-- Module: Mathlib.RingTheory.Kaehler.JacobiZariski
Goal:
R : Type u₁
S : Type u₂
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
T : Type u₃
inst_3 : CommRing T
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
x : Generators.Ring (Q.comp P)
hx' : Membership.mem (γ := Ideal (Extension.Ring (toExtension (Q.comp P)))) (Extension.ker (toExtension (Q.comp P))) x
y : Generators.Ring (Q.comp P)
hy : y ∈ HMul.hMul (α := Ideal (Generators.Ring (Q.comp P))) (ker (Q.comp P)) (ker (Q.comp P))
e :
  Membership.mem (γ := Ideal (Generators.Ring (Q.comp P)))
    (Ideal.map (F := P.Ring →ₐ[R] Generators.Ring (Q.comp P)) (Hom.toAlgHom (Q.toComp P)) P.ker) (x - y)
z : ↥(ker (Q.comp P)) :=
  Subtype.mk (p := fun (x : Generators.Ring (Q.comp P)) ↦
    Membership.mem (γ := Ideal (Generators.Ring (Q.comp P))) (ker (Q.comp P)) x) (x - y) ⋯
hz :
  Membership.mem (γ := Ideal (Generators.Ring (Q.comp P)))
    (Ideal.map (F := P.Ring →+* Generators.Ring (Q.comp P)) (AlgHom.toRingHom (Hom.toAlgHom (Q.toComp P))) P.ker)
    (↑z : (Q.comp P).Ring)
this :
  Eq (α := Extension.Cotangent (toExtension (Q.comp P)))
    ((Extension.Cotangent.mk : (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent)
        (Subtype.mk (p := fun (x : Extension.Ring (toExtension (Q.comp P))) ↦
          Membership.mem (γ := Ideal (Extension.Ring (toExtension (Q.comp P)))) (Extension.ker (toExtension (Q.comp P)))
            x)
          x hx') :
      Extension.Cotangent (toExtension (Q.comp P)))
    ((Extension.Cotangent.mk : (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent) z :
      Extension.Cotangent (toExtension (Q.comp P)))
h2 :
  (↑z : (Q.comp P).toExtension.Ring) ∈
    Submodule.map (Submodule.subtype (Extension.ker (toExtension (Q.comp P))))
      (Submodule.comap Extension.Cotangent.mk
        (Submodule.restrictScalars (Generators.Ring (Q.comp P))
          (Submodule.span T
            (↑(LinearMap.range (Extension.Cotangent.map (Hom.toExtensionHom (Q.toComp P)))) :
              Set (Q.comp P).toExtension.Cotangent))))
⊢ Membership.mem (α := Extension.Cotangent (toExtension (Q.comp P))) (γ :=
    Submodule T (Extension.Cotangent (toExtension (Q.comp P))))
    (Submodule.span T
      (↑(LinearMap.range (Extension.Cotangent.map (Hom.toExtensionHom (Q.toComp P)))) :
        Set (Q.comp P).toExtension.Cotangent))
    ((Extension.Cotangent.mk : (a : ↥(Q.comp P).toExtension.ker) → (Q.comp P).toExtension.Cotangent)
        (Subtype.mk (p := fun (x : Extension.Ring (toExtension (Q.comp P))) ↦
          Membership.mem (γ := Ideal (Extension.Ring (toExtension (Q.comp P)))) (Extension.ker (toExtension (Q.comp P)))
            x)
          x hx') :
      Extension.Cotangent (toExtension (Q.comp P)))

Tactic:
rw [this, ← Submodule.restrictScalars_mem (Q.comp P).Ring, ← Submodule.mem_comap,
      ← Submodule.span_singleton_le_iff_mem, ← Submodule.map_le_map_iff_of_injective
      (f := Submodule.subtype _) Subtype.val_injective, Submodule.map_subtype_span_singleton,
      Submodule.span_singleton_le_iff_mem]

Premises used: Algebra.Generators.Ring, Algebra.Generators.comp, Subtype.val_injective, Submodule.map_le_map_iff_of_injective, Submodule.mem_comap, Submodule.map_subtype_span_singleton, Submodule.restrictScalars_mem, Submodule.span_singleton_le_iff_mem