l3lab/ntp-mathlib huggingface dataset:
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ RingHom.comp (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso (basicOpenSectionsToAffine hU f))))
      (algebraMap ↑(X.presheaf.obj (op U)) ↑(X.presheaf.obj (op (X.basicOpen f)))) =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (op (PrimeSpectrum.basicOpen f)).unop

Next tactic:
rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ CommRingCat.ofHom (algebraMap ↑(X.presheaf.obj (op U)) ↑(X.presheaf.obj (op (X.basicOpen f)))) ≫
      CommRingCat.ofHom (basicOpenSectionsToAffine hU f) =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (op (PrimeSpectrum.basicOpen f)).unop

Next tactic:
dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ X.presheaf.map (homOfLE (_ : X.basicOpen f ≤ U)).op ≫ basicOpenSectionsToAffine hU f =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (PrimeSpectrum.basicOpen f)

Next tactic:
change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ X.presheaf.map (homOfLE (_ : X.basicOpen f ≤ U)).op ≫ basicOpenSectionsToAffine hU f =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (PrimeSpectrum.basicOpen f)

Next tactic:
delta basicOpenSectionsToAffine
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
  delta basicOpenSectionsToAffine
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ X.presheaf.map (homOfLE (_ : X.basicOpen f ≤ U)).op ≫
      (fromSpec hU).val.c.app (op (X.basicOpen f)) ≫
        (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.map
          (eqToHom (_ : PrimeSpectrum.basicOpen f = fromSpec hU⁻¹ᵁ X.basicOpen f)).op =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (PrimeSpectrum.basicOpen f)

Next tactic:
rw [hU.fromSpec.val.c.naturality_assoc, hU.fromSpec_app_self]
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
  delta basicOpenSectionsToAffine
  rw [hU.fromSpec.val.c.naturality_assoc, hU.fromSpec_app_self]
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ (SpecΓIdentity.inv.app (X.presheaf.obj (op U)) ≫
        (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.map (eqToHom (_ : fromSpec hU⁻¹ᵁ U = ⊤)).op) ≫
      ((fromSpec hU).val.base _* (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf).map
          (homOfLE (_ : X.basicOpen f ≤ U)).op ≫
        (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.map
          (eqToHom (_ : PrimeSpectrum.basicOpen f = fromSpec hU⁻¹ᵁ X.basicOpen f)).op =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (PrimeSpectrum.basicOpen f)

Next tactic:
simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
  delta basicOpenSectionsToAffine
  rw [hU.fromSpec.val.c.naturality_assoc, hU.fromSpec_app_self]
  simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
  

Proof state:
case h.e'_3.h.h.e_i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ SpecΓIdentity.inv.app (X.presheaf.obj (op U)) ≫
      (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.map (eqToHom (_ : fromSpec hU⁻¹ᵁ U = ⊤)).op ≫
        ((fromSpec hU).val.base _* (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf).map
            (homOfLE (_ : X.basicOpen f ≤ U)).op ≫
          (Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.map
            (eqToHom (_ : PrimeSpectrum.basicOpen f = fromSpec hU⁻¹ᵁ X.basicOpen f)).op =
    StructureSheaf.toOpen (↑(X.presheaf.obj (op U))) (PrimeSpectrum.basicOpen f)

Next tactic:
apply StructureSheaf.toOpen_res
theorem isLocalization_basicOpen :
    IsLocalization.Away f (X.presheaf.obj (op <| X.basicOpen f)) := by
  apply
    (IsLocalization.isLocalization_iff_of_ringEquiv (Submonoid.powers f)
      (asIso <| basicOpenSectionsToAffine hU f).commRingCatIsoToRingEquiv).mpr
  convert StructureSheaf.IsLocalization.to_basicOpen _ f using 1
  -- Porting note : more hand holding is required here, the next 4 lines were not necessary
  delta StructureSheaf.openAlgebra
  congr 1
  rw [CommRingCat.ringHom_comp_eq_comp, Iso.commRingIsoToRingEquiv_toRingHom, asIso_hom]
  dsimp [CommRingCat.ofHom, RingHom.algebraMap_toAlgebra]
  change X.presheaf.map _ ≫ basicOpenSectionsToAffine hU f = _
  delta basicOpenSectionsToAffine
  rw [hU.fromSpec.val.c.naturality_assoc, hU.fromSpec_app_self]
  simp only [Category.assoc, ← Functor.map_comp, ← op_comp]
  apply StructureSheaf.toOpen_res
  

Proof state:
case h.e'_3.h.h.e_i.i
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
e_1✝ :
  ↑((Scheme.Spec.obj (op (X.presheaf.obj (op U)))).presheaf.obj (op (PrimeSpectrum.basicOpen f))) =
    ↑((structureSheaf ↑(X.presheaf.obj (op U))).val.obj (op (PrimeSpectrum.basicOpen f)))
he✝ : CommRing.toCommSemiring = CommRing.toCommSemiring
⊢ PrimeSpectrum.basicOpen f ⟶ ⊤

Next tactic:
exact homOfLE le_top
theorem isLocalization_of_eq_basicOpen {V : Opens X} (i : V ⟶ U) (e : V = X.basicOpen f) :
    @IsLocalization.Away _ _ f (X.presheaf.obj (op V)) _ (X.presheaf.map i.op).toAlgebra := by
  

Proof state:
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
V : Opens ↑↑X.toPresheafedSpace
i : V ⟶ U
e : V = X.basicOpen f
⊢ IsLocalization.Away f ↑(X.presheaf.obj (op V))

Next tactic:
subst e
theorem isLocalization_of_eq_basicOpen {V : Opens X} (i : V ⟶ U) (e : V = X.basicOpen f) :
    @IsLocalization.Away _ _ f (X.presheaf.obj (op V)) _ (X.presheaf.map i.op).toAlgebra := by
  subst e; 

Proof state:
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
i : X.basicOpen f ⟶ U
⊢ IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))

Next tactic:
convert isLocalization_basicOpen hU f using 3
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  

Proof state:
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
g : ↑(X.presheaf.obj (op (X.basicOpen f)))
⊢ ∃ f', X.basicOpen f' = X.basicOpen g

Next tactic:
have := isLocalization_basicOpen hU f
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  

Proof state:
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
g : ↑(X.presheaf.obj (op (X.basicOpen f)))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
⊢ ∃ f', X.basicOpen f' = X.basicOpen g

Next tactic:
obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  

Proof state:
case intro.intro.mk.intro
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ ∃ f',
    X.basicOpen f' =
      X.basicOpen
        (x •
          ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
              { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) }))

Next tactic:
use f * x
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  

Proof state:
case h
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ X.basicOpen (f * x) =
    X.basicOpen
      (x •
        ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
            { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) }))

Next tactic:
rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
  

Proof state:
case h
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ X.basicOpen f ⊓ X.basicOpen x =
    X.basicOpen ((X.presheaf.map (homOfLE (_ : X.basicOpen f ≤ U)).op) x) ⊓
      X.basicOpen
        ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
            { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) })

Next tactic:
rw [Scheme.basicOpen_res]
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
  rw [Scheme.basicOpen_res]
  

Proof state:
case h
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ X.basicOpen f ⊓ X.basicOpen x =
    X.basicOpen f ⊓ X.basicOpen x ⊓
      X.basicOpen
        ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
            { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) })

Next tactic:
refine' (inf_eq_left.mpr _).symm
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
  rw [Scheme.basicOpen_res]
  refine' (inf_eq_left.mpr _).symm
  -- Porting note : a little help is needed here
  

Proof state:
case h
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ X.basicOpen f ⊓ X.basicOpen x ≤
    X.basicOpen
      ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
          { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) })

Next tactic:
convert inf_le_left (α := Opens X) using 1
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
  rw [Scheme.basicOpen_res]
  refine' (inf_eq_left.mpr _).symm
  -- Porting note : a little help is needed here
  convert inf_le_left (α := Opens X) using 1
  

Proof state:
case h.e'_4
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ X.basicOpen
      ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
          { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) }) =
    X.basicOpen f

Next tactic:
apply Scheme.basicOpen_of_isUnit
theorem basicOpen_basicOpen_is_basicOpen (g : X.presheaf.obj (op <| X.basicOpen f)) :
    ∃ f' : X.presheaf.obj (op U), X.basicOpen f' = X.basicOpen g := by
  have := isLocalization_basicOpen hU f
  obtain ⟨x, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.surj'' (Submonoid.powers f) g
  use f * x
  rw [Algebra.smul_def, Scheme.basicOpen_mul, Scheme.basicOpen_mul, RingHom.algebraMap_toAlgebra]
  rw [Scheme.basicOpen_res]
  refine' (inf_eq_left.mpr _).symm
  -- Porting note : a little help is needed here
  convert inf_le_left (α := Opens X) using 1
  apply Scheme.basicOpen_of_isUnit
  

Proof state:
case h.e'_4.hf
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
this : IsLocalization.Away f ↑(X.presheaf.obj (op (X.basicOpen f)))
x : ↑(X.presheaf.obj (op U))
n : ℕ
⊢ IsUnit
    ↑((IsLocalization.toInvSubmonoid (Submonoid.powers f) ↑(X.presheaf.obj (op (X.basicOpen f))))
        { val := (fun x => f ^ x) n, property := (_ : ∃ y, (fun x => f ^ x) y = (fun x => f ^ x) n) })

Next tactic:
apply
    Submonoid.leftInv_le_isUnit _
      (IsLocalization.toInvSubmonoid (Submonoid.powers f) (X.presheaf.obj (op <| X.basicOpen f))
        _).prop
theorem _root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
    {V : Opens X} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
    ∃ (f : X.presheaf.obj <| op U) (g : X.presheaf.obj <| op V),
      X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f := by
  

Proof state:
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑(X.presheaf.obj (op U))
V : Opens ↑↑X.toPresheafedSpace
hV : IsAffineOpen V
x : ↑↑X.toPresheafedSpace
hx : x ∈ U ⊓ V
⊢ ∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f

Next tactic:
obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basicOpen_le ⟨x, hx.2⟩ hx.1
theorem _root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
    {V : Opens X} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
    ∃ (f : X.presheaf.obj <| op U) (g : X.presheaf.obj <| op V),
      X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f := by
  obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basicOpen_le ⟨x, hx.2⟩ hx.1
  

Proof state:
case intro.intro
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f✝ : ↑(X.presheaf.obj (op U))
V : Opens ↑↑X.toPresheafedSpace
hV : IsAffineOpen V
x : ↑↑X.toPresheafedSpace
hx : x ∈ U ⊓ V
f : ↑(X.presheaf.obj (op U))
hf₁ : X.basicOpen f ≤ V
hf₂ : ↑{ val := x, property := (_ : x ∈ ↑V) } ∈ X.basicOpen f
⊢ ∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f

Next tactic:
obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basicOpen_le ⟨x, hf₂⟩ hx.2
theorem _root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
    {V : Opens X} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
    ∃ (f : X.presheaf.obj <| op U) (g : X.presheaf.obj <| op V),
      X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f := by
  obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basicOpen_le ⟨x, hx.2⟩ hx.1
  obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basicOpen_le ⟨x, hf₂⟩ hx.2
  

Proof state:
case intro.intro.intro.intro
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f✝ : ↑(X.presheaf.obj (op U))
V : Opens ↑↑X.toPresheafedSpace
hV : IsAffineOpen V
x : ↑↑X.toPresheafedSpace
hx : x ∈ U ⊓ V
f : ↑(X.presheaf.obj (op U))
hf₁ : X.basicOpen f ≤ V
hf₂ : ↑{ val := x, property := (_ : x ∈ ↑V) } ∈ X.basicOpen f
g : ↑(X.presheaf.obj (op V))
hg₁ : X.basicOpen g ≤ X.basicOpen f
hg₂ : ↑{ val := x, property := hf₂ } ∈ X.basicOpen g
⊢ ∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f

Next tactic:
obtain ⟨f', hf'⟩ :=
    basicOpen_basicOpen_is_basicOpen hU f (X.presheaf.map (homOfLE hf₁ : _ ⟶ V).op g)
theorem _root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
    {V : Opens X} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
    ∃ (f : X.presheaf.obj <| op U) (g : X.presheaf.obj <| op V),
      X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f := by
  obtain ⟨f, hf₁, hf₂⟩ := hU.exists_basicOpen_le ⟨x, hx.2⟩ hx.1
  obtain ⟨g, hg₁, hg₂⟩ := hV.exists_basicOpen_le ⟨x, hf₂⟩ hx.2
  obtain ⟨f', hf'⟩ :=
    basicOpen_basicOpen_is_basicOpen hU f (X.presheaf.map (homOfLE hf₁ : _ ⟶ V).op g)
  

Proof state:
case intro.intro.intro.intro.intro
X : Scheme
Y : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f✝ : ↑(X.presheaf.obj (op U))
V : Opens ↑↑X.toPresheafedSpace
hV : IsAffineOpen V
x : ↑↑X.toPresheafedSpace
hx : x ∈ U ⊓ V
f : ↑(X.presheaf.obj (op U))
hf₁ : X.basicOpen f ≤ V
hf₂ : ↑{ val := x, property := (_ : x ∈ ↑V) } ∈ X.basicOpen f
g : ↑(X.presheaf.obj (op V))
hg₁ : X.basicOpen g ≤ X.basicOpen f
hg₂ : ↑{ val := x, property := hf₂ } ∈ X.basicOpen g
f' : ↑(X.presheaf.obj (op U))
hf' : X.basicOpen f' = X.basicOpen ((X.presheaf.map (homOfLE hf₁).op) g)
⊢ ∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f

Next tactic:
replace hf' := (hf'.trans (RingedSpace.basicOpen_res _ _ _)).trans (inf_eq_right.mpr hg₁)