Theorem: Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le
File: Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean

Proof steps:
state_before: R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
⊢ ∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i
state_after: R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i
tactic: intro i hi
state_before: R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i
state_after: case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i
tactic: obtain ⟨k, hk⟩ := exists_add_of_le hi
state_before: case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ i
state_after: case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
tactic: rw [hk, pow_add]
state_before: case intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
state_after: case intro.intro.intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
tactic: obtain ⟨y, hy, H⟩ := exists_mem_adjoin_mul_eq_pow_natDegree hx hmo hf
state_before: case intro.intro.intro
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ ∃ y ∈ adjoin R {x}, (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
state_after: case intro.intro.intro.refine_1
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ y * x ^ k ∈ adjoin R {x}

case intro.intro.intro.refine_2
R : Type u
inst✝² : CommRing R
𝓟 : Ideal R
f : R[X]
hf✝ : f.IsWeaklyEisensteinAt 𝓟
S : Type v
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : R
x : S
hx : (aeval x) f = 0
hmo : f.Monic
hf : f.IsWeaklyEisensteinAt (Submodule.span R {p})
i : ℕ
hi : (Polynomial.map (algebraMap R S) f).natDegree ≤ i
k : ℕ
hk : i = (Polynomial.map (algebraMap R S) f).natDegree + k
y : S
hy : y ∈ adjoin R {x}
H : (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree
⊢ (algebraMap R S) p * (y * x ^ k) = x ^ (Polynomial.map (algebraMap R S) f).natDegree * x ^ k
tactic: refine ⟨y * x ^ k, ?_, ?_⟩