Theorem: separableClosure.adjoin_eq_of_isAlgebraic
File: Mathlib/FieldTheory/PurelyInseparable.lean

Proof steps:
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
⊢ adjoin E ↑(separableClosure F K) = separableClosure E K
state_after: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
⊢ adjoin E ↑(separableClosure F K) = S
tactic: set S := separableClosure E K
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
⊢ adjoin E ↑(separableClosure F K) = S
state_after: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : lift (adjoin E ↑(separableClosure F ↥S)) = lift ⊤
⊢ adjoin E ↑(separableClosure F K) = S
tactic: have h := congr_arg lift (adjoin_eq_of_isAlgebraic_of_isSeparable (F := F) S)
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : lift (adjoin E ↑(separableClosure F ↥S)) = lift ⊤
⊢ adjoin E ↑(separableClosure F K) = S
state_after: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
⊢ adjoin E ↑(separableClosure F K) = S
tactic: rw [lift_top, lift_adjoin] at h
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
⊢ adjoin E ↑(separableClosure F K) = S
state_after: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
this : IsScalarTower F (↥S) K
⊢ adjoin E ↑(separableClosure F K) = S
tactic: haveI : IsScalarTower F S K := IsScalarTower.of_algebraMap_eq (congrFun rfl)
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
this : IsScalarTower F (↥S) K
⊢ adjoin E ↑(separableClosure F K) = S
state_after: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
this : IsScalarTower F (↥S) K
⊢ adjoin E
      ↑(IntermediateField.map (IsScalarTower.toAlgHom F (↥(separableClosure E K)) K)
          (separableClosure F ↥(separableClosure E K))) =
    adjoin E (Subtype.val '' ↑(separableClosure F ↥S))
tactic: rw [← h, ← map_eq_of_separableClosure_eq_bot F (separableClosure_eq_bot E K)]
state_before: F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
S : IntermediateField E K := separableClosure E K
h : adjoin E (Subtype.val '' ↑(separableClosure F ↥S)) = S
this : IsScalarTower F (↥S) K
⊢ adjoin E
      ↑(IntermediateField.map (IsScalarTower.toAlgHom F (↥(separableClosure E K)) K)
          (separableClosure F ↥(separableClosure E K))) =
    adjoin E (Subtype.val '' ↑(separableClosure F ↥S))
state_after: no goals
tactic: simp only [coe_map, IsScalarTower.coe_toAlgHom', IntermediateField.algebraMap_apply]