-- 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
inst_56 : IsLocalization.{u_6, u_4} (algebraMapSubmonoid.{u_1, u_6} (R := A) R₂ A⁰) ↥F₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁ ⊔ F₂ = ⊤
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))
this_1 : Algebra.IsSeparable.{u_1, u_6} (FractionRing A) (FractionRing R₂)
this : Dvd.dvd (α := Ideal B) (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)) (differentIdeal R₁ B)
hI :
  ∀
    (this :
      LE.le.{u_4} (α := FractionalIdeal.{u_2, u_4} (R := B) B⁰ L) (dual R₁ (↥F₁) 1)⁻¹
        (↑(Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)) : FractionalIdeal B⁰ L)),
    Ne.{u_4 + 1} (α := FractionalIdeal.{u_2, u_4} (R := B) B⁰ L) (dual R₁ (↥F₁) 1) 0
hJ :
  ∀
    (this :
      LE.le.{u_4} (α := FractionalIdeal.{u_2, u_4} (R := B) B⁰ L) (dual R₁ (↥F₁) 1)⁻¹
        (↑(Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)) : FractionalIdeal B⁰ L)),
    Ne.{u_4 + 1} (α := FractionalIdeal.{u_2, u_4} (R := B) B⁰ L)
      (↑(Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)) : FractionalIdeal B⁰ L) 0
⊢ LE.le (α := Submodule B L) (span B ((⇑(algebraMap (↥F₂) L) : (a : ↥F₂) → L) '' (↑(traceDual A K 1) : Set ↥F₂)))
    (traceDual R₁ (↥F₁) 1)

Tactic:
rwa [Ideal.dvd_iff_le, ← coeIdeal_le_coeIdeal (K := L), coeIdeal_differentIdeal R₁ F₁,
    inv_le_comm, ← extendedHom_coeIdeal_eq_map (K := F₂), coeIdeal_differentIdeal A K, map_inv₀,
    inv_inv, ← coe_le_coe, coe_extendedHom_eq_span, coe_dual_one, ← coeToSet_coeToSubmodule,
    coe_dual_one] at this

Premises used: Ideal.dvd_iff_le, FractionalIdeal.coe_le_coe, Membership.mem, map_inv₀, FractionalIdeal.coe_dual_one, Subtype, coeIdeal_differentIdeal, FractionalIdeal.extendedHom_coeIdeal_eq_map, FractionalIdeal.coeToSet_coeToSubmodule, inv_inv, FractionalIdeal.inv_le_comm, FractionalIdeal.coe_extendedHom_eq_span, FractionalIdeal.coeIdeal_le_coeIdeal