-- Module: Mathlib.Topology.Sheaves.CommRingCat
Goal:
X : TopCat
F : Sheaf CommRingCat X
U V W : Opens (↑X : Type u)
e : W ≤ U ⊔ V
x :
  (↑(CommRingCat.of
        ↥(RingHom.eqLocus
            (RingHom.comp
              (CommRingCat.Hom.hom
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
              (RingHom.fst
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) :
                  Type u)
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) :
                  Type u)))
            (RingHom.comp
              (CommRingCat.Hom.hom
                (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
                  (Quiver.Hom.op (homOfLE ⋯))))
              (RingHom.snd
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) :
                  Type u)
                (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) :
                  Type u))))) :
    Type u)
y : (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) : Type u)
e₁ :
  Eq (α :=
    (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
      Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op U)) (Y :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W)))
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
              (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(F.obj.obj (op U)) : Type u)) → (↑(F.obj.obj (op (U ⊓ W))) : Type u))
        (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).1 :
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
        Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) (Y :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W)))
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
              (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op (U ⊓ W))) : Type u))
        y :
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (U ⊓ W))) :
        Type u))
e₂ :
  Eq (α :=
    (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
      Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op V)) (Y :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W)))
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
              (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(F.obj.obj (op V)) : Type u)) → (↑(F.obj.obj (op (V ⊓ W))) : Type u))
        (↑x : (↑(F.obj.obj (op U)) : Type u) × (↑(F.obj.obj (op V)) : Type u)).2 :
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
        Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op W)) (Y :=
            F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W)))
            (F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).map
              (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(F.obj.obj (op W)) : Type u)) → (↑(F.obj.obj (op (V ⊓ W))) : Type u))
        y :
      (↑(F.obj (P := CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑X : Type u))).obj (op (V ⊓ W))) :
        Type u))
⊢ W ≤ (U ⊔ V) ⊓ W

Tactic:
le_inf e le_rfl

Premises used: le_rfl, le_inf