succ * n))

Tactic:
push_cast

Result:
case h.e'_4
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this✝ : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
this : p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0
⊢ Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n =
    Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) * (-1) ^ (n.succ * n)

Proof state:
case h.e'_4
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this✝ : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
this : p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0
⊢ Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n =
    Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) * (-1) ^ (n.succ * n)

Tactic:
ring_nf

Result:
case h.e'_4
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this✝ : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
this : p ^ n.succ ∣ Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)
hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0
⊢ Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n =
    Q.coeff 0 ^ n.succ * (minpoly R B.gen).coeff 0 ^ n * (-1) ^ (n.succ * n * 2)

Proof state:
case h.a
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
⊢ (algebraMap R K) (Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)) =
    (algebraMap R K) (p ^ n.succ * r)

Tactic:
simp only [_root_.map_mul, map_pow, map_neg, map_one]

Result:
case h.a
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable 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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
⊢ (algebraMap R K) (Q.coeff 0) ^ n.succ * ((-1) ^ (n.succ * n) * (algebraMap R K) ((minpoly R B.gen).coeff 0) ^ n) =
    (algebraMap R K) p ^ n.succ * (algebraMap R K) r