ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.germ_comp_stalkToFiberRingHom
@[simp]
theorem germ_comp_stalkToFiberRingHom (U : Opens (ProjectiveSpectrum.top 𝒜)) (x : U) :
    (Proj.structureSheaf 𝒜).presheaf.germ x ≫ stalkToFiberRingHom 𝒜 x =
      openToLocalization 𝒜 U x x.2 
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)) (x : ↥U)
Goal: CategoryTheory.CategoryStruct.comp ((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ x)
    (AlgebraicGeometry.stalkToFiberRingHom 𝒜 ↑x) =
  AlgebraicGeometry.openToLocalization 𝒜 U ↑x ⋯
-- AlgebraicGeometry.stalkToFiberRingHom_germ'
@[simp]
theorem stalkToFiberRingHom_germ' (U : Opens (ProjectiveSpectrum.top 𝒜))
    (x : ProjectiveSpectrum.top 𝒜) (hx : x ∈ U) (s : (Proj.structureSheaf 𝒜).1.obj (op U)) :
    stalkToFiberRingHom 𝒜 x ((Proj.structureSheaf 𝒜).presheaf.germ ⟨x, hx⟩ s) = (s.1 ⟨x, hx⟩ : _) 
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)) (x : ↑(ProjectiveSpectrum.top 𝒜)) (hx : x ∈ U) (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.stalkToFiberRingHom 𝒜 x)
    (((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ ⟨x, hx⟩) s) =
  ↑s ⟨x, hx⟩
-- AlgebraicGeometry.stalkToFiberRingHom_germ
@[simp]
theorem stalkToFiberRingHom_germ (U : Opens (ProjectiveSpectrum.top 𝒜)) (x : U)
    (s : (Proj.structureSheaf 𝒜).1.obj (op U)) :
    stalkToFiberRingHom 𝒜 x ((Proj.structureSheaf 𝒜).presheaf.germ x s) = s.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] (U : TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜)) (x : ↥U) (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.stalkToFiberRingHom 𝒜 ↑x)
    (((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ x) s) =
  ↑s x
-- AlgebraicGeometry.mem_basicOpen_den
theorem mem_basicOpen_den (x : ProjectiveSpectrum.top 𝒜)
    (f : HomogeneousLocalization.NumDenSameDeg 𝒜 x.asHomogeneousIdeal.toIdeal.primeCompl) :
    x ∈ ProjectiveSpectrum.basicOpen 𝒜 f.den 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (x : ↑(ProjectiveSpectrum.top 𝒜)) (f : HomogeneousLocalization.NumDenSameDeg 𝒜 x.asHomogeneousIdeal.toIdeal.primeCompl)
Goal: x ∈ ProjectiveSpectrum.basicOpen 𝒜 ↑f.den
-- AlgebraicGeometry.homogeneousLocalizationToStalk_stalkToFiberRingHom
lemma homogeneousLocalizationToStalk_stalkToFiberRingHom (x z) :
    homogeneousLocalizationToStalk 𝒜 x (stalkToFiberRingHom 𝒜 x z) = z 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (x : ↑(ProjectiveSpectrum.top 𝒜)) (z : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.stalk x))
Goal: AlgebraicGeometry.homogeneousLocalizationToStalk 𝒜 x ((AlgebraicGeometry.stalkToFiberRingHom 𝒜 x) z) = z
-- AlgebraicGeometry.stalkToFiberRingHom_homogeneousLocalizationToStalk
lemma stalkToFiberRingHom_homogeneousLocalizationToStalk (x z) :
    stalkToFiberRingHom 𝒜 x (homogeneousLocalizationToStalk 𝒜 x z) = z 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (x : ↑(ProjectiveSpectrum.top 𝒜)) (z : HomogeneousLocalization.AtPrime 𝒜 x.asHomogeneousIdeal.toIdeal)
Goal: (AlgebraicGeometry.stalkToFiberRingHom 𝒜 x) (AlgebraicGeometry.homogeneousLocalizationToStalk 𝒜 x z) = z
-- AlgebraicGeometry.Proj.stalkIso'_germ'
@[simp]
theorem Proj.stalkIso'_germ' (U : Opens (ProjectiveSpectrum.top 𝒜))
    (x : ProjectiveSpectrum.top 𝒜) (hx : x ∈ U) (s : (Proj.structureSheaf 𝒜).1.obj (op U)) :
    Proj.stalkIso' 𝒜 x ((Proj.structureSheaf 𝒜).presheaf.germ ⟨x, hx⟩ s) = (s.1 ⟨x, hx⟩ : _) 
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)) (x : ↑(ProjectiveSpectrum.top 𝒜)) (hx : x ∈ U) (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.Proj.stalkIso' 𝒜 x)
    (((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ ⟨x, hx⟩) s) =
  ↑s ⟨x, hx⟩
-- AlgebraicGeometry.Proj.stalkIso'_germ
@[simp]
theorem Proj.stalkIso'_germ (U : Opens (ProjectiveSpectrum.top 𝒜)) (x : U)
    (s : (Proj.structureSheaf 𝒜).1.obj (op U)) :
    Proj.stalkIso' 𝒜 x ((Proj.structureSheaf 𝒜).presheaf.germ x s) = s.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] (U : TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜)) (x : ↥U) (s : ↑((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).val.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.Proj.stalkIso' 𝒜 ↑x)
    (((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ x) s) =
  ↑s x
-- AlgebraicGeometry.Proj.stalkIso'_symm_mk
@[simp]
theorem Proj.stalkIso'_symm_mk (x) (f) :
    (Proj.stalkIso' 𝒜 x).symm (.mk f) = (Proj.structureSheaf 𝒜).presheaf.germ
      ⟨x, mem_basicOpen_den _ x f⟩ (sectionInBasicOpen _ x f) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (x : ↑(ProjectiveSpectrum.top 𝒜)) (f : HomogeneousLocalization.NumDenSameDeg 𝒜 x.asHomogeneousIdeal.toIdeal.primeCompl)
Goal: (AlgebraicGeometry.Proj.stalkIso' 𝒜 x).symm (HomogeneousLocalization.mk f) =
  ((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ ⟨x, ⋯⟩)
    (AlgebraicGeometry.sectionInBasicOpen 𝒜 x f)
-- AlgebraicGeometry.IsAffine.affine
class IsAffine (X : Scheme) : Prop where
  affine : IsIso (ΓSpec.adjunction.unit.app X)
Arguments: {X : AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsAffine X]
Goal: CategoryTheory.IsIso (AlgebraicGeometry.ΓSpec.adjunction.unit.app X)
-- AlgebraicGeometry.Scheme.isoSpec_hom
@[simps! (config := .lemmasOnly) hom]
def Scheme.isoSpec (X : Scheme) [IsAffine X] : X ≅ Spec Γ(X, ⊤) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X]
Goal: X.isoSpec.hom = AlgebraicGeometry.ΓSpec.adjunction.unit.app X
-- AlgebraicGeometry.Scheme.isoSpec_hom_naturality
@[reassoc]
theorem Scheme.isoSpec_hom_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
    X.isoSpec.hom ≫ Spec.map (f.app ⊤) = f ≫ Y.isoSpec.hom 
Arguments: {X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine X] [AlgebraicGeometry.IsAffine Y] (f : X ⟶ Y)
Goal: X.isoSpec.hom ≫ AlgebraicGeometry.Spec.map (AlgebraicGeometry.Scheme.Hom.app f ⊤) = f ≫ Y.isoSpec.hom
-- AlgebraicGeometry.Scheme.isoSpec_inv_naturality
@[reassoc]
theorem Scheme.isoSpec_inv_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
    Spec.map (f.app ⊤) ≫ Y.isoSpec.inv = X.isoSpec.inv ≫ f 
Arguments: {X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine X] [AlgebraicGeometry.IsAffine Y] (f : X ⟶ Y)
Goal: AlgebraicGeometry.Spec.map (AlgebraicGeometry.Scheme.Hom.app f ⊤) ≫ Y.isoSpec.inv = X.isoSpec.inv ≫ f
-- AlgebraicGeometry.AffineScheme.mk_obj
@[simps]
def AffineScheme.mk (X : Scheme) (_ : IsAffine X) : AffineScheme 
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: ∀ (x : AlgebraicGeometry.IsAffine X), (AlgebraicGeometry.AffineScheme.mk X x).obj = X
-- AlgebraicGeometry.mem_Spec_essImage
theorem mem_Spec_essImage (X : Scheme) : X ∈ Scheme.Spec.essImage ↔ IsAffine X 
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: X ∈ AlgebraicGeometry.Scheme.Spec.essImage ↔ AlgebraicGeometry.IsAffine X
-- AlgebraicGeometry.isAffine_of_isIso
theorem isAffine_of_isIso {X Y : Scheme} (f : X ⟶ Y) [IsIso f] [h : IsAffine Y] : IsAffine X 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [CategoryTheory.IsIso f] [h : AlgebraicGeometry.IsAffine Y]
Goal: AlgebraicGeometry.IsAffine X
-- AlgebraicGeometry.AffineScheme.forgetToScheme_map
@[simps!]
def forgetToScheme : AffineScheme ⥤ Scheme 
Goal: ∀ {X Y : CategoryTheory.InducedCategory AlgebraicGeometry.Scheme CategoryTheory.FullSubcategory.obj} (f : X ⟶ Y),
  AlgebraicGeometry.AffineScheme.forgetToScheme.map f = f
-- AlgebraicGeometry.AffineScheme.forgetToScheme_obj
@[simps!]
def forgetToScheme : AffineScheme ⥤ Scheme 
Arguments: (self : CategoryTheory.FullSubcategory AlgebraicGeometry.Scheme.Spec.essImage)
Goal: AlgebraicGeometry.AffineScheme.forgetToScheme.obj self = self.obj
-- AlgebraicGeometry.isAffineOpen_opensRange
theorem isAffineOpen_opensRange {X Y : Scheme} [IsAffine X] (f : X ⟶ Y)
    [H : IsOpenImmersion f] : IsAffineOpen (Scheme.Hom.opensRange f) 
Arguments: {X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine X] (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: AlgebraicGeometry.IsAffineOpen (AlgebraicGeometry.Scheme.Hom.opensRange f)
-- AlgebraicGeometry.isAffineOpen_top
theorem isAffineOpen_top (X : Scheme) [IsAffine X] : IsAffineOpen (⊤ : X.Opens) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X]
Goal: AlgebraicGeometry.IsAffineOpen ⊤
-- AlgebraicGeometry.isBasis_affine_open
theorem isBasis_affine_open (X : Scheme) : Opens.IsBasis X.affineOpens 
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: TopologicalSpace.Opens.IsBasis X.affineOpens
-- AlgebraicGeometry.iSup_affineOpens_eq_top
theorem iSup_affineOpens_eq_top (X : Scheme) : ⨆ i : X.affineOpens, (i : X.Opens) = ⊤ 
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: ⨆ i, ↑i = ⊤
-- AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine
theorem Scheme.map_PrimeSpectrum_basicOpen_of_affine
    (X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) :
    X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X] (f : ↑(AlgebraicGeometry.Scheme.Γ.obj (Opposite.op X)))
Goal: (TopologicalSpace.Opens.map X.isoSpec.hom.val.base).obj (PrimeSpectrum.basicOpen f) = X.basicOpen f
-- AlgebraicGeometry.isBasis_basicOpen
theorem isBasis_basicOpen (X : Scheme) [IsAffine X] :
    Opens.IsBasis (Set.range (X.basicOpen : Γ(X, ⊤) → X.Opens)) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X]
Goal: TopologicalSpace.Opens.IsBasis (Set.range X.basicOpen)
-- AlgebraicGeometry.IsAffineOpen.range_fromSpec
@[simp]
theorem range_fromSpec :
    Set.range hU.fromSpec.1.base = (U : Set X) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: Set.range ⇑hU.fromSpec.val.base = ↑U
-- AlgebraicGeometry.IsAffineOpen.opensRange_fromSpec
@[simp]
theorem opensRange_fromSpec : Scheme.Hom.opensRange hU.fromSpec = U 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: AlgebraicGeometry.Scheme.Hom.opensRange hU.fromSpec = U
-- AlgebraicGeometry.IsAffineOpen.map_fromSpec
@[reassoc (attr := simp)]
theorem map_fromSpec {V : X.Opens} (hV : IsAffineOpen V) (f : op U ⟶ op V) :
    Spec.map (X.presheaf.map f) ≫ hU.fromSpec = hV.fromSpec 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) {V : X.Opens} (hV : AlgebraicGeometry.IsAffineOpen V) (f : Opposite.op U ⟶ Opposite.op V)
Goal: AlgebraicGeometry.Spec.map (X.presheaf.map f) ≫ hU.fromSpec = hV.fromSpec
-- AlgebraicGeometry.IsAffineOpen.isCompact
protected theorem isCompact :
    IsCompact (U : Set X) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: IsCompact ↑U
-- AlgebraicGeometry.IsAffineOpen.image_of_isOpenImmersion
theorem image_of_isOpenImmersion (f : X ⟶ Y) [H : IsOpenImmersion f] :
    IsAffineOpen (f ''ᵁ U) 
Arguments: {X Y : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: AlgebraicGeometry.IsAffineOpen ((AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj U)
-- AlgebraicGeometry.IsAffineOpen.preimage_of_isIso
theorem preimage_of_isIso {U : Y.Opens} (hU : IsAffineOpen U) (f : X ⟶ Y) [IsIso f] :
    IsAffineOpen (f ⁻¹ᵁ U) 
Arguments: {X Y : AlgebraicGeometry.Scheme} {U : Y.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : X ⟶ Y) [CategoryTheory.IsIso f]
Goal: AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.val.base).obj U)
-- AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
theorem _root_.AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
    (f : AlgebraicGeometry.Scheme.Hom X Y) [H : IsOpenImmersion f] {U : X.Opens} :
    IsAffineOpen (f ''ᵁ U) ↔ IsAffineOpen U 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : X.Opens}
Goal: AlgebraicGeometry.IsAffineOpen (f.opensFunctor.obj U) ↔ AlgebraicGeometry.IsAffineOpen U
-- AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_apply_coe_coe
@[simps]
def _root_.AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv (f : X ⟶ Y) [H : IsOpenImmersion f] :
    X.affineOpens ≃ { U : Y.affineOpens // U ≤ f.opensRange } 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : ↑X.affineOpens)
Goal: ↑↑((AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv f) U) = (AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj ↑U
-- AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv_symm_apply_coe
@[simps]
def _root_.AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv (f : X ⟶ Y) [H : IsOpenImmersion f] :
    X.affineOpens ≃ { U : Y.affineOpens // U ≤ f.opensRange } 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] (U : { U // ↑U ≤ AlgebraicGeometry.Scheme.Hom.opensRange f })
Goal: ↑((AlgebraicGeometry.IsOpenImmersion.affineOpensEquiv f).symm U) = (TopologicalSpace.Opens.map f.val.base).obj ↑↑U
-- AlgebraicGeometry.affineOpensRestrict_apply_coe_coe
@[simps! apply_coe_coe]
def _root_.AlgebraicGeometry.affineOpensRestrict {X : Scheme.{u}} (U : X.Opens) :
    U.toScheme.affineOpens ≃ { V : X.affineOpens // V ≤ U } 
Arguments: {X : AlgebraicGeometry.Scheme} (U : X.Opens)
Goal: ∀ (a : ↑(↑U).affineOpens),
  ↑↑((AlgebraicGeometry.affineOpensRestrict U) a) = (AlgebraicGeometry.Scheme.Hom.opensFunctor U.ι).obj ↑a
-- AlgebraicGeometry.affineOpensRestrict_symm_apply_coe
@[simp]
lemma _root_.AlgebraicGeometry.affineOpensRestrict_symm_apply_coe
    {X : Scheme.{u}} (U : X.Opens) (V) :
    ((affineOpensRestrict U).symm V).1 = U.ι ⁻¹ᵁ V 
Arguments: {X : AlgebraicGeometry.Scheme} (U : X.Opens) (V : { V // ↑V ≤ U })
Goal: ↑((AlgebraicGeometry.affineOpensRestrict U).symm V) = (TopologicalSpace.Opens.map U.ι.val.base).obj ↑↑V
-- AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self
@[simp]
theorem fromSpec_preimage_self :
    hU.fromSpec ⁻¹ᵁ U = ⊤ 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: (TopologicalSpace.Opens.map hU.fromSpec.val.base).obj U = ⊤
-- AlgebraicGeometry.IsAffineOpen.SpecΓIdentity_hom_app_fromSpec
theorem SpecΓIdentity_hom_app_fromSpec :
    (Scheme.ΓSpecIso Γ(X, U)).hom ≫ hU.fromSpec.app U =
      (Spec Γ(X, U)).presheaf.map (eqToHom hU.fromSpec_preimage_self).op 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: (AlgebraicGeometry.Scheme.ΓSpecIso (X.presheaf.obj (Opposite.op U))).hom ≫
    AlgebraicGeometry.Scheme.Hom.app hU.fromSpec U =
  (AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).presheaf.map (CategoryTheory.eqToHom ⋯).op
-- AlgebraicGeometry.IsAffineOpen.fromSpec_app_self
@[elementwise]
theorem fromSpec_app_self :
    hU.fromSpec.app U = (Scheme.ΓSpecIso Γ(X, U)).inv ≫
      (Spec Γ(X, U)).presheaf.map (eqToHom hU.fromSpec_preimage_self).op 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U)
Goal: AlgebraicGeometry.Scheme.Hom.app hU.fromSpec U =
  (AlgebraicGeometry.Scheme.ΓSpecIso (X.presheaf.obj (Opposite.op U))).inv ≫
    (AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).presheaf.map (CategoryTheory.eqToHom ⋯).op
-- AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen'
theorem fromSpec_preimage_basicOpen' :
    hU.fromSpec ⁻¹ᵁ X.basicOpen f = (Spec Γ(X, U)).basicOpen ((Scheme.ΓSpecIso Γ(X, U)).inv f) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (TopologicalSpace.Opens.map hU.fromSpec.val.base).obj (X.basicOpen f) =
  (AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).basicOpen
    ((AlgebraicGeometry.Scheme.ΓSpecIso (X.presheaf.obj (Opposite.op U))).inv f)
-- AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen
theorem fromSpec_preimage_basicOpen :
    hU.fromSpec ⁻¹ᵁ X.basicOpen f = PrimeSpectrum.basicOpen f 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (TopologicalSpace.Opens.map hU.fromSpec.val.base).obj (X.basicOpen f) = PrimeSpectrum.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen
theorem fromSpec_image_basicOpen :
    hU.fromSpec ''ᵁ (PrimeSpectrum.basicOpen f) = X.basicOpen f 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.Scheme.Hom.opensFunctor hU.fromSpec).obj (PrimeSpectrum.basicOpen f) = X.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app
@[simp]
theorem basicOpen_fromSpec_app :
    (Spec Γ(X, U)).basicOpen (hU.fromSpec.app U f) = PrimeSpectrum.basicOpen f 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).basicOpen
    ((AlgebraicGeometry.Scheme.Hom.app hU.fromSpec U) f) =
  PrimeSpectrum.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.basicOpen
theorem basicOpen :
    IsAffineOpen (X.basicOpen f) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: AlgebraicGeometry.IsAffineOpen (X.basicOpen f)
-- AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage
theorem ι_basicOpen_preimage (r : Γ(X, ⊤)) :
    IsAffineOpen ((X.basicOpen r).ι ⁻¹ᵁ U) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (r : ↑(X.presheaf.obj (Opposite.op ⊤)))
Goal: AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map (X.basicOpen r).ι.val.base).obj U)
-- AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le
theorem exists_basicOpen_le {V : X.Opens} (x : V) (h : ↑x ∈ U) :
    ∃ f : Γ(X, U), X.basicOpen f ≤ V ∧ ↑x ∈ X.basicOpen f 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) {V : X.Opens} (x : ↥V) (h : ↑x ∈ U)
Goal: ∃ f, X.basicOpen f ≤ V ∧ ↑x ∈ X.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen
theorem isLocalization_basicOpen :
    IsLocalization.Away f Γ(X, X.basicOpen f) 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: IsLocalization.Away f ↑(X.presheaf.obj (Opposite.op (X.basicOpen f)))
-- AlgebraicGeometry.IsAffineOpen.appLE_eq_away_map
lemma appLE_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (hU : IsAffineOpen U)
    {V : X.Opens} (hV : IsAffineOpen V) (e) (r : Γ(Y, U)) :
    letI := hU.isLocalization_basicOpen r
    letI := hV.isLocalization_basicOpen (f.appLE U V e r)
    f.appLE (Y.basicOpen r) (X.basicOpen (f.appLE U V e r)) (by simp [Scheme.Hom.appLE]) =
        IsLocalization.Away.map _ _ (f.appLE U V e) r 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) {U : Y.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) {V : X.Opens} (hV : AlgebraicGeometry.IsAffineOpen V) (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (r : ↑(Y.presheaf.obj (Opposite.op U)))
Goal: AlgebraicGeometry.Scheme.Hom.appLE f (Y.basicOpen r) (X.basicOpen ((AlgebraicGeometry.Scheme.Hom.appLE f U V e) r)) ⋯ =
  IsLocalization.Away.map (↑(Y.presheaf.obj (Opposite.op (Y.basicOpen r))))
    (↑(X.presheaf.obj (Opposite.op (X.basicOpen ((AlgebraicGeometry.Scheme.Hom.appLE f U V e) r)))))
    (AlgebraicGeometry.Scheme.Hom.appLE f U V e) r
-- AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen
theorem isLocalization_of_eq_basicOpen {V : X.Opens} (i : V ⟶ U) (e : V = X.basicOpen f) :
    @IsLocalization.Away _ _ f Γ(X, V) _ (X.presheaf.map i.op).toAlgebra 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U))) {V : X.Opens} (i : V ⟶ U) (e : V = X.basicOpen f)
Goal: IsLocalization.Away f ↑(X.presheaf.obj (Opposite.op V))
-- AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen
theorem basicOpen_basicOpen_is_basicOpen (g : Γ(X, X.basicOpen f)) :
    ∃ f' : Γ(X, U), X.basicOpen f' = X.basicOpen g 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : ↑(X.presheaf.obj (Opposite.op U))) (g : ↑(X.presheaf.obj (Opposite.op (X.basicOpen f))))
Goal: ∃ f', X.basicOpen f' = X.basicOpen g
-- AlgebraicGeometry.exists_basicOpen_le_affine_inter
theorem _root_.AlgebraicGeometry.exists_basicOpen_le_affine_inter
    {V : X.Opens} (hV : IsAffineOpen V) (x : X) (hx : x ∈ U ⊓ V) :
    ∃ (f : Γ(X, U)) (g : Γ(X, V)), X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) {V : X.Opens} (hV : AlgebraicGeometry.IsAffineOpen V) (x : ↑↑X.toPresheafedSpace) (hx : x ∈ U ⊓ V)
Goal: ∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf
theorem fromSpec_primeIdealOf (x : U) :
    hU.fromSpec.val.base (hU.primeIdealOf x) = x.1 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (x : ↥U)
Goal: hU.fromSpec.val.base (hU.primeIdealOf x) = ↑x
-- AlgebraicGeometry.IsAffineOpen.isLocalization_stalk'
theorem isLocalization_stalk' (y : PrimeSpectrum Γ(X, U)) (hy : hU.fromSpec.1.base y ∈ U) :
    @IsLocalization.AtPrime
      (R := Γ(X, U))
      (S := X.presheaf.stalk <| hU.fromSpec.1.base y) _ _
      ((TopCat.Presheaf.algebra_section_stalk X.presheaf _)) y.asIdeal _ 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (y : PrimeSpectrum ↑(X.presheaf.obj (Opposite.op U))) (hy : hU.fromSpec.val.base y ∈ U)
Goal: IsLocalization.AtPrime (↑(X.presheaf.stalk (hU.fromSpec.val.base y))) y.asIdeal
-- AlgebraicGeometry.IsAffineOpen.isLocalization_stalk
theorem isLocalization_stalk (x : U) :
    IsLocalization.AtPrime (X.presheaf.stalk x) (hU.primeIdealOf x).asIdeal 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (x : ↥U)
Goal: IsLocalization.AtPrime (↑(X.presheaf.stalk ↑x)) (hU.primeIdealOf x).asIdeal
-- AlgebraicGeometry.Scheme.affineBasicOpen_coe
@[simps]
def _root_.AlgebraicGeometry.Scheme.affineBasicOpen
    (X : Scheme) {U : X.affineOpens} (f : Γ(X, U)) : X.affineOpens 
Arguments: (X : AlgebraicGeometry.Scheme) {U : ↑X.affineOpens} (f : ↑(X.presheaf.obj (Opposite.op ↑U)))
Goal: ↑(X.affineBasicOpen f) = X.basicOpen f
-- AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff
theorem basicOpen_union_eq_self_iff (s : Set Γ(X, U)) :
    ⨆ f : s, X.basicOpen (f : Γ(X, U)) = U ↔ Ideal.span s = ⊤ 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (s : Set ↑(X.presheaf.obj (Opposite.op U)))
Goal: ⨆ f, X.basicOpen ↑f = U ↔ Ideal.span s = ⊤
-- AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff
theorem self_le_basicOpen_union_iff (s : Set Γ(X, U)) :
    (U ≤ ⨆ f : s, X.basicOpen f.1) ↔ Ideal.span s = ⊤ 
Arguments: {X : AlgebraicGeometry.Scheme} {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (s : Set ↑(X.presheaf.obj (Opposite.op U)))
Goal: U ≤ ⨆ f, X.basicOpen ↑f ↔ Ideal.span s = ⊤