-- Module: Mathlib.AlgebraicGeometry.SpreadingOut
Goal:
X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥X
hx : x ∈ U
f : (↑Γ(X, U) : Type u)
hf : x ∈ X.basicOpen f
H :
  ∀ (x_1 : (↑(X.presheaf.obj (Opposite.op U)) : Type u)),
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx) :
              (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
            x_1 :
          (↑(X.presheaf.stalk x) : Type u))
        (0 : (↑(X.presheaf.stalk x) : Type u)) →
      x_1 = 0
this : IsLocalization.Away f (↑Γ(X, X.basicOpen f) : Type u)
t : (↑Γ(X, U) : Type u)
s : ↥(Submonoid.powers f)
ht :
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
          (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) :
      (↑(X.presheaf.stalk x) : Type u))
    (0 : (↑(X.presheaf.stalk x) : Type u))
rw :
  ∀
    (ht :
      Eq (α := (↑(X.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x ⋯) :
              (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
            t :
          (↑(X.presheaf.stalk x) : Type u))
        (0 : (↑(X.presheaf.stalk x) : Type u))),
    Eq (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
      (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) 0
hy :
  ∀
    (ht :
      Membership.mem (γ := Ideal (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
        (RingHom.ker (F :=
          (fun (R S : CommRingCat.{u}) ↦ RingHom.{u, u} (↑R : Type u) (↑S : Type u))
            (X.presheaf.obj (Opposite.op (X.basicOpen f))) (X.presheaf.stalk x))
          (ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf)))
        (HMul.hMul (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) (β :=
          (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
          ((algebraMap (↑Γ(X, U) : Type u) (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) :
                (a : (↑Γ(X, U) : Type u)) → (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
              t :
            (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
          (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) 1 s :
            (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)))),
    IsUnit (M := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
      (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) 1 s)
⊢ Eq (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
    (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) 0

Tactic:
rw [← RingHom.mem_ker, IsLocalization.mk'_eq_mul_mk'_one, Ideal.mul_unit_mem_iff_mem,
    RingHom.mem_ker, RingHom.algebraMap_toAlgebra, TopCat.Presheaf.germ_res_apply] at ht

Premises used: TopCat.Presheaf.germ_res_apply, IsLocalization.mk'_eq_mul_mk'_one, RingHom.mem_ker, Ideal.mul_unit_mem_iff_mem, RingHom.algebraMap_toAlgebra