Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint

Goal state before:
pp: A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
b₂ : Basis (Free.ChooseBasisIndex A R₂) K ↥F₂ := Basis.localizationLocalization K A⁰ (↥F₂) (Free.chooseBasis A R₂)
B₁ : Basis (Free.ChooseBasisIndex A R₂) (↥F₁) L := h₁.basisOfBasisRight h₂' b₂
h_main : x ∈ span R₁ (Set.range ⇑B₁.traceDual)
⊢ x ∈ span R₁ (⇑(algebraMap (↥F₂) L) '' ↑(traceDual A K 1))

Tactic:
have h : Set.range B₁.traceDual =
      Set.range (IsScalarTower.toAlgHom A F₂ L ∘ b₂.traceDual) := by
    refine congr_arg Set.range <| B₁.traceDual_eq_iff.mpr fun i j ↦ ?_
    rw [LinearDisjoint.basisOfBasisRight_apply, traceForm_apply, Function.comp_apply,
      IsScalarTower.coe_toAlgHom', ← map_mul, h₁.trace_algebraMap h₂, b₂.trace_traceDual_mul,
      MonoidWithZeroHom.map_ite_one_zero]

Goal state after:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst✝³⁴ : CommRing A
inst✝³³ : Field K
inst✝³² : Algebra A K
inst✝³¹ : IsFractionRing A K
inst✝³⁰ : CommRing B
inst✝²⁹ : Field L
inst✝²⁸ : Algebra B L
inst✝²⁷ : Algebra A L
inst✝²⁶ : Algebra K L
inst✝²⁵ : FiniteDimensional K L
inst✝²⁴ : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst✝²³ : CommRing R₁
inst✝²² : CommRing R₂
inst✝²¹ : IsDomain R₁
inst✝²⁰ : Algebra A R₁
inst✝¹⁹ : Algebra A R₂
inst✝¹⁸ : Algebra R₁ B
inst✝¹⁷ : Algebra R₂ B
inst✝¹⁶ : Algebra R₁ L
inst✝¹⁵ : Algebra R₂ L
inst✝¹⁴ : IsScalarTower A R₁ L
inst✝¹³ : IsScalarTower R₁ B L
inst✝¹² : IsScalarTower R₂ B L
inst✝¹¹ : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst✝¹⁰ : Algebra R₁ ↥F₁
inst✝⁹ : Algebra R₂ ↥F₂
inst✝⁸ : IsTorsionFree R₁ ↥F₁
inst✝⁷ : IsScalarTower A (↥F₂) L
inst✝⁶ : IsScalarTower A R₂ ↥F₂
inst✝⁵ : IsScalarTower R₁ (↥F₁) L
inst✝⁴ : IsScalarTower R₂ (↥F₂) L
inst✝³ : Algebra.IsSeparable K ↥F₂
inst✝² : Algebra.IsSeparable (↥F₁) L
inst✝¹ : Free A R₂
inst✝ : IsLocalization (algebraMapSubmonoid R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
x : L
hx : x ∈ restrictScalars R₁ (traceDual R₁ (↥F₁) 1)
h₂' : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
b₂ : Basis (Free.ChooseBasisIndex A R₂) K ↥F₂ := Basis.localizationLocalization K A⁰ (↥F₂) (Free.chooseBasis A R₂)
B₁ : Basis (Free.ChooseBasisIndex A R₂) (↥F₁) L := h₁.basisOfBasisRight h₂' b₂
h_main : x ∈ span R₁ (Set.range ⇑B₁.traceDual)
h : Set.range ⇑B₁.traceDual = Set.range (⇑(IsScalarTower.toAlgHom A (↥F₂) L) ∘ ⇑b₂.traceDual)
⊢ x ∈ span R₁ (⇑(algebraMap (↥F₂) L) '' ↑(traceDual A K 1))