-- Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
ι : Type u_7
b : Basis.{u_7, u_3, u_4} ι K ↥F₂
hb :
  Eq.{u_4 + 1} (α := Submodule A ↥F₂) (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
    (LinearMap.range.{u_1, u_1, u_6, u_4} (M := R₂) (τ₁₂ := RingHom.id A)
      (↑(IsScalarTower.toAlgHom A R₂ ↥F₂) :
        [inst_56 : FunLike (R₂ →ₐ[A] ↥F₂) R₂ ↥F₂] →
          [SemilinearMapClass (R₂ →ₐ[A] ↥F₂) (RingHom.id A) R₂ ↥F₂] → R₂ →ₗ[A] ↥F₂))
h₂' : F₁ ⊔ F₂ = ⊤
this_1 : Finite ι
h_main :
  Eq.{u_4 + 1} (α := Submodule R₁ L)
    (Submodule.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (↑(dual R₁ (↥F₁) (dual R₁ (↥F₁) 1)) : Submodule B L))
    (Submodule.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) (↑1 : Submodule B L))
this :
  Eq (α := (a : ι) → L) (⇑(h₁.basisOfBasisRight h₂ b).traceDual : (a : ι) → L)
    (Function.comp (β := ↥F₂) (⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) (⇑b.traceDual : (a : ι) → ↥F₂))
rw :
  Eq.{u_4 + 1} (α := Submodule R₁ L)
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)) : Submodule R₁ L)
    (LinearMap.BilinForm.dualSubmodule.{u_5, u_4, u_4} (S := ↥F₁) (M := L) (traceForm (↥F₁) L)
        (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b).traceDual : (a : ι) → L))) :
      Submodule R₁ L)
rw₁ :
  Eq.{u_4 + 1} (α := Submodule R₁ L)
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)))
    (span R₁
      (Set.range.{u_4, u_7 + 1} (ι := ι)
        (⇑(LinearMap.BilinForm.dualBasis.{u_4, u_4, u_7} (V := L) (K := ↥F₁) (ι := ι) (traceForm (↥F₁) L) ⋯
                (h₁.basisOfBasisRight h₂ b).traceDual :
              Basis ι (↥F₁) L) :
          (a : ι) → L)))
rw₂ :
  Eq.{u_4 + 1} (α := Submodule R₁ L)
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)))
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b).traceDual.traceDual : (a : ι) → L)))
rw₃ rw₄ :
  Eq.{u_4 + 1} (α := Submodule R₁ L)
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)))
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)))
⊢ Eq (α := Submodule R₁ L)
    (span R₁ (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑(h₁.basisOfBasisRight h₂ b) : (a : ι) → L)) : Submodule R₁ L)
    (LinearMap.BilinForm.dualSubmodule (S := ↥F₁) (M := L) (traceForm (↥F₁) L)
        (span R₁
          (Set.range.{u_4, u_7 + 1} (ι := ι)
            (Function.comp (β := ↥F₂) (⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) (⇑b.traceDual : (a : ι) → ↥F₂)))) :
      Submodule R₁ L)

Tactic:
rw [← this, (traceForm F₁ L).dualSubmodule_span_of_basis (traceForm_nondegenerate F₁ L),
      ← Basis.traceDual_def, Basis.traceDual_traceDual]

Premises used: LinearMap.BilinForm.dualSubmodule_span_of_basis, Module.Basis.traceDual_def, Membership.mem, Module.Basis.traceDual_traceDual, Subtype, traceForm_nondegenerate, Algebra.traceForm