ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.pi_val
@[simp]
theorem pi_val (s : Finset α) (t : ∀ a, Finset (β a)) : (s.pi t).1 = s.1.pi fun a => (t a).1 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] (s : Finset α) (t : (a : α) → Finset (β a))
Goal: (s.pi t).val = s.val.pi fun a => (t a).val
-- Finset.mem_pi
@[simp]
theorem mem_pi {s : Finset α} {t : ∀ a, Finset (β a)} {f : ∀ a ∈ s, β a} :
    f ∈ s.pi t ↔ ∀ (a) (h : a ∈ s), f a h ∈ t a 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] {s : Finset α} {t : (a : α) → Finset (β a)} {f : (a : α) → a ∈ s → β a}
Goal: f ∈ s.pi t ↔ ∀ (a : α) (h : a ∈ s), f a h ∈ t a
-- Finset.Pi.cons_same
@[simp]
theorem Pi.cons_same (s : Finset α) (a : α) (b : δ a) (f : ∀ a, a ∈ s → δ a) (h : a ∈ insert a s) :
    Pi.cons s a b f a h = b 
Arguments: {α : Type u_1} {δ : α → Sort v} [DecidableEq α] (s : Finset α) (a : α) (b : δ a) (f : (a : α) → a ∈ s → δ a) (h : a ∈ insert a s)
Goal: Finset.Pi.cons s a b f a h = b
-- Finset.Pi.cons_ne
theorem Pi.cons_ne {s : Finset α} {a a' : α} {b : δ a} {f : ∀ a, a ∈ s → δ a} {h : a' ∈ insert a s}
    (ha : a ≠ a') : Pi.cons s a b f a' h = f a' ((mem_insert.1 h).resolve_left ha.symm) 
Arguments: {α : Type u_1} {δ : α → Sort v} [DecidableEq α] {s : Finset α} {a a' : α} {b : δ a} {f : (a : α) → a ∈ s → δ a} {h : a' ∈ insert a s} (ha : a ≠ a')
Goal: Finset.Pi.cons s a b f a' h = f a' ⋯
-- Finset.Pi.cons_injective
theorem Pi.cons_injective {a : α} {b : δ a} {s : Finset α} (hs : a ∉ s) :
    Function.Injective (Pi.cons s a b) 
Arguments: {α : Type u_1} {δ : α → Sort v} [DecidableEq α] {a : α} {b : δ a} {s : Finset α} (hs : a ∉ s)
Goal: Function.Injective (Finset.Pi.cons s a b)
-- Finset.pi_empty
@[simp]
theorem pi_empty {t : ∀ a : α, Finset (β a)} : pi (∅ : Finset α) t = singleton (Pi.empty β) 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] {t : (a : α) → Finset (β a)}
Goal: ∅.pi t = {Finset.Pi.empty β}
-- Finset.pi_nonempty
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
lemma pi_nonempty : (s.pi t).Nonempty ↔ ∀ a ∈ s, (t a).Nonempty 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] {s : Finset α} {t : (a : α) → Finset (β a)}
Goal: (s.pi t).Nonempty ↔ ∀ a ∈ s, (t a).Nonempty
-- Finset.pi_insert
@[simp]
theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
    (ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] [(a : α) → DecidableEq (β a)] {s : Finset α} {t : (a : α) → Finset (β a)} {a : α} (ha : a ∉ s)
Goal: (insert a s).pi t = (t a).biUnion fun b => Finset.image (Finset.Pi.cons s a b) (s.pi t)
-- Finset.pi_singletons
theorem pi_singletons {β : Type*} (s : Finset α) (f : α → β) :
    (s.pi fun a => ({f a} : Finset β)) = {fun a _ => f a} 
Arguments: {α : Type u_1} [DecidableEq α] {β : Type u_2} (s : Finset α) (f : α → β)
Goal: (s.pi fun a => {f a}) = {fun a x => f a}
-- Finset.pi_const_singleton
theorem pi_const_singleton {β : Type*} (s : Finset α) (i : β) :
    (s.pi fun _ => ({i} : Finset β)) = {fun _ _ => i} 
Arguments: {α : Type u_1} [DecidableEq α] {β : Type u_2} (s : Finset α) (i : β)
Goal: (s.pi fun x => {i}) = {fun x x => i}
-- Finset.pi_subset
theorem pi_subset {s : Finset α} (t₁ t₂ : ∀ a, Finset (β a)) (h : ∀ a ∈ s, t₁ a ⊆ t₂ a) :
    s.pi t₁ ⊆ s.pi t₂ 
Arguments: {α : Type u_1} {β : α → Type u} [DecidableEq α] {s : Finset α} (t₁ t₂ : (a : α) → Finset (β a)) (h : ∀ a ∈ s, t₁ a ⊆ t₂ a)
Goal: s.pi t₁ ⊆ s.pi t₂
-- Finset.pi_disjoint_of_disjoint
theorem pi_disjoint_of_disjoint {δ : α → Type*} {s : Finset α} (t₁ t₂ : ∀ a, Finset (δ a)) {a : α}
    (ha : a ∈ s) (h : Disjoint (t₁ a) (t₂ a)) : Disjoint (s.pi t₁) (s.pi t₂) 
Arguments: {α : Type u_1} [DecidableEq α] {δ : α → Type u_2} {s : Finset α} (t₁ t₂ : (a : α) → Finset (δ a)) {a : α} (ha : a ∈ s) (h : Disjoint (t₁ a) (t₂ a))
Goal: Disjoint (s.pi t₁) (s.pi t₂)
-- Finset.mem_piDiag
@[simp] lemma mem_piDiag : f ∈ s.piDiag ι ↔ ∃ a ∈ s, const ι a = f 
Arguments: {α : Type u_1} {ι : Type u_2} [DecidableEq (ι → α)] {s : Finset α} {f : ι → α}
Goal: f ∈ s.piDiag ι ↔ ∃ a ∈ s, Function.const ι a = f
-- Finset.card_piDiag
@[simp] lemma card_piDiag (s : Finset α) (ι : Type*) [DecidableEq (ι → α)] [Nonempty ι] :
    (s.piDiag ι).card = s.card 
Arguments: {α : Type u_1} (s : Finset α) (ι : Type u_3) [DecidableEq (ι → α)] [Nonempty ι]
Goal: (s.piDiag ι).card = s.card
-- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier
@[simp]
theorem mk_mem_carrier (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
    HomogeneousLocalization.mk z ∈ carrier x ↔ z.num.1 ∈ x.1.asHomogeneousIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} (x : ↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toTopCat) (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f))
Goal: HomogeneousLocalization.mk z ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier x ↔
  ↑z.num ∈ (↑x).asHomogeneousIdeal
-- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.isPrime_carrier
theorem isPrime_carrier : Ideal.IsPrime (carrier x) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} (x : ↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toTopCat)
Goal: (AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier x).IsPrime
-- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun_asIdeal
@[simps (config := .lemmasOnly)]
def toFun (x : Proj.T| pbo f) : Spec.T A⁰_ f 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (f : A) (x : ↑↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toPresheafedSpace)
Goal: (AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun f x).asIdeal =
  AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier x
-- AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen
theorem preimage_basicOpen (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
    toFun f ⁻¹' (sbo (HomogeneousLocalization.mk z) : Set (PrimeSpectrum (A⁰_ f))) =
      Subtype.val ⁻¹' (pbo z.num.1 : Set (ProjectiveSpectrum 𝒜)) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] (f : A) (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f))
Goal: AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.toFun f ⁻¹' ↑(PrimeSpectrum.basicOpen (HomogeneousLocalization.mk z)) =
  Subtype.val ⁻¹' ↑(ProjectiveSpectrum.basicOpen 𝒜 ↑z.num)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_apply_asIdeal
@[simps! (config := .lemmasOnly) apply_asIdeal]
def toSpec (f : A) : (Proj.T| pbo f) ⟶ Spec.T A⁰_ f 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (f : A) (x : ↑↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toPresheafedSpace)
Goal: ((AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f) x).asIdeal =
  AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier x
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_preimage_basicOpen
lemma toSpec_preimage_basicOpen {f} (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
    toSpec 𝒜 f ⁻¹' (sbo (HomogeneousLocalization.mk z) : Set (PrimeSpectrum (A⁰_ f))) =
      Subtype.val ⁻¹' (pbo z.num.1 : Set (ProjectiveSpectrum 𝒜)) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f))
Goal: ⇑(AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f) ⁻¹' ↑(PrimeSpectrum.basicOpen (HomogeneousLocalization.mk z)) =
  Subtype.val ⁻¹' ↑(ProjectiveSpectrum.basicOpen 𝒜 ↑z.num)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff
theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔ ∀ i, (HomogeneousLocalization.mk ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
      ⟨f ^ i, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.1 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (a : A)
Goal: a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔
  ∀ (i : ℕ),
    HomogeneousLocalization.mk
        { deg := m * i, num := ⟨(GradedAlgebra.proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔
      ∀ i, (Localization.mk (proj 𝒜 i a ^ m) ⟨f ^ i, ⟨i, rfl⟩⟩ : Localization.Away f) ∈
          algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f) '' { s | s ∈ q.1 } 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (a : A)
Goal: a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔
  ∀ (i : ℕ),
    Localization.mk ((GradedAlgebra.proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈
      ⇑(algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem
theorem mem_carrier_iff_of_mem (hm : 0 < m) (q : Spec.T A⁰_ f) (a : A) {n} (hn : a ∈ 𝒜 n) :
    a ∈ carrier f_deg q ↔
      (HomogeneousLocalization.mk ⟨m * n, ⟨a ^ m, pow_mem_graded m hn⟩,
        ⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (a : A) {n : ℕ} (hn : a ∈ 𝒜 n)
Goal: a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔
  HomogeneousLocalization.mk { deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul
theorem mem_carrier_iff_of_mem_mul (hm : 0 < m)
    (q : Spec.T A⁰_ f) (a : A) {n} (hn : a ∈ 𝒜 (n * m)) :
    a ∈ carrier f_deg q ↔ (HomogeneousLocalization.mk ⟨m * n, ⟨a, mul_comm n m ▸ hn⟩,
        ⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (a : A) {n : ℕ} (hn : a ∈ 𝒜 (n * m))
Goal: a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔
  HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.num_mem_carrier_iff
theorem num_mem_carrier_iff (hm : 0 < m) (q : Spec.T A⁰_ f)
    (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
    z.num.1 ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f))
Goal: ↑z.num ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) {a b : A} (ha : a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q) (hb : b ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q)
Goal: a + b ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
theorem carrier.zero_mem : (0 : A) ∈ carrier f_deg q 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: 0 ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem
theorem carrier.smul_mem (c x : A) (hx : x ∈ carrier f_deg q) : c • x ∈ carrier f_deg q 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace) (c x : A) (hx : x ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q)
Goal: c • x ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.homogeneous
theorem carrier.asIdeal.homogeneous : (carrier.asIdeal f_deg hm q).IsHomogeneous 𝒜 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: Ideal.IsHomogeneous 𝒜 (AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal f_deg hm q)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_not_mem
theorem carrier.denom_not_mem : f ∉ carrier.asIdeal f_deg hm q 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: f ∉ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal f_deg hm q
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.relevant
theorem carrier.relevant : ¬HomogeneousIdeal.irrelevant 𝒜 ≤ carrier.asHomogeneousIdeal f_deg hm q 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: ¬HomogeneousIdeal.irrelevant 𝒜 ≤
    AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asHomogeneousIdeal f_deg hm q
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.ne_top
theorem carrier.asIdeal.ne_top : carrier.asIdeal f_deg hm q ≠ ⊤ 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal f_deg hm q ≠ ⊤
-- AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (q :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: (AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal f_deg hm q).IsPrime
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec
lemma toSpec_fromSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : Spec.T (A⁰_ f)) :
    toSpec 𝒜 f (FromSpec.toFun f_deg hm x) = x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x :
  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj
          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace)
Goal: (AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f)
    (AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.toFun f_deg hm x) =
  x
-- AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec
lemma fromSpec_toSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : Proj.T| pbo f) :
    FromSpec.toFun f_deg hm (toSpec 𝒜 f x) = x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : ↑↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toPresheafedSpace)
Goal: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.toFun f_deg hm
    ((AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f) x) =
  x
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_injective
lemma toSpec_injective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
    Function.Injective (toSpec 𝒜 f) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)
Goal: Function.Injective ⇑(AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_surjective
lemma toSpec_surjective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
    Function.Surjective (toSpec 𝒜 f) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)
Goal: Function.Surjective ⇑(AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_bijective
lemma toSpec_bijective {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
    Function.Bijective (toSpec (𝒜 := 𝒜) (f := f)) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)
Goal: Function.Bijective ⇑(AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f)
-- AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec.image_basicOpen_eq_basicOpen
lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
    toSpec 𝒜 f '' (Subtype.val ⁻¹' (pbo (decompose 𝒜 a i) : Set (ProjectiveSpectrum 𝒜))) =
    (PrimeSpectrum.basicOpen (R := A⁰_ f) <|
      HomogeneousLocalization.mk
        ⟨m * i, ⟨decompose 𝒜 a i ^ m, SetLike.pow_mem_graded _ (Submodule.coe_mem _)⟩,
          ⟨f^i, by rw [mul_comm]; exact SetLike.pow_mem_graded _ f_deg⟩, ⟨i, rfl⟩⟩).1 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {𝒜 : ℕ → Submodule R A} [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (a : A) (i : ℕ)
Goal: ⇑(AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f) ''
    (Subtype.val ⁻¹' ↑(ProjectiveSpectrum.basicOpen 𝒜 ↑(((DirectSum.decompose 𝒜) a) i))) =
  (PrimeSpectrum.basicOpen
      (HomogeneousLocalization.mk
        { deg := m * i, num := ⟨↑(((DirectSum.decompose 𝒜) a) i) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ })).carrier
-- AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ
lemma awayToSection_germ (f x) :
    awayToSection 𝒜 f ≫ (structureSheaf 𝒜).presheaf.germ x =
      (HomogeneousLocalization.mapId 𝒜 (Submonoid.powers_le.mpr x.2)) ≫
        (Proj.stalkIso' 𝒜 x).toCommRingCatIso.inv 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (f : A) (x : ↥(ProjectiveSpectrum.basicOpen 𝒜 f))
Goal: CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection 𝒜 f)
    ((AlgebraicGeometry.ProjectiveSpectrum.Proj.structureSheaf 𝒜).presheaf.germ x) =
  CategoryTheory.CategoryStruct.comp (HomogeneousLocalization.mapId 𝒜 ⋯)
    (AlgebraicGeometry.Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
-- AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk
lemma awayToΓ_ΓToStalk (f) (x) :
    awayToΓ 𝒜 f ≫ (Proj| pbo f).presheaf.Γgerm x =
      HomogeneousLocalization.mapId 𝒜 (Submonoid.powers_le.mpr x.2) ≫
      (Proj.stalkIso' 𝒜 x.1).toCommRingCatIso.inv ≫
      ((Proj.toLocallyRingedSpace 𝒜).restrictStalkIso (Opens.openEmbedding _) x).inv 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] (f : A) (x : ↑↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toPresheafedSpace)
Goal: CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ 𝒜 f)
    (((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).presheaf.Γgerm x) =
  CategoryTheory.CategoryStruct.comp (HomogeneousLocalization.mapId 𝒜 ⋯)
    (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Proj.stalkIso' 𝒜 ↑x).toCommRingCatIso.inv
      ((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrictStalkIso ⋯ x).inv)
-- AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap
lemma toSpec_base_apply_eq_comap {f} (x : Proj| pbo f) :
    (toSpec 𝒜 f).1.base x = PrimeSpectrum.comap (mapId 𝒜 (Submonoid.powers_le.mpr x.2))
      (closedPoint (AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal)) 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} (x : ↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toTopCat)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec 𝒜 f).val.base x =
  (PrimeSpectrum.comap (HomogeneousLocalization.mapId 𝒜 ⋯))
    (LocalRing.closedPoint (HomogeneousLocalization.AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal))
-- AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq
lemma toSpec_base_apply_eq {f} (x : Proj| pbo f) :
    (toSpec 𝒜 f).1.base x = ProjIsoSpecTopComponent.toSpec 𝒜 f x 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} (x : ↑((AlgebraicGeometry.Proj.toLocallyRingedSpace 𝒜).restrict ⋯).toTopCat)
Goal: (AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec 𝒜 f).val.base x =
  (AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec 𝒜 f) x
-- AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_isIso
lemma toSpec_base_isIso {f} {m} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
    IsIso (toSpec 𝒜 f).1.base 
Arguments: {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m)
Goal: CategoryTheory.IsIso (AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec 𝒜 f).val.base