-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
theorem mem_carrier_iff' (q : Spec.T A⁰_ f) (a : A) :
    a ∈ carrier f_deg q ↔
      ∀ i, (Localization.mk (proj 𝒜 i a ^ m) ⟨f ^ i, ⟨i, rfl⟩⟩ : Localization.Away f) ∈
          algebraMap (HomogeneousLocalization.Away 𝒜 f) (Localization.Away f) '' { s | s ∈ q.1 } :=
  (mem_carrier_iff f_deg q a).trans
    (by
      constructor <;> intro h i <;> specialize h i
      · rw [Set.mem_image]; refine ⟨_, h, rfl⟩
      · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
        change x ∈ q.asIdeal at h
        convert h
        rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk]
        dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)

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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
⊢ (∀ (i : ℕ),
      HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
        q.asIdeal) ↔
    ∀ (i : ℕ),
      Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}

Tactic:
constructor <;> intro h i <;> specialize h i

Result:
case mp
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h :
  HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
⊢ Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}

case mpr
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h : Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Proof state:
case mp
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h :
  HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
⊢ ∃ x ∈ {s | s ∈ q.asIdeal},
    (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩

Tactic:
refine ⟨_, h, rfl⟩

Result:
no goals

Proof state:
case 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ = HomogeneousLocalization.val x

Tactic:
rw [← hx]

Result:
case 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ (algebraMap (A⁰_ f) (Localization.Away f)) x = HomogeneousLocalization.val x

Proof state:
case mp
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h :
  HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
⊢ Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}

Tactic:
rw [Set.mem_image]

Result:
case mp
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h :
  HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
⊢ ∃ x ∈ {s | s ∈ q.asIdeal},
    (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩

Proof state:
case mpr.intro.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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
h : x ∈ {s | s ∈ q.asIdeal}
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Tactic:
change x ∈ q.asIdeal at h

Result:
case mpr.intro.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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Proof state:
case 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } = x

Tactic:
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk]

Result:
case 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ Localization.mk ↑{ deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    HomogeneousLocalization.val x

Proof state:
case mpr
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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
h :
  ∃ x ∈ {s | s ∈ q.asIdeal},
    (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Tactic:
rcases h with ⟨x, h, hx⟩

Result:
case mpr.intro.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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
h : x ∈ {s | s ∈ q.asIdeal}
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal