Theorem: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff_of_mem_mul
File: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean

Proof steps:
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ f ^ n ∈ 𝒜 (m * n)
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ f ^ n ∈ 𝒜 (n * m)
tactic: rw [mul_comm]
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ f ^ n ∈ 𝒜 (n * m)
state_after: no goals
tactic: mem_tac
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ a ∈ carrier f_deg q ↔
    HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, 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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ (HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m ∈ q.asIdeal) =
    (HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal)
tactic: rw [mem_carrier_iff_of_mem f_deg hm q a hn, iff_iff_eq, eq_comm,
    ← Ideal.IsPrime.pow_mem_iff_mem (α := A⁰_ f) inferInstance m hm]
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ (HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m ∈ q.asIdeal) =
    (HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ } ∈
      q.asIdeal)
state_after: case e_a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m =
    HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ }
tactic: congr 1
state_before: case e_a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m =
    HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ }
state_after: case e_a.a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ HomogeneousLocalization.val
      (HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m) =
    (HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ }).val
tactic: apply HomogeneousLocalization.val_injective
state_before: case e_a.a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ HomogeneousLocalization.val
      (HomogeneousLocalization.mk { deg := m * n, num := ⟨a, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ^ m) =
    (HomogeneousLocalization.mk { deg := m * (n * m), num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ (n * m), ⋯⟩, den_mem := ⋯ }).val
state_after: case e_a.a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ Localization.mk (a ^ m) (⟨f ^ n, ⋯⟩ ^ m) = Localization.mk (a ^ m) ⟨(f ^ n) ^ m, ⋯⟩
tactic: simp only [HomogeneousLocalization.val_mk, HomogeneousLocalization.val_pow,
    Localization.mk_pow, pow_mul]
state_before: case e_a.a
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
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ Localization.mk (a ^ m) (⟨f ^ n, ⋯⟩ ^ m) = Localization.mk (a ^ m) ⟨(f ^ n) ^ m, ⋯⟩
state_after: no goals
tactic: rfl