ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.Scheme.ofRestrict_app
@[simp]
lemma Scheme.ofRestrict_app (V) :
    (X.ofRestrict h).app V = X.presheaf.map (h.isOpenMap.adjunction.counit.app V).op  
Arguments: {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : X.Opens)
Goal: AlgebraicGeometry.Scheme.Hom.app (X.ofRestrict h) V = X.presheaf.map (⋯.adjunction.counit.app V).op
-- AlgebraicGeometry.Scheme.ofRestrict_appLE
@[simp]
lemma Scheme.ofRestrict_appLE (V W e) :
    (X.ofRestrict h).appLE V W e = X.presheaf.map
      (homOfLE (show X.ofRestrict h ''ᵁ _ ≤ _ by exact Set.image_subset_iff.mpr e)).op 
Arguments: {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V : X.Opens) (W : (X.restrict h).Opens) (e : W ≤ (TopologicalSpace.Opens.map (X.ofRestrict h).val.base).obj V)
Goal: AlgebraicGeometry.Scheme.Hom.appLE (X.ofRestrict h) V W e = X.presheaf.map (CategoryTheory.homOfLE ⋯).op
-- AlgebraicGeometry.Scheme.ofRestrict_appIso
@[simp]
lemma Scheme.ofRestrict_appIso (U) :
    (X.ofRestrict h).appIso U = Iso.refl _ 
Arguments: {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f)
Goal: ∀ (U_1 : (X.restrict h).Opens),
  AlgebraicGeometry.Scheme.Hom.appIso (X.ofRestrict h) U_1 =
    CategoryTheory.Iso.refl
      (X.presheaf.obj (Opposite.op ((AlgebraicGeometry.Scheme.Hom.opensFunctor (X.ofRestrict h)).obj U_1)))
-- AlgebraicGeometry.Scheme.restrict_presheaf_map
@[simp]
lemma Scheme.restrict_presheaf_map (V W) (i : V ⟶ W) :
    (X.restrict h).presheaf.map i = X.presheaf.map (homOfLE (show X.ofRestrict h ''ᵁ W.unop ≤
      X.ofRestrict h ''ᵁ V.unop from Set.image_subset _ i.unop.le)).op 
Arguments: {U : TopCat} (X : AlgebraicGeometry.Scheme) {f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace} (h : OpenEmbedding ⇑f) (V W : (TopologicalSpace.Opens ↑↑(X.restrict h).toPresheafedSpace)ᵒᵖ) (i : V ⟶ W)
Goal: (X.restrict h).presheaf.map i = X.presheaf.map (CategoryTheory.homOfLE ⋯).op
-- AlgebraicGeometry.IsOpenImmersion.to_iso
theorem to_iso {X Y : Scheme.{u}} (f : X ⟶ Y) [h : IsOpenImmersion f] [Epi f.1.base] : IsIso f 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [h : AlgebraicGeometry.IsOpenImmersion f] [CategoryTheory.Epi f.val.base]
Goal: CategoryTheory.IsIso f
-- AlgebraicGeometry.IsOpenImmersion.of_stalk_iso
theorem of_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) (hf : OpenEmbedding f.1.base)
    [∀ x, IsIso (f.stalkMap x)] : IsOpenImmersion f 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (hf : OpenEmbedding ⇑f.val.base) [∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.Scheme.Hom.stalkMap f x)]
Goal: AlgebraicGeometry.IsOpenImmersion f
-- AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso
theorem iff_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) :
    IsOpenImmersion f ↔ OpenEmbedding f.1.base ∧ ∀ x, IsIso (f.stalkMap x) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
Goal: AlgebraicGeometry.IsOpenImmersion f ↔
  OpenEmbedding ⇑f.val.base ∧
    ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.Scheme.Hom.stalkMap f x)
-- AlgebraicGeometry.isIso_iff_isOpenImmersion
theorem _root_.AlgebraicGeometry.isIso_iff_isOpenImmersion {X Y : Scheme.{u}} (f : X ⟶ Y) :
    IsIso f ↔ IsOpenImmersion f ∧ Epi f.1.base 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
Goal: CategoryTheory.IsIso f ↔ AlgebraicGeometry.IsOpenImmersion f ∧ CategoryTheory.Epi f.val.base
-- AlgebraicGeometry.isIso_iff_stalk_iso
theorem _root_.AlgebraicGeometry.isIso_iff_stalk_iso {X Y : Scheme.{u}} (f : X ⟶ Y) :
    IsIso f ↔ IsIso f.1.base ∧ ∀ x, IsIso (f.stalkMap x) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
Goal: CategoryTheory.IsIso f ↔
  CategoryTheory.IsIso f.val.base ∧
    ∀ (x : ↑↑X.toPresheafedSpace), CategoryTheory.IsIso (AlgebraicGeometry.Scheme.Hom.stalkMap f x)
-- AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left
theorem range_pullback_snd_of_left :
    Set.range (pullback.snd f g).1.base = (g ⁻¹ᵁ f.opensRange).1 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: Set.range ⇑(CategoryTheory.Limits.pullback.snd f g).val.base =
  ((TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)).carrier
-- AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_snd_of_left
theorem opensRange_pullback_snd_of_left :
    (pullback.snd f g).opensRange = g ⁻¹ᵁ f.opensRange 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: AlgebraicGeometry.Scheme.Hom.opensRange (CategoryTheory.Limits.pullback.snd f g) =
  (TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)
-- AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right
theorem range_pullback_fst_of_right :
    Set.range (pullback.fst g f).1.base =
      ((Opens.map g.1.base).obj ⟨Set.range f.1.base, H.base_open.isOpen_range⟩).1 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: Set.range ⇑(CategoryTheory.Limits.pullback.fst g f).val.base =
  ((TopologicalSpace.Opens.map g.val.base).obj { carrier := Set.range ⇑f.val.base, is_open' := ⋯ }).carrier
-- AlgebraicGeometry.IsOpenImmersion.opensRange_pullback_fst_of_right
theorem opensRange_pullback_fst_of_right :
    (pullback.fst g f).opensRange = g ⁻¹ᵁ f.opensRange 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: AlgebraicGeometry.Scheme.Hom.opensRange (CategoryTheory.Limits.pullback.fst g f) =
  (TopologicalSpace.Opens.map g.val.base).obj (AlgebraicGeometry.Scheme.Hom.opensRange f)
-- AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left
theorem range_pullback_to_base_of_left :
    Set.range (pullback.fst f g ≫ f).1.base =
      Set.range f.1.base ∩ Set.range g.1.base 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: Set.range ⇑(CategoryTheory.Limits.pullback.fst f g ≫ f).val.base = Set.range ⇑f.val.base ∩ Set.range ⇑g.val.base
-- AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right
theorem range_pullback_to_base_of_right :
    Set.range (pullback.fst g f ≫ g).1.base =
      Set.range g.1.base ∩ Set.range f.1.base 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f]
Goal: Set.range ⇑(CategoryTheory.Limits.pullback.fst g f ≫ g).val.base = Set.range ⇑g.val.base ∩ Set.range ⇑f.val.base
-- AlgebraicGeometry.IsOpenImmersion.lift_fac
@[simp, reassoc]
theorem lift_fac (H' : Set.range g.1.base ⊆ Set.range f.1.base) : lift f g H' ≫ f = g 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base)
Goal: AlgebraicGeometry.IsOpenImmersion.lift f g H' ≫ f = g
-- AlgebraicGeometry.IsOpenImmersion.lift_uniq
theorem lift_uniq (H' : Set.range g.1.base ⊆ Set.range f.1.base) (l : Y ⟶ X) (hl : l ≫ f = g) :
    l = lift f g H' 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [H : AlgebraicGeometry.IsOpenImmersion f] (H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) (l : Y ⟶ X) (hl : l ≫ f = g)
Goal: l = AlgebraicGeometry.IsOpenImmersion.lift f g H'
-- AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac
@[simp, reassoc]
lemma isoOfRangeEq_hom_fac {X Y Z : Scheme.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)
    [IsOpenImmersion f] [IsOpenImmersion g] (e : Set.range f.1.base = Set.range g.1.base) :
    (isoOfRangeEq f g e).hom ≫ g = f 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base)
Goal: (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).hom ≫ g = f
-- AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_inv_fac
@[simp, reassoc]
lemma isoOfRangeEq_inv_fac {X Y Z : Scheme.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)
    [IsOpenImmersion f] [IsOpenImmersion g] (e : Set.range f.1.base = Set.range g.1.base) :
    (isoOfRangeEq f g e).inv ≫ f = g 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsOpenImmersion g] (e : Set.range ⇑f.val.base = Set.range ⇑g.val.base)
Goal: (AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq f g e).inv ≫ f = g
-- AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux
theorem app_eq_invApp_app_of_comp_eq_aux {X Y U : Scheme.{u}} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X)
    (H : fg = f ≫ g) [h : IsOpenImmersion g] (V : U.Opens) :
    f ⁻¹ᵁ V = fg ⁻¹ᵁ (g ''ᵁ V) 
Arguments: {X Y U : AlgebraicGeometry.Scheme} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X) (H : fg = f ≫ g) [h : AlgebraicGeometry.IsOpenImmersion g] (V : U.Opens)
Goal: (TopologicalSpace.Opens.map f.val.base).obj V =
  (TopologicalSpace.Opens.map fg.val.base).obj ((AlgebraicGeometry.Scheme.Hom.opensFunctor g).obj V)
-- AlgebraicGeometry.IsOpenImmersion.app_eq_appIso_inv_app_of_comp_eq
theorem app_eq_appIso_inv_app_of_comp_eq {X Y U : Scheme.{u}} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X)
    (H : fg = f ≫ g) [h : IsOpenImmersion g] (V : U.Opens) :
    f.app V = (g.appIso V).inv ≫ fg.app (g ''ᵁ V) ≫ Y.presheaf.map
      (eqToHom <| IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux f g fg H V).op 
Arguments: {X Y U : AlgebraicGeometry.Scheme} (f : Y ⟶ U) (g : U ⟶ X) (fg : Y ⟶ X) (H : fg = f ≫ g) [h : AlgebraicGeometry.IsOpenImmersion g] (V : U.Opens)
Goal: AlgebraicGeometry.Scheme.Hom.app f V =
  (AlgebraicGeometry.Scheme.Hom.appIso g V).inv ≫
    AlgebraicGeometry.Scheme.Hom.app fg ((AlgebraicGeometry.Scheme.Hom.opensFunctor g).obj V) ≫
      Y.presheaf.map (CategoryTheory.eqToHom ⋯).op
-- AlgebraicGeometry.IsOpenImmersion.lift_app
theorem lift_app {X Y U : Scheme.{u}} (f : U ⟶ Y) (g : X ⟶ Y) [IsOpenImmersion f] (H)
    (V : U.Opens) :
    (IsOpenImmersion.lift f g H).app V = (f.appIso V).inv ≫ g.app (f ''ᵁ V) ≫
      X.presheaf.map (eqToHom <| IsOpenImmersion.app_eq_invApp_app_of_comp_eq_aux _ _ _
        (IsOpenImmersion.lift_fac f g H).symm V).op 
Arguments: {X Y U : AlgebraicGeometry.Scheme} (f : U ⟶ Y) (g : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (H : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base) (V : U.Opens)
Goal: AlgebraicGeometry.Scheme.Hom.app (AlgebraicGeometry.IsOpenImmersion.lift f g H) V =
  (AlgebraicGeometry.Scheme.Hom.appIso f V).inv ≫
    AlgebraicGeometry.Scheme.Hom.app g ((AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj V) ≫
      X.presheaf.map (CategoryTheory.eqToHom ⋯).op
-- AlgebraicGeometry.IsOpenImmersion.ΓIso_inv
@[simp]
lemma ΓIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
    (ΓIso f U).inv = f.appLE (f.opensRange ⊓ U) (f⁻¹ᵁ U)
      (by rw [← f.image_preimage_eq_opensRange_inter, f.preimage_image_eq]) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (U : Y.Opens)
Goal: (AlgebraicGeometry.IsOpenImmersion.ΓIso f U).inv =
  AlgebraicGeometry.Scheme.Hom.appLE f (AlgebraicGeometry.Scheme.Hom.opensRange f ⊓ U)
    ((TopologicalSpace.Opens.map f.val.base).obj U) ⋯
-- AlgebraicGeometry.IsOpenImmersion.map_ΓIso_inv
@[reassoc, elementwise]
lemma map_ΓIso_inv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
    Y.presheaf.map (homOfLE inf_le_right).op ≫ (ΓIso f U).inv = f.app U 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (U : Y.Opens)
Goal: Y.presheaf.map (CategoryTheory.homOfLE ⋯).op ≫ (AlgebraicGeometry.IsOpenImmersion.ΓIso f U).inv =
  AlgebraicGeometry.Scheme.Hom.app f U
-- AlgebraicGeometry.IsOpenImmersion.ΓIso_hom_map
@[reassoc, elementwise]
lemma ΓIso_hom_map {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] (U : Y.Opens) :
    f.app U ≫ (ΓIso f U).hom = Y.presheaf.map (homOfLE inf_le_right).op 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsOpenImmersion f] (U : Y.Opens)
Goal: AlgebraicGeometry.Scheme.Hom.app f U ≫ (AlgebraicGeometry.IsOpenImmersion.ΓIso f U).hom =
  Y.presheaf.map (CategoryTheory.homOfLE ⋯).op
-- AlgebraicGeometry.Scheme.image_basicOpen
theorem image_basicOpen {X Y : Scheme.{u}} (f : X ⟶ Y) [H : IsOpenImmersion f] {U : X.Opens}
    (r : Γ(X, U)) :
    f ''ᵁ X.basicOpen r = Y.basicOpen ((f.appIso U).inv r) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : X.Opens} (r : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj (X.basicOpen r) =
  Y.basicOpen ((AlgebraicGeometry.Scheme.Hom.appIso f U).inv r)
-- AlgebraicGeometry.germ_injective_of_isIntegral
theorem germ_injective_of_isIntegral [IsIntegral X] {U : X.Opens} (x : U) :
    Function.Injective (X.presheaf.germ x) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] {U : X.Opens} (x : ↥U)
Goal: Function.Injective ⇑(X.presheaf.germ x)
-- AlgebraicGeometry.Scheme.germToFunctionField_injective
theorem Scheme.germToFunctionField_injective [IsIntegral X] (U : X.Opens) [Nonempty U] :
    Function.Injective (X.germToFunctionField U) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] (U : X.Opens) [Nonempty ↑↑(↑U).toPresheafedSpace]
Goal: Function.Injective ⇑(X.germToFunctionField U)
-- AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion
theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
    [hX : IrreducibleSpace X] [IrreducibleSpace Y] :
    f.1.base (genericPoint X) = genericPoint Y 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] [hX : IrreducibleSpace ↑↑X.toPresheafedSpace] [IrreducibleSpace ↑↑Y.toPresheafedSpace]
Goal: f.val.base (genericPoint ↑↑X.toPresheafedSpace) = genericPoint ↑↑Y.toPresheafedSpace
-- AlgebraicGeometry.genericPoint_eq_bot_of_affine
@[simp]
theorem genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
    genericPoint (Spec R) = (⊥ : PrimeSpectrum R) 
Arguments: (R : CommRingCat) [IsDomain ↑R]
Goal: genericPoint ↑↑(AlgebraicGeometry.Spec R).toPresheafedSpace = ⊥
-- AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint
theorem IsAffineOpen.primeIdealOf_genericPoint {X : Scheme} [IsIntegral X] {U : X.Opens}
    (hU : IsAffineOpen U) [h : Nonempty U] :
    hU.primeIdealOf
        ⟨genericPoint X,
          ((genericPoint_spec X).mem_open_set_iff U.isOpen).mpr (by simpa using h)⟩ =
      genericPoint (Spec Γ(X, U)) 
Arguments: {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral X] {U : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) [h : Nonempty ↑↑(↑U).toPresheafedSpace]
Goal: hU.primeIdealOf ⟨genericPoint ↑↑X.toPresheafedSpace, ⋯⟩ =
  genericPoint ↑↑(AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).toPresheafedSpace
-- AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen
theorem functionField_isFractionRing_of_isAffineOpen [IsIntegral X] (U : X.Opens)
    (hU : IsAffineOpen U) [Nonempty U] :
    IsFractionRing Γ(X, U) X.functionField 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] (U : X.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) [Nonempty ↑↑(↑U).toPresheafedSpace]
Goal: IsFractionRing ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField
-- ProjectiveSpectrum.isPrime
@[ext]
structure ProjectiveSpectrum where
  asHomogeneousIdeal : HomogeneousIdeal 𝒜
  isPrime : asHomogeneousIdeal.toIdeal.IsPrime
  not_irrelevant_le : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ asHomogeneousIdeal
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (self : ProjectiveSpectrum 𝒜)
Goal: self.asHomogeneousIdeal.toIdeal.IsPrime
-- ProjectiveSpectrum.not_irrelevant_le
@[ext]
structure ProjectiveSpectrum where
  asHomogeneousIdeal : HomogeneousIdeal 𝒜
  isPrime : asHomogeneousIdeal.toIdeal.IsPrime
  not_irrelevant_le : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ asHomogeneousIdeal
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (self : ProjectiveSpectrum 𝒜)
Goal: ¬HomogeneousIdeal.irrelevant 𝒜 ≤ self.asHomogeneousIdeal
-- ProjectiveSpectrum.mem_zeroLocus
@[simp]
theorem mem_zeroLocus (x : ProjectiveSpectrum 𝒜) (s : Set A) :
    x ∈ zeroLocus 𝒜 s ↔ s ⊆ x.asHomogeneousIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (x : ProjectiveSpectrum 𝒜) (s : Set A)
Goal: x ∈ ProjectiveSpectrum.zeroLocus 𝒜 s ↔ s ⊆ ↑x.asHomogeneousIdeal
-- ProjectiveSpectrum.zeroLocus_span
@[simp]
theorem zeroLocus_span (s : Set A) : zeroLocus 𝒜 (Ideal.span s) = zeroLocus 𝒜 s 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (s : Set A)
Goal: ProjectiveSpectrum.zeroLocus 𝒜 ↑(Ideal.span s) = ProjectiveSpectrum.zeroLocus 𝒜 s
-- ProjectiveSpectrum.coe_vanishingIdeal
theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
    (vanishingIdeal t : Set A) =
      { f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (t : Set (ProjectiveSpectrum 𝒜))
Goal: ↑(ProjectiveSpectrum.vanishingIdeal t) = {f | ∀ x ∈ t, f ∈ x.asHomogeneousIdeal}
-- ProjectiveSpectrum.mem_vanishingIdeal
theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
    f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (t : Set (ProjectiveSpectrum 𝒜)) (f : A)
Goal: f ∈ ProjectiveSpectrum.vanishingIdeal t ↔ ∀ x ∈ t, f ∈ x.asHomogeneousIdeal
-- ProjectiveSpectrum.vanishingIdeal_singleton
@[simp]
theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
    vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (x : ProjectiveSpectrum 𝒜)
Goal: ProjectiveSpectrum.vanishingIdeal {x} = x.asHomogeneousIdeal
-- ProjectiveSpectrum.subset_zeroLocus_iff_le_vanishingIdeal
theorem subset_zeroLocus_iff_le_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (I : Ideal A) :
    t ⊆ zeroLocus 𝒜 I ↔ I ≤ (vanishingIdeal t).toIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (t : Set (ProjectiveSpectrum 𝒜)) (I : Ideal A)
Goal: t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 ↑I ↔ I ≤ (ProjectiveSpectrum.vanishingIdeal t).toIdeal
-- ProjectiveSpectrum.gc_ideal
theorem gc_ideal :
    @GaloisConnection (Ideal A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
      (fun I => zeroLocus 𝒜 I) fun t => (vanishingIdeal t).toIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
Goal: GaloisConnection (fun I => ProjectiveSpectrum.zeroLocus 𝒜 ↑I) fun t => (ProjectiveSpectrum.vanishingIdeal t).toIdeal
-- ProjectiveSpectrum.gc_set
theorem gc_set :
    @GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
      (fun s => zeroLocus 𝒜 s) fun t => vanishingIdeal t 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
Goal: GaloisConnection (fun s => ProjectiveSpectrum.zeroLocus 𝒜 s) fun t => ↑(ProjectiveSpectrum.vanishingIdeal t)
-- ProjectiveSpectrum.gc_homogeneousIdeal
theorem gc_homogeneousIdeal :
    @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
      (fun I => zeroLocus 𝒜 I) fun t => vanishingIdeal t 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
Goal: GaloisConnection (fun I => ProjectiveSpectrum.zeroLocus 𝒜 ↑I) fun t => ProjectiveSpectrum.vanishingIdeal t
-- ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal
theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A) :
    t ⊆ zeroLocus 𝒜 s ↔ s ⊆ vanishingIdeal t 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (t : Set (ProjectiveSpectrum 𝒜)) (s : Set A)
Goal: t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 s ↔ s ⊆ ↑(ProjectiveSpectrum.vanishingIdeal t)
-- ProjectiveSpectrum.subset_vanishingIdeal_zeroLocus
theorem subset_vanishingIdeal_zeroLocus (s : Set A) : s ⊆ vanishingIdeal (zeroLocus 𝒜 s) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (s : Set A)
Goal: s ⊆ ↑(ProjectiveSpectrum.vanishingIdeal (ProjectiveSpectrum.zeroLocus 𝒜 s))
-- ProjectiveSpectrum.ideal_le_vanishingIdeal_zeroLocus
theorem ideal_le_vanishingIdeal_zeroLocus (I : Ideal A) :
    I ≤ (vanishingIdeal (zeroLocus 𝒜 I)).toIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (I : Ideal A)
Goal: I ≤ (ProjectiveSpectrum.vanishingIdeal (ProjectiveSpectrum.zeroLocus 𝒜 ↑I)).toIdeal
-- ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
theorem homogeneousIdeal_le_vanishingIdeal_zeroLocus (I : HomogeneousIdeal 𝒜) :
    I ≤ vanishingIdeal (zeroLocus 𝒜 I) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (I : HomogeneousIdeal 𝒜)
Goal: I ≤ ProjectiveSpectrum.vanishingIdeal (ProjectiveSpectrum.zeroLocus 𝒜 ↑I)
-- ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal
theorem subset_zeroLocus_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
    t ⊆ zeroLocus 𝒜 (vanishingIdeal t) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (t : Set (ProjectiveSpectrum 𝒜))
Goal: t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 ↑(ProjectiveSpectrum.vanishingIdeal t)
-- ProjectiveSpectrum.zeroLocus_anti_mono
theorem zeroLocus_anti_mono {s t : Set A} (h : s ⊆ t) : zeroLocus 𝒜 t ⊆ zeroLocus 𝒜 s 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {s t : Set A} (h : s ⊆ t)
Goal: ProjectiveSpectrum.zeroLocus 𝒜 t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 s
-- ProjectiveSpectrum.zeroLocus_anti_mono_ideal
theorem zeroLocus_anti_mono_ideal {s t : Ideal A} (h : s ≤ t) :
    zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {s t : Ideal A} (h : s ≤ t)
Goal: ProjectiveSpectrum.zeroLocus 𝒜 ↑t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 ↑s
-- ProjectiveSpectrum.zeroLocus_anti_mono_homogeneousIdeal
theorem zeroLocus_anti_mono_homogeneousIdeal {s t : HomogeneousIdeal 𝒜} (h : s ≤ t) :
    zeroLocus 𝒜 (t : Set A) ⊆ zeroLocus 𝒜 (s : Set A) 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {s t : HomogeneousIdeal 𝒜} (h : s ≤ t)
Goal: ProjectiveSpectrum.zeroLocus 𝒜 ↑t ⊆ ProjectiveSpectrum.zeroLocus 𝒜 ↑s
-- ProjectiveSpectrum.vanishingIdeal_anti_mono
theorem vanishingIdeal_anti_mono {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t) :
    vanishingIdeal t ≤ vanishingIdeal s 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {s t : Set (ProjectiveSpectrum 𝒜)} (h : s ⊆ t)
Goal: ProjectiveSpectrum.vanishingIdeal t ≤ ProjectiveSpectrum.vanishingIdeal s
-- ProjectiveSpectrum.zeroLocus_bot
theorem zeroLocus_bot : zeroLocus 𝒜 ((⊥ : Ideal A) : Set A) = Set.univ 
Arguments: {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜]
Goal: ProjectiveSpectrum.zeroLocus 𝒜 ↑⊥ = Set.univ