Theorem: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
File: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean

tactic: rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ (∀ (i : ℕ),
      HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) x ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal) ∨
    ∀ (i : ℕ),
      HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) y ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 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, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal

R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ ∀ (b : ℕ),
    b ≠ ny →
      HomogeneousLocalization.mk { deg := m * b, num := ⟨(proj 𝒜 b) y ^ m, ⋯⟩, den := ⟨f ^ b, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal

R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ ∀ (b : ℕ),
    b ≠ nx →
      HomogeneousLocalization.mk { deg := m * b, num := ⟨(proj 𝒜 b) x ^ m, ⋯⟩, den := ⟨f ^ b, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal

tactic: apply q.2.mem_or_mem
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 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, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal ∨
    HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 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, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal

tactic: convert hxy (nx + ny) using 1
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 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, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk { deg := m * nx, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨(proj 𝒜 (nx + ny)) (x * y) ^ m, ⋯⟩, den := ⟨f ^ (nx + ny), ⋯⟩, den_mem := ⋯ }

tactic: dsimp
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk { deg := m * nx, num := ⟨(proj 𝒜 nx) x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨(proj 𝒜 ny) y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨(proj 𝒜 (nx + ny)) (x * y) ^ m, ⋯⟩, den := ⟨f ^ (nx + ny), ⋯⟩, den_mem := ⋯ }
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk
        { deg := m * nx, num := ⟨↑(((decompose 𝒜) x) nx) ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk
        { deg := m * ny, num := ⟨↑(((decompose 𝒜) y) ny) ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨↑(((decompose 𝒜) (x * y)) (nx + ny)) ^ m, ⋯⟩, den := ⟨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]
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk
        { deg := m * nx, num := ⟨↑(((decompose 𝒜) x) nx) ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk
        { deg := m * ny, num := ⟨↑(((decompose 𝒜) y) ny) ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨↑(((decompose 𝒜) (x * y)) (nx + ny)) ^ m, ⋯⟩, den := ⟨f ^ (nx + ny), ⋯⟩,
        den_mem := ⋯ }
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk { deg := m * nx, num := ⟨x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨x ^ m * y ^ m, ⋯⟩, den := ⟨f ^ nx * f ^ ny, ⋯⟩, den_mem := ⋯ }

tactic: simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
  HomogeneousLocalization.val_mul, Localization.mk_mul]
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ HomogeneousLocalization.mk { deg := m * nx, num := ⟨x ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * ny, num := ⟨y ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨x ^ m * y ^ m, ⋯⟩, den := ⟨f ^ nx * f ^ ny, ⋯⟩, den_mem := ⋯ }
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ Localization.mk (x ^ m * y ^ m) (⟨f ^ nx, ⋯⟩ * ⟨f ^ ny, ⋯⟩) = Localization.mk (x ^ m * y ^ m) ⟨f ^ nx * f ^ ny, ⋯⟩

tactic: simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
e_1✝ : ↑(CommRingCat.of (A⁰_ f)) = HomogeneousLocalization 𝒜 (Submonoid.powers f)
⊢ Localization.mk (x ^ m * y ^ m) (⟨f ^ nx, ⋯⟩ * ⟨f ^ ny, ⋯⟩) = Localization.mk (x ^ m * y ^ m) ⟨f ^ nx * f ^ ny, ⋯⟩
state_after: no goals

tactic: intro n hn
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ ∀ (b : ℕ),
    b ≠ nx →
      HomogeneousLocalization.mk { deg := m * b, num := ⟨(proj 𝒜 b) x ^ m, ⋯⟩, den := ⟨f ^ b, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

tactic: convert q.1.zero_mem using 1
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } = 0

tactic: rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
  HomogeneousLocalization.val_zero]
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } = 0
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk ↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    0

tactic: simp_rw [proj_apply]
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk ↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    0
state_after: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk (↑(((decompose 𝒜) x) n) ^ m) ⟨f ^ n, ⋯⟩ = 0

tactic: convert mk_zero (S := Submonoid.powers f) _
state_before: case h.e'_4.h
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk (↑(((decompose 𝒜) x) n) ^ m) ⟨f ^ n, ⋯⟩ = 0
state_after: case h.e'_2.h.e'_4
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ ↑(((decompose 𝒜) x) n) ^ m = 0

tactic: rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm.ne']
state_before: case h.e'_2.h.e'_4
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ ↑(((decompose 𝒜) x) n) ^ m = 0
state_after: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ x ∈ 𝒜 nx

tactic: first | exact hnx | exact hny
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ x ∈ 𝒜 nx
state_after: no goals

tactic: exact hnx
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ x ∈ 𝒜 nx
state_after: no goals

tactic: exact hny
state_before: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : Homogeneous 𝒜 x
x✝ : Homogeneous 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ ny
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ y ∈ 𝒜 ny
state_after: no goals