-- 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
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))
inst_55 : Free A R₂
B₁ : Basis.{u_6, u_5, u_2} (Free.ChooseBasisIndex A R₂) R₁ B :=
  Basis.ofIsCoprimeDifferentIdeal A B R₁ R₂ h₁ h₂ h₃ (Free.chooseBasis A R₂)
x : B
refine :
  ∀ (i : Free.ChooseBasisIndex A R₂),
    Membership.mem (γ := Finset (Free.ChooseBasisIndex A R₂)) Finset.univ i →
      Membership.mem (γ := Subalgebra A B)
        (AlgHom.range (IsScalarTower.toAlgHom A R₁ B) ⊔ AlgHom.range (IsScalarTower.toAlgHom A R₂ B))
        (HSMul.hSMul.{u_5, u_2, u_2} (α := R₁) (β := B)
          ((((B₁.repr : (a : B) → Free.ChooseBasisIndex A R₂ →₀ R₁) x :
                  Finsupp.{u_6, u_5} (Free.ChooseBasisIndex A R₂) R₁) :
                (a : Free.ChooseBasisIndex A R₂) → R₁)
              i :
            R₁)
          ((B₁ : (a : Free.ChooseBasisIndex A R₂) → B) i : B))
⊢ Membership.mem (γ := Subalgebra A B)
    (AlgHom.range (IsScalarTower.toAlgHom A R₁ B) ⊔ AlgHom.range (IsScalarTower.toAlgHom A R₂ B))
    (∑ i : Free.ChooseBasisIndex A R₂,
      HSMul.hSMul.{u_5, u_2, u_2} (α := R₁) (β := B)
        ((((B₁.repr : (a : B) → Free.ChooseBasisIndex A R₂ →₀ R₁) x :
                Finsupp.{u_6, u_5} (Free.ChooseBasisIndex A R₂) R₁) :
              (a : Free.ChooseBasisIndex A R₂) → R₁)
            i :
          R₁)
        ((B₁ : (a : Free.ChooseBasisIndex A R₂) → B) i : B))

Tactic:
refine Subalgebra.sum_mem _ fun i _ ↦ ?_

Premises used: Subalgebra.sum_mem