Theorem: AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem
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
i : ℕ
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, 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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } = 0
tactic: convert Submodule.zero_mem q.1 using 1
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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, 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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk ↑{ deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk ↑{ deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * i, num := ⟨(proj 𝒜 i) 0 ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, 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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk 0 ⟨f ^ i, ⋯⟩ = 0
tactic: simp_rw [map_zero, zero_pow hm.ne']
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
i : ℕ
e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))
⊢ Localization.mk 0 ⟨f ^ i, ⋯⟩ = 0
state_after: no goals
tactic: convert Localization.mk_zero (S := Submonoid.powers f) _ using 1