-- Module: Mathlib.AlgebraicGeometry.AffineScheme
Goal:
X Y : Scheme
U_1 : X.Opens
hU_1 : IsAffineOpen U_1
f_1 : (↑Γ(X, U_1) : Type u)
f : X ⟶ Y
x : ↥X
U : Y.Opens
hU : IsAffineOpen U
V : X.Opens
hV : IsAffineOpen V
hVU : V ≤ f ⁻¹ᵁ U
hx : x ∈ V
this_1 : Algebra (↑(Y.presheaf.obj (op U)) : Type u)
  (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u) :=
  Y.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)
this_2 :
  IsLocalization.AtPrime (↑(Y.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯) : ↥Y)) : Type u)
    (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
this_3 : Algebra (↑(X.presheaf.obj (op V)) : Type u)
  (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u) :=
  X.presheaf.algebra_section_stalk (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)
this :
  IsLocalization.AtPrime (↑(X.presheaf.stalk (↑(Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx) : ↥X)) : Type u)
    (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
a : (↑Γ(Y, U) : Type u)
refine_3 :
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((algebraMap (↑Γ(X, V) : Type u) (↑(X.presheaf.stalk x) : Type u) :
          (a : (↑Γ(X, V) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        ((CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU) : (a : (↑Γ(Y, U) : Type u)) → (↑Γ(X, V) : Type u)) a :
          (↑Γ(X, V) : Type u)) :
      (↑(X.presheaf.stalk x) : Type u))
    ((CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x) :
          (a : (↑(Y.presheaf.stalk (f x)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        ((algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u) :
              (a : (↑Γ(Y, U) : Type u)) → (↑(Y.presheaf.stalk (f x)) : Type u))
            a :
          (↑(Y.presheaf.stalk (f x)) : Type u)) :
      (↑(X.presheaf.stalk x) : Type u))
⊢ Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((IsLocalization.map (R := (↑Γ(X, V) : Type u)) (M :=
            (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl) (T :=
            (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal.primeCompl)
            (↑(X.presheaf.stalk x) : Type u) (RingHom.id (↑Γ(X, V) : Type u)) ⋯ :
          (a : Localization.AtPrime (hV.primeIdealOf ⟨x, hx⟩).asIdeal) → (↑(X.presheaf.stalk x) : Type u))
        ((Localization.localRingHom (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal
                (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal
                (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) ⋯ :
              (a : Localization.AtPrime (hU.primeIdealOf ⟨f x, ⋯⟩).asIdeal) →
                Localization.AtPrime (hV.primeIdealOf ⟨x, hx⟩).asIdeal)
            ((IsLocalization.map (R := (↑Γ(Y, U) : Type u)) (M :=
                  (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal.primeCompl) (T :=
                  (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal.primeCompl)
                  (Localization.AtPrime (hU.primeIdealOf (Subtype.mk (p := fun (x : ↥Y) ↦ x ∈ U) (f x) ⋯)).asIdeal)
                  (RingHom.id (↑Γ(Y, U) : Type u)) ⋯ :
                (a : (↑(Y.presheaf.stalk (f x)) : Type u)) → Localization.AtPrime (hU.primeIdealOf ⟨f x, ⋯⟩).asIdeal)
              ((algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u) :
                  (a : (↑Γ(Y, U) : Type u)) → (↑(Y.presheaf.stalk (f x)) : Type u))
                a)) :
          Localization.AtPrime (hV.primeIdealOf (Subtype.mk (p := fun (x : ↥X) ↦ x ∈ V) x hx)).asIdeal) :
      (↑(X.presheaf.stalk x) : Type u))
    ((CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x) :
          (a : (↑(Y.presheaf.stalk (f x)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        ((algebraMap (↑Γ(Y, U) : Type u) (↑(Y.presheaf.stalk (f x)) : Type u) :
              (a : (↑Γ(Y, U) : Type u)) → (↑(Y.presheaf.stalk (f x)) : Type u))
            a :
          (↑(Y.presheaf.stalk (f x)) : Type u)) :
      (↑(X.presheaf.stalk x) : Type u))

Tactic:
simp only [IsLocalization.map_eq, RingHom.id_apply, Localization.localRingHom_to_map,
      RingHomCompTriple.comp_apply]

Premises used: RingHom.id_apply, Localization.localRingHom_to_map, IsLocalization.map_eq, RingHomCompTriple.comp_apply