-- Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean
theorem pow_natDegree_le_of_aeval_zero_of_monic_mem_map (hf : f.IsWeaklyEisensteinAt 𝓟)
    {x : S} (hx : aeval x f = 0) (hmo : f.Monic) :
    ∀ i, (f.map (algebraMap R S)).natDegree ≤ i → x ^ i ∈ 𝓟.map (algebraMap R S)

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
rw [aeval_def, eval₂_eq_eval_map, ← IsRoot.def] at hx

Result:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (Polynomial.map (algebraMap R S) f).IsRoot x
hmo : f.Monic
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (Polynomial.map (algebraMap R S) f).IsRoot x
hmo : f.Monic
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
exact pow_natDegree_le_of_root_of_monic_mem (hf.map _) hx (hmo.map _) _ rfl.le

Result:
no goals

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
⊢ ∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
intro i hi

Result:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
exact mul_mem_right _ _ this

Result:
no goals

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
⊢ ∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
suffices x ^ (f.map (algebraMap R S)).natDegree ∈ 𝓟.map (algebraMap R S) by
  intro i hi
  obtain ⟨k, hk⟩ := exists_add_of_le hi
  rw [hk, pow_add]
  exact mul_mem_right _ _ this

Result:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟

Proof state:
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
obtain ⟨k, hk⟩ := exists_add_of_le hi

Result:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Proof state:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ x ^ i ∈ Ideal.map (algebraMap R S) 𝓟

Tactic:
rw [hk, pow_add]

Result:
case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
hf : f.IsWeaklyEisensteinAt 𝓟
x : S
hx : (aeval x) f = 0
hmo : f.Monic
this : x ^ (Polynomial.map (algebraMap R S) f).natDegree ∈ Ideal.map (algebraMap R S) 𝓟
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k ∈ Ideal.map (algebraMap R S) 𝓟