Theorem: Normal.exists_isSplittingField
File: Mathlib/FieldTheory/Normal.lean

Proof steps:
state_before: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
⊢ ∃ p, IsSplittingField F K p
state_after: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ ∃ p, IsSplittingField F K p
tactic: let s := Basis.ofVectorSpace F K
state_before: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ ∃ p, IsSplittingField F K p
state_after: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x)).rootSet K)) =
    Subalgebra.toSubmodule ⊤
tactic: refine
    ⟨∏ x, minpoly F (s x), splits_prod _ fun x _ => h.splits (s x),
      Subalgebra.toSubmodule.injective ?_⟩
state_before: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x)).rootSet K)) =
    Subalgebra.toSubmodule ⊤
state_after: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ ∀ (y : ↑(Basis.ofVectorSpaceIndex F K)),
    s y ∈
      ↑(Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x)).rootSet K)))
tactic: rw [Algebra.top_toSubmodule, eq_top_iff, ← s.span_eq, Submodule.span_le, Set.range_subset_iff]
state_before: F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
⊢ ∀ (y : ↑(Basis.ofVectorSpaceIndex F K)),
    s y ∈
      ↑(Subalgebra.toSubmodule (Algebra.adjoin F ((∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x)).rootSet K)))
state_after: case refine_1
F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
x✝¹ x : ↑(Basis.ofVectorSpaceIndex F K)
x✝ : x ∈ Finset.univ
⊢ minpoly F (s x) ≠ 0

case refine_2
F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
x : ↑(Basis.ofVectorSpaceIndex F K)
⊢ (map (algebraMap F K) (∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x))).IsRoot (s x)
tactic: refine fun x =>
    Algebra.subset_adjoin
      (Multiset.mem_toFinset.mpr <|
        (mem_roots <|
              mt (Polynomial.map_eq_zero <| algebraMap F K).1 <|
                Finset.prod_ne_zero_iff.2 fun x _ => ?_).2 ?_)
state_before: case refine_2
F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
x : ↑(Basis.ofVectorSpaceIndex F K)
⊢ (map (algebraMap F K) (∏ x : ↑(Basis.ofVectorSpaceIndex F K), minpoly F (s x))).IsRoot (s x)
state_after: case refine_2
F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
x : ↑(Basis.ofVectorSpaceIndex F K)
⊢ ∏ x_1 : ↑(Basis.ofVectorSpaceIndex F K), (aeval (s x)) (minpoly F (s x_1)) = 0
tactic: rw [IsRoot.def, eval_map, ← aeval_def, map_prod]
state_before: case refine_2
F : Type u_1
K : Type u_2
inst✝³ : Field F
inst✝² : Field K
inst✝¹ : Algebra F K
h : Normal F K
inst✝ : FiniteDimensional F K
s : Basis (↑(Basis.ofVectorSpaceIndex F K)) F K := Basis.ofVectorSpace F K
x : ↑(Basis.ofVectorSpaceIndex F K)
⊢ ∏ x_1 : ↑(Basis.ofVectorSpaceIndex F K), (aeval (s x)) (minpoly F (s x_1)) = 0
state_after: no goals
tactic: exact Finset.prod_eq_zero (Finset.mem_univ _) (minpoly.aeval _ _)