-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
theorem num_mem_carrier_iff (hm : 0 < m) (q : Spec.T A⁰_ f)
    (z : HomogeneousLocalization.NumDenSameDeg 𝒜 (.powers f)) :
    z.num.1 ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

Proof state:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ ↑z.num ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

Tactic:
convert mem_carrier_iff_of_mem_mul f_deg hm q z.num.1 (n := n) ?_ using 2

Result:
case h.e'_2.h.e'_5
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ HomogeneousLocalization.mk z =
    HomogeneousLocalization.mk { deg := m * n, num := ⟨↑z.num, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }

case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ ↑z.num ∈ 𝒜 (n * m)

Proof state:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this✝ : f ^ n ≠ 0
this : n • m = z.deg
⊢ ↑z.num ∈ 𝒜 z.deg

Tactic:
exact z.num.2

Result:
no goals

Proof state:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
⊢ ↑z.num ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

Tactic:
have : f ^ n ≠ 0 := fun e ↦ by
  have := HomogeneousLocalization.subsingleton 𝒜 (x := .powers f) ⟨n, e⟩
  exact IsEmpty.elim (inferInstanceAs (IsEmpty (PrimeSpectrum (A⁰_ f)))) q

Result:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ ↑z.num ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

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).toPresheafedSpace
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
e : f ^ n = 0
⊢ False

Tactic:
have := HomogeneousLocalization.subsingleton 𝒜 (x := .powers f) ⟨n, e⟩

Result:
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
e : f ^ n = 0
this : Subsingleton (HomogeneousLocalization 𝒜 (Submonoid.powers f))
⊢ False

Proof state:
case h.e'_2.h.e'_5.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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ (HomogeneousLocalization.mk z).val =
    (HomogeneousLocalization.mk { deg := m * n, num := ⟨↑z.num, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }).val

Tactic:
simp only [hn, HomogeneousLocalization.val_mk]

Result:
no 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).toPresheafedSpace
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ ↑z.num ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

Tactic:
obtain ⟨n, hn : f ^ n = _⟩ := z.den_mem

Result:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
⊢ ↑z.num ∈ carrier f_deg q ↔ HomogeneousLocalization.mk z ∈ q.asIdeal

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).toPresheafedSpace
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
e : f ^ n = 0
this : Subsingleton (HomogeneousLocalization 𝒜 (Submonoid.powers f))
⊢ False

Tactic:
exact IsEmpty.elim (inferInstanceAs (IsEmpty (PrimeSpectrum (A⁰_ f)))) q

Result:
no goals

Proof state:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ ↑z.num ∈ 𝒜 (n * m)

Tactic:
have := degree_eq_of_mem_mem 𝒜 (SetLike.pow_mem_graded n f_deg) (hn.symm ▸ z.den.2) this

Result:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this✝ : f ^ n ≠ 0
this : n • m = z.deg
⊢ ↑z.num ∈ 𝒜 (n * m)

Proof state:
case h.e'_2.h.e'_5
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ HomogeneousLocalization.mk z =
    HomogeneousLocalization.mk { deg := m * n, num := ⟨↑z.num, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }

Tactic:
apply HomogeneousLocalization.val_injective

Result:
case h.e'_2.h.e'_5.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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this : f ^ n ≠ 0
⊢ (HomogeneousLocalization.mk z).val =
    (HomogeneousLocalization.mk { deg := m * n, num := ⟨↑z.num, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }).val

Proof state:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this✝ : f ^ n ≠ 0
this : n • m = z.deg
⊢ ↑z.num ∈ 𝒜 (n * m)

Tactic:
rw [← smul_eq_mul, this]

Result:
case intro
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
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
n : ℕ
hn : f ^ n = ↑z.den
this✝ : f ^ n ≠ 0
this : n • m = z.deg
⊢ ↑z.num ∈ 𝒜 z.deg