-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor
Goal:
A : Type u_1
B : Type u_2
σ : Type u_4
τ : Type u_5
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
inst_3 : CommRing B
inst_4 : SetLike τ B
inst_5 : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst_6 : GradedRing 𝒜
inst_7 : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : LE.le (α := HomogeneousIdeal ℬ) (ℬ₊) (HomogeneousIdeal.map f (𝒜₊))
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV :
  V.carrier ⊆
    (⇑(ProjectiveSpectrum.comap f hf : ContinuousMap.{u_2, u_1} (ProjectiveSpectrum ℬ) (ProjectiveSpectrum 𝒜)) :
        (a : ProjectiveSpectrum ℬ) → ProjectiveSpectrum 𝒜) ⁻¹'
      U.carrier
s : (x : ↥U) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toSubmodule
hs :
  TopCat.PrelocalPredicate.pred (X := top 𝒜) (T := fun (x : (↑(top 𝒜) : Type u_1)) ↦
    AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
    (TopCat.LocalPredicate.toPrelocalPredicate (T := fun (x : (↑(top 𝒜) : Type u_1)) ↦
        AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal) (ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜) :
      TopCat.PrelocalPredicate (X := top 𝒜) fun (x : (↑(top 𝒜) : Type u_1)) ↦ AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
    s
p : (↑(top ℬ) : Type u_2)
hpV : p ∈ V
W : Opens (↑(top 𝒜) : Type u_1)
m :
  Membership.mem (α := (↑(top 𝒜) : Type u_1)) W
    ((↑(Subtype.mk (p := fun (x : (↑(top 𝒜) : Type u_1)) ↦ x ∈ U)
            (((ProjectiveSpectrum.comap f hf : ContinuousMap.{u_2, u_1} (ProjectiveSpectrum ℬ) (ProjectiveSpectrum 𝒜)) :
                (a : ProjectiveSpectrum ℬ) → ProjectiveSpectrum 𝒜)
              p)
            ⋯) :
        (↑(top 𝒜) : Type u_1)) :
      (↑(top 𝒜) : Type u_1))
iWU : W ⟶ U
i : ℕ
a b : ↥(𝒜 i)
hb : ∀ (x : ↥W), (↑b : A) ∉ (↑x : (↑(top 𝒜) : Type u_1)).asHomogeneousIdeal
h_frac :
  ∀ (x : ↥W),
    Eq (α := AtPrime 𝒜 (↑x : (↑(top 𝒜) : Type u_1)).asHomogeneousIdeal.toIdeal)
      ((fun (x : ↥W) ↦
          (s ((iWU : (a : ↥W) → ↥U) x) :
            AtPrime 𝒜
              (HomogeneousSubmodule.toSubmodule.{0, 0, u_4, u_4, u_1, u_1} (σA := σ) (𝒜 := 𝒜)
                (asHomogeneousIdeal (↑((iWU : (a : ↥W) → ↥U) x) : ProjectiveSpectrum 𝒜)))))
        x)
      (HomogeneousLocalization.mk { deg := i, num := a, den := b, den_mem := ⋯ })
refine :
  ∀
    (x :
      ↥(((Opens.comap (ProjectiveSpectrum.comap f hf) :
                  FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
              W ⊓
            V :
          Opens (↑(top ℬ) : Type u_2)))
    (q : (↑(top ℬ) : Type u_2))
    (hqW :
      q ∈
        (↑(((Opens.comap (ProjectiveSpectrum.comap f hf) :
                    FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                  (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                W :
              Opens (ProjectiveSpectrum ℬ)) :
          Set (↑(top ℬ) : Type u_2)))
    (hqV : q ∈ (↑V : Set (↑(top ℬ) : Type u_2))),
    Eq (α :=
      AtPrime ℬ
        (↑(Subtype.mk (p := fun (x : (↑(top ℬ) : Type u_2)) ↦
                  Membership.mem (γ := Opens (↑(top ℬ) : Type u_2))
                    (((Opens.comap (ProjectiveSpectrum.comap f hf) :
                              FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                            (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                          W ⊓
                        V :
                      Opens (↑(top ℬ) : Type u_2))
                    x)
                  q ⋯) :
              (↑(top ℬ) : Type u_2)).asHomogeneousIdeal.toIdeal)
      ((fun
            (x :
              ↥(((Opens.comap (ProjectiveSpectrum.comap f hf) :
                          FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                        (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                      W ⊓
                    V :
                  Opens (↑(top ℬ) : Type u_2))) ↦
          (comapStructureSheafFun f hf U V hUV s
              ((Opens.infLERight
                    (((Opens.comap (ProjectiveSpectrum.comap f hf) :
                          FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                        (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                      W)
                    V :
                  (a :
                      ↥((Opens.comap (ProjectiveSpectrum.comap f hf) :
                              (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                            W ⊓
                          V)) →
                    ↥V)
                x) :
            AtPrime ℬ
              (HomogeneousSubmodule.toSubmodule.{0, 0, u_5, u_5, u_2, u_2} (σA := τ) (𝒜 := ℬ)
                (asHomogeneousIdeal
                  (↑((Opens.infLERight
                            (((Opens.comap (ProjectiveSpectrum.comap f hf) :
                                  FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                                (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                              W)
                            V :
                          (a :
                              ↥((Opens.comap (ProjectiveSpectrum.comap f hf) :
                                      (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                                    W ⊓
                                  V)) →
                            ↥V)
                        x) :
                    ProjectiveSpectrum ℬ)))))
        (Subtype.mk (p := fun (x : (↑(top ℬ) : Type u_2)) ↦
          Membership.mem (γ := Opens (↑(top ℬ) : Type u_2))
            (((Opens.comap (ProjectiveSpectrum.comap f hf) :
                      FrameHom.{u_1, u_2} (Opens (ProjectiveSpectrum 𝒜)) (Opens (ProjectiveSpectrum ℬ))) :
                    (a : Opens (ProjectiveSpectrum 𝒜)) → Opens (ProjectiveSpectrum ℬ))
                  W ⊓
                V :
              Opens (↑(top ℬ) : Type u_2))
            x)
          q ⋯))
      (HomogeneousLocalization.mk
        { deg := i, num := ((f.gradedAddHom i : AddMonoidHom.{u_1, u_2} ↥(𝒜 i) ↥(ℬ i)) : (a : ↥(𝒜 i)) → ↥(ℬ i)) a,
          den := ((f.gradedAddHom i : AddMonoidHom.{u_1, u_2} ↥(𝒜 i) ↥(ℬ i)) : (a : ↥(𝒜 i)) → ↥(ℬ i)) b, den_mem := ⋯ })
⊢ ∃ (V_1 : Opens (↑(top ℬ) : Type u_2)) (_ :
    (↑(Subtype.mk (p := fun (x : (↑(top ℬ) : Type u_2)) ↦ x ∈ V) p hpV) : (↑(top ℬ) : Type u_2)) ∈ V_1) (i : V_1 ⟶ V),
    TopCat.PrelocalPredicate.pred (X := top ℬ) (T := fun (x : (↑(top ℬ) : Type u_2)) ↦
      AtPrime ℬ x.asHomogeneousIdeal.toIdeal)
      (ProjectiveSpectrum.StructureSheaf.isFractionPrelocal ℬ :
        TopCat.PrelocalPredicate (X := top ℬ) fun (x : (↑(top ℬ) : Type u_2)) ↦ AtPrime ℬ x.asHomogeneousIdeal.toIdeal)
      (U := V_1) fun (x : ↥V_1) ↦
      (comapStructureSheafFun f hf U V hUV s ((i : (a : ↥V_1) → ↥V) x) :
        AtPrime ℬ
          (HomogeneousSubmodule.toSubmodule.{0, 0, u_5, u_5, u_2, u_2} (σA := τ) (𝒜 := ℬ)
            (asHomogeneousIdeal (↑((i : (a : ↥V_1) → ↥V) x) : ProjectiveSpectrum ℬ))))

Tactic:
refine ⟨W.comap (ProjectiveSpectrum.comap f hf) ⊓ V, ⟨m, hpV⟩, Opens.infLERight _ _, i,
    f.gradedAddHom i a, f.gradedAddHom i b, fun ⟨q, ⟨hqW, hqV⟩⟩ ↦ hb ⟨_, hqW⟩,
    fun ⟨q, ⟨hqW, hqV⟩⟩ ↦ ?_⟩