mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
L₁ L₂ : Lifts F E K
⊢ L₁ = L₂ ↔ ∃ (h : L₁.carrier = L₂.carrier), L₂.emb.comp (inclusion ⋯) = L₁.emb

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : IsAlgClosed 𝕜
a : A
n : ℕ
hn : 0 < n
⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝³ : Field 𝕜
inst✝² : Ring A
inst✝¹ : Algebra 𝕜 A
inst✝ : IsAlgClosed 𝕜
a : A
ha : (σ a).Nonempty
n : ℕ
⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝⁴ : Field 𝕜
inst✝³ : Ring A
inst✝² : Algebra 𝕜 A
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Nontrivial A
I : FiniteDimensional 𝕜 A
a : A
p : 𝕜[X]
h_mon : p.Monic
h_eval_p : eval₂ ↑ₐ a p = 0
⊢ ¬IsUnit ((aeval a) p)

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝² : Field k
k' : Type u
inst✝¹ : Field k'
e : k ≃+* k'
inst✝ : IsAlgClosed k
⊢ IsAlgClosed k'

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
L₁ L₂ : Lifts F E K
⊢ L₁ < L₂ ↔ L₁ ≤ L₂ ∧ L₁.carrier ≠ L₂.carrier

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u
A : Type v
inst✝⁴ : Field 𝕜
inst✝³ : Ring A
inst✝² : Algebra 𝕜 A
inst✝¹ : IsAlgClosed 𝕜
inst✝ : Nontrivial A
I : FiniteDimensional 𝕜 A
a : A
⊢ (σ a).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝² : Field 𝕜
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
p : A
hp : IsIdempotentElem p
⊢ spectrum 𝕜 p ⊆ {0, 1}

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝² : Field 𝕜
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
p : A
hp : IsIdempotentElem p
⊢ 0 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IsAlgClosed.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
A : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : NonUnitalRing A
inst✝² : Module 𝕜 A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : SMulCommClass 𝕜 A A
p : A
hp : IsIdempotentElem p
⊢ 0 ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
L₁ L₂ : Lifts F E K
⊢ L₁ < L₂ ↔ ∃ (h : L₁.carrier < L₂.carrier), L₂.emb.comp (inclusion ⋯) = L₁.emb

Tactic:
by

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

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Ring K
inst✝² : IsDomain K
hk : IsAlgClosed k
inst✝¹ : Algebra k K
inst✝ : Algebra.IsIntegral k K
⊢ Function.Bijective ⇑(algebraMap k K)

Tactic:
by

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

Goal state before:
pp: k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Field K
inst✝² : IsAlgClosed k
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : Algebra.IsAlgebraic k ↥L
x : K
hx : x ∈ L
⊢ x ∈ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_1
L : Type u_2
E : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field E
inst✝³ : Algebra K L
inst✝² : Algebra K E
inst✝¹ : Algebra L E
inst✝ : IsScalarTower K L E
E' : IntermediateField L E
H : (restrictScalars K E').FG
⊢ E'.FG

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_1
L : Type u_2
E : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field E
inst✝³ : Algebra K L
inst✝² : Algebra K E
inst✝¹ : Algebra L E
inst✝ : IsScalarTower K L E
E' : IntermediateField L E
H : (restrictScalars K E').FG
⊢ E'.FG

Tactic:
by
  obtain ⟨s, hs⟩ := H
  refine ⟨s, le_antisymm ?_ ?_⟩
  · rw [adjoin_le_iff]
    exact (subset_adjoin K _).trans_eq congr(($hs : Set E))
  · rw [← restrictScalars_le_iff K, ← hs, adjoin_le_iff]
    exact subset_adjoin L _

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
ι : Type u_3
i : ι → E
x : E
⊢ x ∈ adjoin F (Set.range i) ↔ ∃ r s, x = (MvPolynomial.aeval i) r / (MvPolynomial.aeval i) s

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
⊢ x ∈ adjoin F S ↔ ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α x : E
⊢ x ∈ F⟮α⟯ ↔ ∃ r s, x = (aeval α) r / (aeval α) s

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝⁵ : Field F
E : Type u_2
inst✝⁴ : Field E
inst✝³ : Algebra F E
S : Set E
M : Type u_3
inst✝² : Monoid M
inst✝¹ : MulSemiringAction M E
inst✝ : SMulCommClass M F E
m : M
⊢ (∀ x ∈ adjoin F S, m • x = x) ↔ ∀ x ∈ S, m • x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial M
f : End R M
k : ℕ∞
hf : ⨆ μ, (f.genEigenspace μ) k = ⊤
this : ∃ μ, f.HasUnifEigenvalue μ k
⊢ ∃ μ, f.HasEigenvalue μ

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝³ : Field k
K : Type v
inst✝² : Field K
inst✝¹ : IsAlgClosed K
inst✝ : Algebra k K
x : k[X]
hu : x ∈ nonunits k[X]
h0 : x ≠ 0
w✝¹ w✝ : k[X]
h : ∀ (a : K), (aeval a) (x * w✝¹) ≠ 0 ∨ (aeval a) (x * w✝) ≠ 0
⊢ (map (algebraMap k K) x).degree ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial M
f : End R M
k : ℕ∞
hf : ⨆ μ, (f.genEigenspace μ) k = ⊤
⊢ ∃ μ, f.HasEigenvalue μ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
inst✝² : IsAlgClosed K
inst✝¹ : FiniteDimensional K V
inst✝ : Nontrivial V
f : End K V
⊢ ∃ c, f.HasEigenvalue c

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : IsAlgClosed K
inst✝ : FiniteDimensional K V
f : End K V
this : ∀ (n : ℕ), finrank K V = n → ⨆ μ, f.maxGenEigenspace μ = ⊤
⊢ ⨆ μ, f.maxGenEigenspace μ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝⁷ : Field F
E : Type u_2
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
S : Set E
K : Type u_3
L : Type u_4
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E1 E2 : IntermediateField K L
inst✝¹ : FiniteDimensional K ↥E1
inst✝ : FiniteDimensional K ↥E2
g : TensorProduct K ↥E1 ↥E2 →ₐ[K] L := Algebra.TensorProduct.productMap E1.val E2.val
this : g.range = (E1 ⊔ E2).toSubalgebra
⊢ FiniteDimensional K ↥(E1 ⊔ E2)

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝³ : Field k
K : Type v
inst✝² : Field K
inst✝¹ : IsAlgClosed K
inst✝ : Algebra k K
x : k[X]
hu : x ∈ nonunits k[X]
h0 : x ≠ 0
w✝¹ w✝ : k[X]
h : ∀ (a : K), (aeval a) (x * w✝¹) ≠ 0 ∨ (aeval a) (x * w✝) ≠ 0
a : K
ha : eval a (map (algebraMap k K) x) = 0
⊢ (aeval a) (x * w✝¹) = 0 ∧ (aeval a) (x * w✝) = 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
S : Set E
L₁ L₂ : Lifts F E K
c : Set (Lifts F E K)
hc✝ : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
t : ↑(insert ⊥ c) → IntermediateField F E := fun i ↦ (↑i).carrier
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) (insert ⊥ c)
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) t
i j : ↑(insert ⊥ c)
h : (t i).toSubalgebra ≤ (t j).toSubalgebra
x : ↥(↑i).carrier
hji : ↑j ≤ ↑i
⊢ (↑i).emb x = ((↑j).emb.comp (Subalgebra.inclusion h)) x

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝³ : Field k
K : Type v
inst✝² : Field K
inst✝¹ : IsAlgClosed K
inst✝ : Algebra k K
p q : k[X]
⊢ IsCoprime p q ↔ ∀ (a : K), (aeval a) p ≠ 0 ∨ (aeval a) q ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
σ : Lifts F E K
hσ✝ : σ ∈ c
hσ : σ ∈ insert ⊥ c
t : ↑(insert ⊥ c) → IntermediateField F E := fun i ↦ (↑i).carrier
x : ↥σ.carrier
⊢ (union c hc).emb ((inclusion ⋯) x) = σ.emb x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
⊢ ∀ (i : ↑(insert ⊥ c)), (↑i).carrier ≤ ⨆ i, (↑i).carrier

Tactic:
by

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

Goal state before:
pp: M : Type w
inst✝¹ : Field M
inst✝ : IsAlgClosed M
p : M[X]
hp : 0 < p.degree
x : M
⊢ (p - C x).degree ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝⁷ : Field F
E : Type u_2
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
S : Set E
K : Type u_3
L : Type u_4
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
E1 E2 : IntermediateField K L
inst✝¹ : FiniteDimensional K ↥E1
inst✝ : FiniteDimensional K ↥E2
⊢ FiniteDimensional K ↥(E1 ⊔ E2)

Tactic:
by

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

Goal state before:
pp: M : Type w
inst✝¹ : Field M
inst✝ : IsAlgClosed M
p : M[X]
hp : 0 < p.degree
x y : M
hy : eval y (p - C x) = 0
⊢ (fun x ↦ eval x p) y = x

Tactic:
by

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

Goal state before:
pp: M : Type w
inst✝¹ : Field M
inst✝ : IsAlgClosed M
p : M[X]
hp : p.natDegree ≠ 0
x : M
⊢ ∃ a, (fun x ↦ eval x p) a = x

Tactic:
rw [← Nat.pos_iff_ne_zero, natDegree_pos_iff_degree_pos] at hp
    have ⟨y, hy⟩ := (IsAlgClosed.splits (p - C x)).exists_eval_eq_zero <| by
      simpa only [degree_sub_C hp] using hp.ne'
    exact ⟨y, by simpa [eval_sub, sub_eq_zero] using hy⟩

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

Goal state before:
pp: M : Type w
inst✝¹ : Field M
inst✝ : IsAlgClosed M
p : M[X]
hp : p.natDegree ≠ 0
x : M
⊢ ∃ a, (fun x ↦ eval x p) a = x

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
E1 E2 : IntermediateField K L
halg : Algebra.IsAlgebraic K ↥E1 ∨ Algebra.IsAlgebraic K ↥E2
⊢ Module.rank K ↥(E1 ⊔ E2) ≤ Module.rank K ↥E1 * Module.rank K ↥E2

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
⊢ 0 < finrank ?m.80 V

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁶ : Field k
K : Type u
inst✝¹⁵ : Field K
L : Type v
M : Type w
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : IsAlgClosed M
inst✝⁹ : Algebra.IsAlgebraic K L
R : Type u
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type v
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
inst✝³ : Algebra R M
inst✝² : IsTorsionFree R S
inst✝¹ : IsTorsionFree R M
inst✝ : Algebra.IsAlgebraic R S
⊢ Algebra.IsAlgebraic (FractionRing R) (FractionRing S)

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.Extension

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Field K
inst✝² : Algebra F E
inst✝¹ : Algebra F K
c : Set (Lifts F E K)
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
alg : Algebra.IsAlgebraic F E
inst✝ : Nonempty ↑c
hext : ∀ σ ∈ c, σ.IsExtendible
S : Finset E
Ω : Type (max u_2 u_3) := ↥(adjoin F ↑S) →ₐ[F] K
⊢ ∃ ω, ∀ (π : ↑c), ∃ θ ≥ ↑π, { carrier := adjoin F ↑S, emb := ω } ≤ θ ∧ θ.carrier = (↑π).carrier ⊔ adjoin F ↑S

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁶ : Field k
K : Type u
inst✝¹⁵ : Field K
L : Type v
M : Type w
inst✝¹⁴ : Field L
inst✝¹³ : Algebra K L
inst✝¹² : Field M
inst✝¹¹ : Algebra K M
inst✝¹⁰ : IsAlgClosed M
inst✝⁹ : Algebra.IsAlgebraic K L
R : Type u
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type v
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
inst✝³ : Algebra R M
inst✝² : IsTorsionFree R S
inst✝¹ : IsTorsionFree R M
inst✝ : Algebra.IsAlgebraic R S
⊢ S →ₐ[R] M

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
⊢ 0 < finrank K ↥ES

Tactic:
by

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

Goal state before:
pp: F : Type u_1
F' : Type u_2
E : Type u_3
inst✝⁶ : Field F
inst✝⁵ : Field F'
inst✝⁴ : Field E
inst✝³ : Algebra F F'
inst✝² : Algebra F E
inst✝¹ : Algebra.IsAlgebraic F E
inst✝ : IsAlgClosed F'
h : finrank F F' = 2
⊢ Nonempty (E ≃ₐ[F] F) ∨ Nonempty (E ≃ₐ[F] F')

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁷ : Field k
K : Type u
inst✝¹⁶ : Field K
L : Type v
M : Type w
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra K L
inst✝¹³ : Field M
inst✝¹² : Algebra K M
inst✝¹¹ : IsAlgClosed M
inst✝¹⁰ : Algebra.IsAlgebraic K L
R : Type u
inst✝⁹ : CommRing R
inst✝⁸ : IsDomain R
S : Type v
inst✝⁷ : CommRing S
inst✝⁶ : IsDomain S
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R M
inst✝³ : IsTorsionFree R S
inst✝² : IsTorsionFree R M
inst✝¹ : Algebra.IsAlgebraic R S
inst✝ : IsAlgClosed k
⊢ PerfectField k

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
E1 E2 : IntermediateField K L
⊢ finrank K ↥(E1 ⊔ E2) ≤ finrank K ↥E1 * finrank K ↥E2

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁸ : Field k
K✝ : Type u
inst✝¹⁷ : Field K✝
L : Type v
M : Type w
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra K✝ L
inst✝¹⁴ : Field M
inst✝¹³ : Algebra K✝ M
inst✝¹² : IsAlgClosed M
inst✝¹¹ : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type v
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R M
inst✝⁴ : IsTorsionFree R S
inst✝³ : IsTorsionFree R M
inst✝² : Algebra.IsAlgebraic R S
K : Type u_1
inst✝¹ : Field K
inst✝ : IsAlgClosed K
hfin : Fintype K
n : ℕ := Fintype.card K
f : K[X] := X ^ (n + 1) - 1
⊢ ↑(n + 1) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
⊢ finrank K ↥ER + finrank K ↥ES = finrank K V

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
h_dim_add : finrank K ↥ER + finrank K ↥ES = finrank K V
⊢ finrank K ↥ER < n.succ

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: K : Type u_3
L : Type u_4
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
ι : Type u_5
t : ι → IntermediateField K L
dir : Directed (fun x1 x2 ↦ x1 ≤ x2) t
⊢ (iSup t).toSubalgebra = ⨆ i, (t i).toSubalgebra

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
h_dim_add : finrank K ↥ER + finrank K ↥ES = finrank K V
h_dim_ER : finrank K ↥ER < n.succ
ih_ER : ⨆ μ, f'.maxGenEigenspace μ = ⊤
⊢ ⨆ μ, Submodule.map ER.subtype (f'.maxGenEigenspace μ) = ER

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁸ : Field k
K✝ : Type u
inst✝¹⁷ : Field K✝
L : Type v
M : Type w
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra K✝ L
inst✝¹⁴ : Field M
inst✝¹³ : Algebra K✝ M
inst✝¹² : IsAlgClosed M
inst✝¹¹ : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type v
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R M
inst✝⁴ : IsTorsionFree R S
inst✝³ : IsTorsionFree R M
inst✝² : Algebra.IsAlgebraic R S
K : Type u_1
inst✝¹ : Field K
inst✝ : IsAlgClosed K
hfin : Fintype K
n : ℕ := Fintype.card K
f : K[X] := X ^ (n + 1) - 1
hfsep : f.Separable
⊢ n.succ = Fintype.card ↑(f.rootSet K)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
h_dim_add : finrank K ↥ER + finrank K ↥ES = finrank K V
h_dim_ER : finrank K ↥ER < n.succ
ih_ER : ⨆ μ, f'.maxGenEigenspace μ = ⊤
ih_ER' : ⨆ μ, Submodule.map ER.subtype (f'.maxGenEigenspace μ) = ER
⊢ ∀ (μ : K), Submodule.map ER.subtype (f'.maxGenEigenspace μ) ≤ f.maxGenEigenspace μ

Tactic:
by

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

Goal state before:
pp: k : Type u
inst✝¹⁸ : Field k
K✝ : Type u
inst✝¹⁷ : Field K✝
L : Type v
M : Type w
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra K✝ L
inst✝¹⁴ : Field M
inst✝¹³ : Algebra K✝ M
inst✝¹² : IsAlgClosed M
inst✝¹¹ : Algebra.IsAlgebraic K✝ L
R : Type u
inst✝¹⁰ : CommRing R
inst✝⁹ : IsDomain R
S : Type v
inst✝⁸ : CommRing S
inst✝⁷ : IsDomain S
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R M
inst✝⁴ : IsTorsionFree R S
inst✝³ : IsTorsionFree R M
inst✝² : Algebra.IsAlgebraic R S
K : Type u_1
inst✝¹ : Field K
inst✝ : IsAlgClosed K
⊢ Infinite K

Tactic:
by

Goal state after:
Module: Mathlib.FieldTheory.IntermediateField.Adjoin.Basic

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
E : Type u_2
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
S : Set E
K : Type u_3
L : Type u_4
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
E1 E2 : IntermediateField K L
ι : Type u_5
t : ι → IntermediateField K L
h : Finite ι
inst✝ : ∀ (i : ι), FiniteDimensional K ↥(t i)
⊢ FiniteDimensional K ↥(⨆ i, t i)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : IsAlgClosed K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
n : ℕ
ih :
  ∀ m < n + 1,
    ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),
      finrank K V = m → ⨆ μ, f.maxGenEigenspace μ = ⊤
h_dim : finrank K V = n + 1
this : Nontrivial V
μ₀ : K
hμ₀ : f.HasEigenvalue μ₀
ES : Submodule K V := (f.genEigenspace μ₀) ↑(finrank K V)
ER : Submodule K V := f.genEigenrange μ₀ ↑(finrank K V)
h_f_ER : ∀ x ∈ ER, f x ∈ ER
f' : End K ↥ER := LinearMap.restrict f h_f_ER
h_dim_ES_pos : 0 < finrank K ↥ES
h_dim_add : finrank K ↥ER + finrank K ↥ES = finrank K V
h_dim_ER : finrank K ↥ER < n.succ
ih_ER : ⨆ μ, f'.maxGenEigenspace μ = ⊤
ih_ER' : ⨆ μ, Submodule.map ER.subtype (f'.maxGenEigenspace μ) = ER
hff' : ∀ (μ : K), Submodule.map ER.subtype (f'.maxGenEigenspace μ) ≤ f.maxGenEigenspace μ
⊢ ER ≤ ⨆ μ, f.maxGenEigenspace μ

Tactic:
by

Goal state after: