-- 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))
have : F₁ ⊔ F₂ = ⊤
have₁ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤),
    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.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) 1)
have₂ : F₁ ⊔ F₂ = ⊤
have₃ have₄ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤),
    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.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) 1)
have₅ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤) (this : Finite ι),
    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.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) 1)
have₆ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤) (this : 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))),
    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.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) 1)
have₇ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤) (this : 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))),
    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)
have₈ have₉ :
  ∀ (h₂' : F₁ ⊔ F₂ = ⊤) (this : 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))),
    Eq.{u_4 + 1} (α := Submodule A ↥F₂) (Submodule.restrictScalars.{u_1, u_6, u_4} A (R := R₂) 1)
      (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
⊢ 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.restrictScalars.{u_5, u_2, u_4} R₁ (R := B) 1)

Tactic:
classical
  have h₂' : F₁ ⊔ F₂ = ⊤ := by
    rwa [← sup_toSubalgebra_of_isAlgebraic_right, ← top_toSubalgebra, toSubalgebra_inj] at h₂
  have : Finite ι := Module.Finite.finite_basis b
  have h_main := congr_arg (Submodule.restrictScalars R₁) <|
    congr_arg coeToSubmodule <| (1 : FractionalIdeal B⁰ L).dual_dual R₁ F₁
  rw [← coe_one, ← h_main, coe_dual _ _ (by simp), coe_dual_one, restrictScalars_traceDual,
    ← traceDual_eq_span_map_traceDual_of_linearDisjoint A B R₁ R₂ h₁ h₂' h₃,
    ← coe_restrictScalars A, traceDual_span_of_basis A (1 : Submodule R₂ F₂) b,
    ← IsScalarTower.coe_toAlgHom' A F₂ L, ← AlgHom.coe_toLinearMap, ← map_coe, map_span,
    span_span_of_tower, AlgHom.coe_toLinearMap, IsScalarTower.coe_toAlgHom', ← Set.range_comp]
  · have : (h₁.basisOfBasisRight h₂ b).traceDual = algebraMap F₂ L ∘ b.traceDual := by
      refine Basis.traceDual_eq_iff.mpr fun i j ↦ ?_
      rw [Function.comp_apply, h₁.basisOfBasisRight_apply, traceForm_apply, ← map_mul,
        h₁.trace_algebraMap h₂', b.trace_traceDual_mul i j, MonoidWithZeroHom.map_ite_one_zero]
    rw [← this, (traceForm F₁ L).dualSubmodule_span_of_basis (traceForm_nondegenerate F₁ L),
      ← Basis.traceDual_def, Basis.traceDual_traceDual]
  · rw [hb]
    ext; simp

Premises used: AlgHom.coe_toLinearMap, Set.range_comp, Submodule.span_span_of_tower, IsScalarTower.coe_toAlgHom', LinearMap.BilinForm.dualSubmodule_span_of_basis, Module.Basis.traceDual_def, Membership.mem, Module.Basis.traceDual_traceDual, FractionalIdeal.coe_dual, FractionalIdeal.coe_dual_one, Submodule.coe_restrictScalars, Subtype, Submodule.restrictScalars_traceDual, FractionalIdeal.coe_one, traceForm_nondegenerate, Algebra.traceForm, Submodule.map_span, Submodule.map_coe