l3lab/ntp-mathlib huggingface dataset:
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        

Proof state:
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).toSheafedSpace.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)) =
    Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f)))
⊢ Quotient.mk''
        { deg := m * nx,
          num :=
            { val := ↑(((decompose 𝒜) x) nx) ^ m,
              property :=
                (_ :
                  ↑((LinearMap.comp (DFinsupp.lapply nx) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) x) ^ m ∈
                    𝒜 (m • nx)) },
          den := { val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) },
          den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) }) } *
      Quotient.mk''
        { deg := m * ny,
          num :=
            { val := ↑(((decompose 𝒜) y) ny) ^ m,
              property :=
                (_ :
                  ↑((LinearMap.comp (DFinsupp.lapply ny) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                    𝒜 (m • ny)) },
          den := { val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) },
          den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) }) } =
    Quotient.mk''
      { deg := m * (nx + ny),
        num :=
          { val := ↑(((decompose 𝒜) (x * y)) (nx + ny)) ^ m,
            property :=
              (_ :
                ↑((LinearMap.comp (DFinsupp.lapply (nx + ny)) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) (x * y)) ^
                    m ∈
                  𝒜 (m • (nx + ny))) },
        den := { val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) ∈ 𝒜 (m * (nx + ny))) },
        den_mem :=
          (_ :
            ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ (nx + ny), property := (_ : f ^ (nx + ny) ∈ 𝒜 (m * (nx + ny))) }) }

Next 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]
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        

Proof state:
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).toSheafedSpace.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)) =
    Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f)))
⊢ Quotient.mk''
        { deg := m * nx, num := { val := x ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * nx)) (x ^ m)) },
          den := { val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) },
          den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) }) } *
      Quotient.mk''
        { deg := m * ny, num := { val := y ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * ny)) (y ^ m)) },
          den := { val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) },
          den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) }) } =
    Quotient.mk''
      { deg := m * (nx + ny),
        num := { val := x ^ m * y ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (x ^ m * y ^ m)) },
        den := { val := f ^ nx * f ^ ny, property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny)) },
        den_mem :=
          (_ :
            ↑{ val := f ^ nx * f ^ ny, property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny)) } ∈
              Submonoid.powers f) }

Next tactic:
simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        

Proof state:
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).toSheafedSpace.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)) =
    Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f)))
⊢ Localization.mk (x ^ m * y ^ m)
      ({ val := f ^ nx,
          property :=
            (_ :
              ↑{ deg := m * nx, num := { val := x ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * nx)) (x ^ m)) },
                      den := { val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) },
                      den_mem :=
                        (_ :
                          ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ nx, property := (_ : f ^ nx ∈ 𝒜 (m * nx)) }) }.den ∈
                Submonoid.powers f) } *
        { val := f ^ ny,
          property :=
            (_ :
              ↑{ deg := m * ny, num := { val := y ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * ny)) (y ^ m)) },
                      den := { val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) },
                      den_mem :=
                        (_ :
                          ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ ny, property := (_ : f ^ ny ∈ 𝒜 (m * ny)) }) }.den ∈
                Submonoid.powers f) }) =
    Localization.mk (x ^ m * y ^ m)
      { val := f ^ nx * f ^ ny,
        property :=
          (_ :
            ↑{ deg := m * (nx + ny),
                    num := { val := x ^ m * y ^ m, property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (x ^ m * y ^ m)) },
                    den :=
                      { val := f ^ nx * f ^ ny, property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny)) },
                    den_mem :=
                      (_ :
                        ↑{ val := f ^ nx * f ^ ny,
                              property := (_ : (fun x => x ∈ 𝒜 (m * (nx + ny))) (f ^ nx * f ^ ny)) } ∈
                          Submonoid.powers f) }.den ∈
              Submonoid.powers f) }

Next tactic:
simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      

Proof state:
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).toSheafedSpace.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 →
      Quotient.mk''
          { deg := m * b,
            num :=
              { val := (proj 𝒜 b) y ^ m,
                property :=
                  (_ :
                    ↑((LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                      𝒜 (m • b)) },
            den := { val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) },
            den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) }) } ∈
        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).toSheafedSpace.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 →
      Quotient.mk''
          { deg := m * b,
            num :=
              { val := (proj 𝒜 b) x ^ m,
                property :=
                  (_ :
                    ↑((LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) x) ^ m ∈
                      𝒜 (m • b)) },
            den := { val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) },
            den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) }) } ∈
        q.asIdeal

Next tactic:
all_goals
        intro n hn; convert q.1.zero_mem using 1
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
        convert mk_zero (S := Submonoid.powers f) _
        rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
        · first | exact hnx | exact hny
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        

Proof state:
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).toSheafedSpace.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 →
      Quotient.mk''
          { deg := m * b,
            num :=
              { val := (proj 𝒜 b) y ^ m,
                property :=
                  (_ :
                    ↑((LinearMap.comp (DFinsupp.lapply b) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                      𝒜 (m • b)) },
            den := { val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) },
            den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ b, property := (_ : f ^ b ∈ 𝒜 (m * b)) }) } ∈
        q.asIdeal

Next tactic:
intro n hn
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; 

Proof state:
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).toSheafedSpace.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
⊢ Quotient.mk''
      { deg := m * n,
        num :=
          { val := (proj 𝒜 n) y ^ m,
            property :=
              (_ :
                ↑((LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                  𝒜 (m • n)) },
        den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) }) } ∈
    q.asIdeal

Next tactic:
convert q.1.zero_mem using 1
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; convert q.1.zero_mem using 1
        

Proof state:
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).toSheafedSpace.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✝ :
  Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) =
    ↑(CommRingCat.of (A⁰_ f))
⊢ Quotient.mk''
      { deg := m * n,
        num :=
          { val := (proj 𝒜 n) y ^ m,
            property :=
              (_ :
                ↑((LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                  𝒜 (m • n)) },
        den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) },
        den_mem := (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) }) } =
    0

Next tactic:
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; convert q.1.zero_mem using 1
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
        

Proof state:
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).toSheafedSpace.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✝ :
  Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) =
    ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk (↑(((decompose 𝒜) y) n) ^ m)
      { val := f ^ n,
        property :=
          (_ :
            ↑{ deg := m * n,
                    num :=
                      { val := (proj 𝒜 n) y ^ m,
                        property :=
                          (_ :
                            ↑((LinearMap.comp (DFinsupp.lapply n) (AlgHom.toLinearMap ↑(decomposeAlgEquiv 𝒜))) y) ^ m ∈
                              𝒜 (m • n)) },
                    den := { val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) },
                    den_mem :=
                      (_ : ∃ y, (fun x => f ^ x) y = ↑{ val := f ^ n, property := (_ : f ^ n ∈ 𝒜 (m * n)) }) }.den ∈
              Submonoid.powers f) } =
    0

Next tactic:
convert mk_zero (S := Submonoid.powers f) _
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; convert q.1.zero_mem using 1
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
        convert mk_zero (S := Submonoid.powers f) _
        

Proof state:
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).toSheafedSpace.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✝ :
  Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) =
    ↑(CommRingCat.of (A⁰_ f))
⊢ ↑(((decompose 𝒜) y) n) ^ m = 0

Next tactic:
rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; convert q.1.zero_mem using 1
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
        convert mk_zero (S := Submonoid.powers f) _
        rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
        · 

Proof state:
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).toSheafedSpace.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✝ :
  Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) =
    ↑(CommRingCat.of (A⁰_ f))
⊢ y ∈ 𝒜 ny

Next tactic:
first | exact hnx | exact hny
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
  (carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
    (carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
    show (∀ i, _ ∈ _) ∨ ∀ i, _ ∈ _ by
      rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
      · apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
        dsimp
        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]
        simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.mul_val, mk_mul]
        simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
      all_goals
        intro n hn; convert q.1.zero_mem using 1
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk'',
          HomogeneousLocalization.zero_val]; simp_rw [proj_apply]
        convert mk_zero (S := Submonoid.powers f) _
        rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm]
        · first | 

Proof state:
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).toSheafedSpace.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✝ :
  Quotient (Setoid.ker (HomogeneousLocalization.NumDenSameDeg.embedding 𝒜 (Submonoid.powers f))) =
    ↑(CommRingCat.of (A⁰_ f))
⊢ y ∈ 𝒜 ny

Next tactic:
exact hnx