ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.Spec.topMap_comp
@[simp]
theorem Spec.topMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
    Spec.topMap (f ≫ g) = Spec.topMap g ≫ Spec.topMap f 
Arguments: {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
Goal: AlgebraicGeometry.Spec.topMap (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.topMap g) (AlgebraicGeometry.Spec.topMap f)
-- AlgebraicGeometry.Spec.toTop_obj
@[simps! obj map]
def Spec.toTop : CommRingCat.{u}ᵒᵖ ⥤ TopCat 
Arguments: (R : CommRingCatᵒᵖ)
Goal: AlgebraicGeometry.Spec.toTop.obj R = AlgebraicGeometry.Spec.topObj (Opposite.unop R)
-- AlgebraicGeometry.Spec.toTop_map
@[simps! obj map]
def Spec.toTop : CommRingCat.{u}ᵒᵖ ⥤ TopCat 
Arguments: {R S : CommRingCatᵒᵖ} (f : R ⟶ S)
Goal: AlgebraicGeometry.Spec.toTop.map f = AlgebraicGeometry.Spec.topMap f.unop
-- AlgebraicGeometry.Spec.sheafedSpaceObj_presheaf
@[simps]
def Spec.sheafedSpaceObj (R : CommRingCat.{u}) : SheafedSpace CommRingCat 
Arguments: (R : CommRingCat)
Goal: (AlgebraicGeometry.Spec.sheafedSpaceObj R).presheaf = (AlgebraicGeometry.Spec.structureSheaf ↑R).val
-- AlgebraicGeometry.Spec.sheafedSpaceObj_carrier
@[simps]
def Spec.sheafedSpaceObj (R : CommRingCat.{u}) : SheafedSpace CommRingCat 
Arguments: (R : CommRingCat)
Goal: ↑(AlgebraicGeometry.Spec.sheafedSpaceObj R).toPresheafedSpace = AlgebraicGeometry.Spec.topObj R
-- AlgebraicGeometry.Spec.sheafedSpaceMap_base
@[simps]
def Spec.sheafedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
    Spec.sheafedSpaceObj S ⟶ Spec.sheafedSpaceObj R 
Arguments: {R S : CommRingCat} (f : R ⟶ S)
Goal: (AlgebraicGeometry.Spec.sheafedSpaceMap f).base = AlgebraicGeometry.Spec.topMap f
-- AlgebraicGeometry.Spec.sheafedSpaceMap_c_app
@[simps]
def Spec.sheafedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
    Spec.sheafedSpaceObj S ⟶ Spec.sheafedSpaceObj R 
Arguments: {R S : CommRingCat} (f : R ⟶ S) (U : (TopologicalSpace.Opens ↑↑(AlgebraicGeometry.Spec.sheafedSpaceObj R).toPresheafedSpace)ᵒᵖ)
Goal: (AlgebraicGeometry.Spec.sheafedSpaceMap f).c.app U =
  AlgebraicGeometry.StructureSheaf.comap f (Opposite.unop U)
    ((TopologicalSpace.Opens.map (AlgebraicGeometry.Spec.topMap f)).obj (Opposite.unop U)) ⋯
-- AlgebraicGeometry.Spec.sheafedSpaceMap_id
@[simp]
theorem Spec.sheafedSpaceMap_id {R : CommRingCat.{u}} :
    Spec.sheafedSpaceMap (𝟙 R) = 𝟙 (Spec.sheafedSpaceObj R) 
Arguments: {R : CommRingCat}
Goal: AlgebraicGeometry.Spec.sheafedSpaceMap (CategoryTheory.CategoryStruct.id R) =
  CategoryTheory.CategoryStruct.id (AlgebraicGeometry.Spec.sheafedSpaceObj R)
-- AlgebraicGeometry.Spec.sheafedSpaceMap_comp
theorem Spec.sheafedSpaceMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
    Spec.sheafedSpaceMap (f ≫ g) = Spec.sheafedSpaceMap g ≫ Spec.sheafedSpaceMap f 
Arguments: {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
Goal: AlgebraicGeometry.Spec.sheafedSpaceMap (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.sheafedSpaceMap g)
    (AlgebraicGeometry.Spec.sheafedSpaceMap f)
-- AlgebraicGeometry.Spec.toSheafedSpace_obj
@[simps]
def Spec.toSheafedSpace : CommRingCat.{u}ᵒᵖ ⥤ SheafedSpace CommRingCat 
Arguments: (R : CommRingCatᵒᵖ)
Goal: AlgebraicGeometry.Spec.toSheafedSpace.obj R = AlgebraicGeometry.Spec.sheafedSpaceObj (Opposite.unop R)
-- AlgebraicGeometry.Spec.toSheafedSpace_map
@[simps]
def Spec.toSheafedSpace : CommRingCat.{u}ᵒᵖ ⥤ SheafedSpace CommRingCat 
Goal: ∀ {X Y : CommRingCatᵒᵖ} (f : X ⟶ Y),
  AlgebraicGeometry.Spec.toSheafedSpace.map f = AlgebraicGeometry.Spec.sheafedSpaceMap f.unop
-- AlgebraicGeometry.Spec.toPresheafedSpace_obj
@[simp]
theorem Spec.toPresheafedSpace_obj (R : CommRingCat.{u}ᵒᵖ) :
    Spec.toPresheafedSpace.obj R = (Spec.sheafedSpaceObj (unop R)).toPresheafedSpace 
Arguments: (R : CommRingCatᵒᵖ)
Goal: AlgebraicGeometry.Spec.toPresheafedSpace.obj R =
  (AlgebraicGeometry.Spec.sheafedSpaceObj (Opposite.unop R)).toPresheafedSpace
-- AlgebraicGeometry.Spec.toPresheafedSpace_obj_op
theorem Spec.toPresheafedSpace_obj_op (R : CommRingCat.{u}) :
    Spec.toPresheafedSpace.obj (op R) = (Spec.sheafedSpaceObj R).toPresheafedSpace 
Arguments: (R : CommRingCat)
Goal: AlgebraicGeometry.Spec.toPresheafedSpace.obj (Opposite.op R) =
  (AlgebraicGeometry.Spec.sheafedSpaceObj R).toPresheafedSpace
-- AlgebraicGeometry.Spec.toPresheafedSpace_map
@[simp]
theorem Spec.toPresheafedSpace_map (R S : CommRingCat.{u}ᵒᵖ) (f : R ⟶ S) :
    Spec.toPresheafedSpace.map f = Spec.sheafedSpaceMap f.unop 
Arguments: (R S : CommRingCatᵒᵖ) (f : R ⟶ S)
Goal: AlgebraicGeometry.Spec.toPresheafedSpace.map f = AlgebraicGeometry.Spec.sheafedSpaceMap f.unop
-- AlgebraicGeometry.Spec.toPresheafedSpace_map_op
theorem Spec.toPresheafedSpace_map_op (R S : CommRingCat.{u}) (f : R ⟶ S) :
    Spec.toPresheafedSpace.map f.op = Spec.sheafedSpaceMap f 
Arguments: (R S : CommRingCat) (f : R ⟶ S)
Goal: AlgebraicGeometry.Spec.toPresheafedSpace.map f.op = AlgebraicGeometry.Spec.sheafedSpaceMap f
-- AlgebraicGeometry.Spec.basicOpen_hom_ext
theorem Spec.basicOpen_hom_ext {X : RingedSpace.{u}} {R : CommRingCat.{u}}
    {α β : X ⟶ Spec.sheafedSpaceObj R} (w : α.base = β.base)
    (h : ∀ r : R,
      let U := PrimeSpectrum.basicOpen r
      (toOpen R U ≫ α.c.app (op U)) ≫ X.presheaf.map (eqToHom (by rw [w])) =
        toOpen R U ≫ β.c.app (op U)) :
    α = β 
Arguments: {X : AlgebraicGeometry.RingedSpace} {R : CommRingCat} {α β : X ⟶ AlgebraicGeometry.Spec.sheafedSpaceObj R} (w : α.base = β.base) (h :
  ∀ (r : ↑R),
    let U := PrimeSpectrum.basicOpen r;
    CategoryTheory.CategoryStruct.comp
        (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toOpen (↑R) U) (α.c.app (Opposite.op U)))
        (X.presheaf.map (CategoryTheory.eqToHom ⋯)) =
      CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toOpen (↑R) U) (β.c.app (Opposite.op U)))
Goal: α = β
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf
@[simps! toSheafedSpace presheaf]
def Spec.locallyRingedSpaceObj (R : CommRingCat.{u}) : LocallyRingedSpace 
Arguments: (R : CommRingCat)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj R).presheaf = (AlgebraicGeometry.Spec.structureSheaf ↑R).val
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_toSheafedSpace
@[simps! toSheafedSpace presheaf]
def Spec.locallyRingedSpaceObj (R : CommRingCat.{u}) : LocallyRingedSpace 
Arguments: (R : CommRingCat)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj R).toSheafedSpace = AlgebraicGeometry.Spec.sheafedSpaceObj R
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf
lemma Spec.locallyRingedSpaceObj_sheaf (R : CommRingCat.{u}) :
    (Spec.locallyRingedSpaceObj R).sheaf = structureSheaf R 
Arguments: (R : CommRingCat)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj R).sheaf = AlgebraicGeometry.Spec.structureSheaf ↑R
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf'
lemma Spec.locallyRingedSpaceObj_sheaf' (R : Type u) [CommRing R] :
    (Spec.locallyRingedSpaceObj <| CommRingCat.of R).sheaf = structureSheaf R 
Arguments: (R : Type u) [CommRing R]
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj (CommRingCat.of R)).sheaf = AlgebraicGeometry.Spec.structureSheaf R
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map
lemma Spec.locallyRingedSpaceObj_presheaf_map (R : CommRingCat.{u}) {U V} (i : U ⟶ V) :
    (Spec.locallyRingedSpaceObj R).presheaf.map i =
    (structureSheaf R).1.map i 
Arguments: (R : CommRingCat) {U V : (TopologicalSpace.Opens ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj R).toPresheafedSpace)ᵒᵖ} (i : U ⟶ V)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj R).presheaf.map i = (AlgebraicGeometry.Spec.structureSheaf ↑R).val.map i
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf'
lemma Spec.locallyRingedSpaceObj_presheaf' (R : Type u) [CommRing R] :
    (Spec.locallyRingedSpaceObj <| CommRingCat.of R).presheaf = (structureSheaf R).1 
Arguments: (R : Type u) [CommRing R]
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj (CommRingCat.of R)).presheaf =
  (AlgebraicGeometry.Spec.structureSheaf R).val
-- AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map'
lemma Spec.locallyRingedSpaceObj_presheaf_map' (R : Type u) [CommRing R] {U V} (i : U ⟶ V) :
    (Spec.locallyRingedSpaceObj <| CommRingCat.of R).presheaf.map i =
    (structureSheaf R).1.map i 
Arguments: (R : Type u) [CommRing R] {U V : (TopologicalSpace.Opens ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj (CommRingCat.of R)).toPresheafedSpace)ᵒᵖ} (i : U ⟶ V)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceObj (CommRingCat.of R)).presheaf.map i =
  (AlgebraicGeometry.Spec.structureSheaf R).val.map i
-- AlgebraicGeometry.stalkMap_toStalk
@[elementwise]
theorem stalkMap_toStalk {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
    toStalk R (PrimeSpectrum.comap f p) ≫ (Spec.sheafedSpaceMap f).stalkMap p =
      f ≫ toStalk S p 
Arguments: {R S : CommRingCat} (f : R ⟶ S) (p : PrimeSpectrum ↑S)
Goal: CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toStalk (↑R) ((PrimeSpectrum.comap f) p))
    (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (AlgebraicGeometry.Spec.sheafedSpaceMap f) p) =
  CategoryTheory.CategoryStruct.comp f (AlgebraicGeometry.StructureSheaf.toStalk (↑S) p)
-- AlgebraicGeometry.localRingHom_comp_stalkIso
@[elementwise]
theorem localRingHom_comp_stalkIso {R S : CommRingCat.{u}} (f : R ⟶ S) (p : PrimeSpectrum S) :
    (stalkIso R (PrimeSpectrum.comap f p)).hom ≫
        @CategoryStruct.comp _ _
          (CommRingCat.of (Localization.AtPrime (PrimeSpectrum.comap f p).asIdeal))
          (CommRingCat.of (Localization.AtPrime p.asIdeal)) _
          (Localization.localRingHom (PrimeSpectrum.comap f p).asIdeal p.asIdeal f rfl)
          (stalkIso S p).inv =
      (Spec.sheafedSpaceMap f).stalkMap p 
Arguments: {R S : CommRingCat} (f : R ⟶ S) (p : PrimeSpectrum ↑S)
Goal: CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.stalkIso (↑R) ((PrimeSpectrum.comap f) p)).hom
    (CategoryTheory.CategoryStruct.comp (Localization.localRingHom ((PrimeSpectrum.comap f) p).asIdeal p.asIdeal f ⋯)
      (AlgebraicGeometry.StructureSheaf.stalkIso (↑S) p).inv) =
  AlgebraicGeometry.PresheafedSpace.Hom.stalkMap (AlgebraicGeometry.Spec.sheafedSpaceMap f) p
-- AlgebraicGeometry.Spec.locallyRingedSpaceMap_val
@[simps]
def Spec.locallyRingedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
    Spec.locallyRingedSpaceObj S ⟶ Spec.locallyRingedSpaceObj R 
Arguments: {R S : CommRingCat} (f : R ⟶ S)
Goal: (AlgebraicGeometry.Spec.locallyRingedSpaceMap f).val = AlgebraicGeometry.Spec.sheafedSpaceMap f
-- AlgebraicGeometry.Spec.locallyRingedSpaceMap_id
@[simp]
theorem Spec.locallyRingedSpaceMap_id (R : CommRingCat.{u}) :
    Spec.locallyRingedSpaceMap (𝟙 R) = 𝟙 (Spec.locallyRingedSpaceObj R) 
Arguments: (R : CommRingCat)
Goal: AlgebraicGeometry.Spec.locallyRingedSpaceMap (CategoryTheory.CategoryStruct.id R) =
  CategoryTheory.CategoryStruct.id (AlgebraicGeometry.Spec.locallyRingedSpaceObj R)
-- AlgebraicGeometry.Spec.locallyRingedSpaceMap_comp
theorem Spec.locallyRingedSpaceMap_comp {R S T : CommRingCat.{u}} (f : R ⟶ S) (g : S ⟶ T) :
    Spec.locallyRingedSpaceMap (f ≫ g) =
      Spec.locallyRingedSpaceMap g ≫ Spec.locallyRingedSpaceMap f 
Arguments: {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
Goal: AlgebraicGeometry.Spec.locallyRingedSpaceMap (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.locallyRingedSpaceMap g)
    (AlgebraicGeometry.Spec.locallyRingedSpaceMap f)
-- AlgebraicGeometry.Spec.toLocallyRingedSpace_map
@[simps]
def Spec.toLocallyRingedSpace : CommRingCat.{u}ᵒᵖ ⥤ LocallyRingedSpace 
Goal: ∀ {X Y : CommRingCatᵒᵖ} (f : X ⟶ Y),
  AlgebraicGeometry.Spec.toLocallyRingedSpace.map f = AlgebraicGeometry.Spec.locallyRingedSpaceMap f.unop
-- AlgebraicGeometry.Spec.toLocallyRingedSpace_obj
@[simps]
def Spec.toLocallyRingedSpace : CommRingCat.{u}ᵒᵖ ⥤ LocallyRingedSpace 
Arguments: (R : CommRingCatᵒᵖ)
Goal: AlgebraicGeometry.Spec.toLocallyRingedSpace.obj R = AlgebraicGeometry.Spec.locallyRingedSpaceObj (Opposite.unop R)
-- AlgebraicGeometry.toSpecΓ_apply_coe
@[simps!]
def toSpecΓ (R : CommRingCat.{u}) : R ⟶ Γ.obj (op (Spec.toLocallyRingedSpace.obj (op R))) 
Arguments: (R : CommRingCat) (f : ↑(CommRingCat.of ↑R)) (x : ↥(Opposite.unop (Opposite.op ⊤)))
Goal: ↑((AlgebraicGeometry.toSpecΓ R) f) x = (algebraMap (↑R) (AlgebraicGeometry.StructureSheaf.Localizations ↑R ↑x)) f
-- AlgebraicGeometry.Spec_Γ_naturality
@[reassoc]
theorem Spec_Γ_naturality {R S : CommRingCat.{u}} (f : R ⟶ S) :
    f ≫ toSpecΓ S = toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op 
Arguments: {R S : CommRingCat} (f : R ⟶ S)
Goal: CategoryTheory.CategoryStruct.comp f (AlgebraicGeometry.toSpecΓ S) =
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.toSpecΓ R)
    (AlgebraicGeometry.LocallyRingedSpace.Γ.map (AlgebraicGeometry.Spec.toLocallyRingedSpace.map f.op).op)
-- AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app
@[simps! hom_app inv_app]
def LocallyRingedSpace.SpecΓIdentity : Spec.toLocallyRingedSpace.rightOp ⋙ Γ ≅ 𝟭 _ 
Arguments: (X : CommRingCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity.hom.app X = CategoryTheory.inv (AlgebraicGeometry.toSpecΓ X)
-- AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_inv_app
@[simps! hom_app inv_app]
def LocallyRingedSpace.SpecΓIdentity : Spec.toLocallyRingedSpace.rightOp ⋙ Γ ≅ 𝟭 _ 
Arguments: (X : CommRingCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity.inv.app X = AlgebraicGeometry.toSpecΓ X
-- AlgebraicGeometry.Spec_map_localization_isIso
theorem Spec_map_localization_isIso (R : CommRingCat.{u}) (M : Submonoid R)
    (x : PrimeSpectrum (Localization M)) :
    IsIso
      ((Spec.toPresheafedSpace.map
        (CommRingCat.ofHom (algebraMap R (Localization M))).op).stalkMap x) 
Arguments: (R : CommRingCat) (M : Submonoid ↑R) (x : PrimeSpectrum (Localization M))
Goal: CategoryTheory.IsIso
  (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap
    (AlgebraicGeometry.Spec.toPresheafedSpace.map (CommRingCat.ofHom (algebraMap (↑R) (Localization M))).op) x)
-- AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp
@[reassoc]
theorem toPushforwardStalk_comp :
    f ≫ StructureSheaf.toPushforwardStalk f p =
      StructureSheaf.toStalk R p ≫
        (TopCat.Presheaf.stalkFunctor _ _).map (Spec.sheafedSpaceMap f).c 
Arguments: {R S : CommRingCat} (f : R ⟶ S) (p : PrimeSpectrum ↑R)
Goal: CategoryTheory.CategoryStruct.comp f (AlgebraicGeometry.StructureSheaf.toPushforwardStalk f p) =
  CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.StructureSheaf.toStalk (↑R) p)
    ((TopCat.Presheaf.stalkFunctor CommRingCat p).map (AlgebraicGeometry.Spec.sheafedSpaceMap f).c)
-- AlgebraicGeometry.StructureSheaf.algebraMap_pushforward_stalk
theorem algebraMap_pushforward_stalk :
    algebraMap R ((Spec.topMap f _* (structureSheaf S).1).stalk p) =
      f ≫ StructureSheaf.toPushforwardStalk f p 
Arguments: {R S : CommRingCat} (f : R ⟶ S) (p : PrimeSpectrum ↑R)
Goal: algebraMap ↑R
    ↑(((TopCat.Presheaf.pushforward CommRingCat (AlgebraicGeometry.Spec.topMap f)).obj
            (AlgebraicGeometry.Spec.structureSheaf ↑S).val).stalk
        p) =
  CategoryTheory.CategoryStruct.comp f (AlgebraicGeometry.StructureSheaf.toPushforwardStalk f p)
-- AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply
@[simps!]
def toPushforwardStalkAlgHom :
    S →ₐ[R] (Spec.topMap (algebraMap R S) _* (structureSheaf S).1).stalk p 
Arguments: (R S : CommRingCat) (p : PrimeSpectrum ↑R) [Algebra ↑R ↑S]
Goal: ∀ (a : ↑S),
  (AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom R S p) a =
    (((TopCat.Presheaf.pushforward CommRingCat (AlgebraicGeometry.Spec.topMap (algebraMap ↑R ↑S))).obj
            (AlgebraicGeometry.Spec.structureSheaf ↑S).val).germ
        ⟨p, trivial⟩)
      ((AlgebraicGeometry.StructureSheaf.toOpen ↑S ⊤) a)
-- AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux
theorem isLocalizedModule_toPushforwardStalkAlgHom_aux (y) :
    ∃ x : S × p.asIdeal.primeCompl, x.2 • y = toPushforwardStalkAlgHom R S p x.1 
Arguments: (R S : CommRingCat) (p : PrimeSpectrum ↑R) [Algebra ↑R ↑S] (y :
  ↑(((TopCat.Presheaf.pushforward CommRingCat (AlgebraicGeometry.Spec.topMap (algebraMap ↑R ↑S))).obj
          (AlgebraicGeometry.Spec.structureSheaf ↑S).val).stalk
      p))
Goal: ∃ x, x.2 • y = (AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom R S p) x.1
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.zero_mem'
theorem zero_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) :
    (isLocallyFraction 𝒜).pred (0 : ∀ x : U.unop, at x.1) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred 0
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.one_mem'
theorem one_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) :
    (isLocallyFraction 𝒜).pred (1 : ∀ x : U.unop, at x.1) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred 1
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.add_mem'
theorem add_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a + b) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : (x : ↥(Opposite.unop U)) → HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) (ha : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred a) (hb : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred b)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred (a + b)
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.neg_mem'
theorem neg_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : (x : ↥(Opposite.unop U)) → HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) (ha : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred a)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred (-a)
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.mul_mem'
theorem mul_mem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : (x : ↥(Opposite.unop U)) → HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) (ha : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred a) (hb : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred b)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred (a * b)
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_map_apply
@[simps]
def structurePresheafInCommRing : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
Goal: ∀ {X Y : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (i : X ⟶ Y)
  (a : (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structureSheafInType 𝒜).val.obj X),
  ((AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing 𝒜).map i) a =
    (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structureSheafInType 𝒜).val.map i a
-- AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing_obj
@[simps]
def structurePresheafInCommRing : Presheaf CommRingCat (ProjectiveSpectrum.top 𝒜) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structurePresheafInCommRing 𝒜).obj U =
  CommRingCat.of ((AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.structureSheafInType 𝒜).val.obj U)
-- AlgebraicGeometry.Proj.res_apply
@[simp]
theorem Proj.res_apply (x) : ((Proj.structureSheaf 𝒜).1.map i s).1 x = s.1 (i.unop x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {U V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (i : V ⟶ U) (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (x : ↥(Opposite.unop U))
Goal: ↑(((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.map i) s) x = ↑s ((fun x => ⟨↑x, ⋯⟩) x)
-- AlgebraicGeometry.Proj.ext
@[ext] theorem Proj.ext (h : s.1 = t.1) : s = t 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (s t : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (h : ↑s = ↑t)
Goal: s = t
-- AlgebraicGeometry.Proj.add_apply
@[simp] theorem Proj.add_apply : (s + t).1 x = s.1 x + t.1 x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (s t : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (x : ↥(Opposite.unop V))
Goal: ↑(s + t) x = ↑s x + ↑t x
-- AlgebraicGeometry.Proj.mul_apply
@[simp] theorem Proj.mul_apply : (s * t).1 x = s.1 x * t.1 x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (s t : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (x : ↥(Opposite.unop V))
Goal: ↑(s * t) x = ↑s x * ↑t x
-- AlgebraicGeometry.Proj.sub_apply
@[simp] theorem Proj.sub_apply : (s - t).1 x = s.1 x - t.1 x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (s t : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (x : ↥(Opposite.unop V))
Goal: ↑(s - t) x = ↑s x - ↑t x
-- AlgebraicGeometry.Proj.pow_apply
@[simp] theorem Proj.pow_apply (n : ℕ) : (s ^ n).1 x = s.1 x ^ n 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj V)) (x : ↥(Opposite.unop V)) (n : ℕ)
Goal: ↑(s ^ n) x = ↑s x ^ n
-- AlgebraicGeometry.Proj.zero_apply
@[simp] theorem Proj.zero_apply : (0 : (Proj.structureSheaf 𝒜).1.obj V).1 x = 0 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (x : ↥(Opposite.unop V))
Goal: ↑0 x = 0
-- AlgebraicGeometry.Proj.one_apply
@[simp] theorem Proj.one_apply : (1 : (Proj.structureSheaf 𝒜).1.obj V).1 x = 1 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {V : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ} (x : ↥(Opposite.unop V))
Goal: ↑1 x = 1