awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
⊢ ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → m_1 • x + m' • x ∈ carrier f_deg q

Tactic:
fun _ _ => carrier.add_mem f_deg q

Premises used: Set.Mem, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x : A
hx : x ∈ carrier f_deg q
⊢ ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → m_1 • x + m' • x ∈ carrier f_deg q

Tactic:
exact fun _ _ => carrier.add_mem f_deg q

Premises used: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
i : ℕ
a : A
ha : a ∈ asIdeal f_deg hm q
j : ℕ
h : i = j
⊢ HomogeneousLocalization.mk
      { deg := m * i,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i))
            (HPow.hPow (α := A)
              ((proj 𝒜 i : (a : A) → A)
                  (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) a : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) i) : A) :
                A)
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * i)) (f ^ i) ⋯, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
simpa only [proj_apply, decompose_coe, of_eq_same] using ha _

Premises used: DirectSum.of_eq_same, DirectSum.decompose_coe, GradedRing.proj_apply
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
i : ℕ
a : A
ha : a ∈ asIdeal f_deg hm q
j : ℕ
h : ¬i = j
⊢ HomogeneousLocalization.mk
      { deg := m * j,
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * j))
            (HPow.hPow (α := A)
              ((proj 𝒜 j : (a : A) → A)
                  (↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) a : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) i) : A) :
                A)
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * j)) (f ^ j) ⋯, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
simpa only [proj_apply, decompose_of_mem_ne 𝒜 (SetLike.coe_mem (decompose 𝒜 a i)) h,
      zero_pow hm.ne', map_zero] using carrier.zero_mem f_deg hm q j

Premises used: SetLike.coe_mem, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem, map_zero, LT.lt.ne', DirectSum.decompose_of_mem_ne, DFunLike.coe, zero_pow, GradedRing.proj_apply, DirectSum.decompose
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
⊢ HomogeneousLocalization.mk
      { deg := m * m,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
rid m
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
h :
  Eq (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) 1
    (HomogeneousLocalization.mk
      { deg := m * m,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ })
⊢ 1 ∈ q.asIdeal

Tactic:
convert! rid m
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
h :
  Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      (↑(NumDenSameDeg.num
            { deg := m * m,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
        A)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
        (↑(NumDenSameDeg.den
              { deg := m * m,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
          A)
        ⋯))
⊢ Eq (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1)) 1
    (HomogeneousLocalization.mk
      { deg := m * m,
        num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ })

Tactic:
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_one,
          HomogeneousLocalization.val_mk]

Premises used: HomogeneousLocalization.val_one, HomogeneousLocalization.val_mk, HomogeneousLocalization.ext_iff_val
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
h :
  Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) f : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) m) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) ⋯))
⊢ Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      (↑(NumDenSameDeg.num
            { deg := m * m,
              num :=
                Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m) ⋯,
              den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
        A)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x)
        (↑(NumDenSameDeg.den
              { deg := m * m,
                num :=
                  Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (HPow.hPow (α := A) ((proj 𝒜 m : (a : A) → A) f : A) m)
                    ⋯,
                den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * m)) (f ^ m) ⋯, den_mem := ⋯ }) :
          A)
        ⋯))

Tactic:
dsimp
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
h :
  Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk (f ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) ⋯))
⊢ Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk
      ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) f : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) m) : A) ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) ⋯))

Tactic:
simp_rw [decompose_of_mem_same _ f_deg]

Premises used: DirectSum.decompose_of_mem_same
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
rid : f ∈ asIdeal f_deg hm q
⊢ Eq (α := Localization (Submonoid.powers f)) 1
    (Localization.mk (f ^ m)
      (Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := Submonoid A) (Submonoid.powers f) x) (f ^ m) ⋯))

Tactic:
simp
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
⊢ ¬HomogeneousIdeal.irrelevant 𝒜 ≤ asHomogeneousIdeal f_deg hm q

Tactic:
fun rid => carrier.denom_notMem f_deg hm q <| rid <| DirectSum.decompose_of_mem_ne 𝒜 f_deg hm.ne'

Premises used: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_notMem, LT.lt.ne', DirectSum.decompose_of_mem_ne, Not
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
⊢ asIdeal f_deg hm q ≠ ⊤

Tactic:
fun rid =>
  carrier.denom_notMem f_deg hm q (rid.symm ▸ Submodule.mem_top)

Premises used: Eq.rec, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_notMem, Eq.symm, Submodule.mem_top, Not
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
rw :
  HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ } ∈
      q.asIdeal
rw₁ :
  ∀ (b : ℕ),
    b ≠ ny →
      HomogeneousLocalization.mk
          { deg := m * b,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (HPow.hPow (α := A) ((proj 𝒜 b : (a : A) → A) y : A) m) ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (f ^ b) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
rw₂ :
  ∀ (b : ℕ),
    b ≠ nx →
      HomogeneousLocalization.mk
          { deg := m * b,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (HPow.hPow (α := A) ((proj 𝒜 b : (a : A) → A) x : A) m) ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * b)) (f ^ b) ⋯, den_mem := ⋯ } ∈
        q.asIdeal
⊢ (∀ (x_1 : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x_1,
            num :=
              Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (HPow.hPow (α := A) ((proj 𝒜 x_1 : (a : A) → A) x : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * x_1)) (f ^ x_1) ⋯, den_mem := ⋯ } ∈
        q.asIdeal) ∨
    ∀ (x : ℕ),
      HomogeneousLocalization.mk
          { deg := m * x,
            num :=
              Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (HPow.hPow (α := A) ((proj 𝒜 x : (a : A) → A) y : A) m)
                ⋯,
            den := Subtype.mk (p := fun (x_1 : A) ↦ x_1 ∈ 𝒜 (m * x)) (f ^ x) ⋯, den_mem := ⋯ } ∈
        q.asIdeal

Tactic:
rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]

Premises used: and_forall_ne, and_iff_left
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
apply :
  HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) ∈
    q.asIdeal
⊢ HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ } ∈
      q.asIdeal

Tactic:
apply q.2.mem_or_mem

Premises used: PrimeSpectrum.isPrime, Ideal.IsPrime.mem_or_mem
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h : IsHomogeneousElem 𝒜 x
h_1 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            (HPow.hPow (α := A) ((proj 𝒜 (nx + ny) : (a : A) → A) (x * y) : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
hxy (nx + ny)

Premises used: HAdd.hAdd
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h_1 : IsHomogeneousElem 𝒜 x
h_2 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
h :
  HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            (HPow.hPow (α := A) ((proj 𝒜 (nx + ny) : (a : A) → A) (x * y) : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }
⊢ HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) ∈
    q.asIdeal

Tactic:
convert! hxy (nx + ny) using 1

Premises used: HAdd.hAdd
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h_1 : IsHomogeneousElem 𝒜 x
h_2 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
h :
  HMul.hMul (α := A⁰_ f)
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) nx) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) y : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) ny) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) (x * y) : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a))
                    (nx + ny)) :
                A) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }
⊢ HMul.hMul (α := (↑(CommRingCat.of (A⁰_ f)) : Type u_1))
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (HPow.hPow (α := A) ((proj 𝒜 nx : (a : A) → A) x : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (HPow.hPow (α := A) ((proj 𝒜 ny : (a : A) → A) y : A) m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            (HPow.hPow (α := A) ((proj 𝒜 (nx + ny) : (a : A) → A) (x * y) : A) m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }

Tactic:
dsimp
-- Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)
x y : A
h_1 : IsHomogeneousElem 𝒜 x
h_2 : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
h :
  HMul.hMul (α := A⁰_ f)
      (HomogeneousLocalization.mk
        { deg := m * nx, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (x ^ m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny, num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (y ^ m) ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (x ^ m * y ^ m) ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny) ⋯, den_mem := ⋯ }
⊢ HMul.hMul (α := A⁰_ f)
      (HomogeneousLocalization.mk
        { deg := m * nx,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) x : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) nx) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * nx)) (f ^ nx) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) })
      (HomogeneousLocalization.mk
        { deg := m * ny,
          num :=
            Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny))
              ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) y : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a)) ny) : A) ^ m)
              ⋯,
          den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * ny)) (f ^ ny) ⋯, den_mem := ⋯ :
          NumDenSameDeg 𝒜 (Submonoid.powers f) }) =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny),
        num :=
          Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny)))
            ((↑((((decompose 𝒜 : (a : A) → ⨁ (i : ℕ), ↥(𝒜 i)) (x * y) : ⨁ (i : ℕ), ↥(𝒜 i)) : (a : ℕ) → ↥(𝒜 a))
                    (nx + ny)) :
                A) ^
              m)
            ⋯,
        den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 (m * (nx + ny))) (f ^ (nx + ny)) ⋯, den_mem := ⋯ }

Tactic:
simp_rw [decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
          decompose_of_mem_same 𝒜 (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny),
          mul_pow, pow_add]

Premises used: DirectSum.decompose_of_mem_same, pow_add, mul_pow, SetLike.GradedMul.mul_mem