-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg q)
    (hb : b ∈ carrier f_deg q) : a + b ∈ carrier f_deg q

Proof state:
case refine_3
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 b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun j =>
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
⊢ HomogeneousLocalization.mk
      { deg := m * i + m * i,
        num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩,
        den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
convert_to ∑ i ∈ range (m + m + 1), g i ∈ q.1

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 b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun j =>
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
⊢ HomogeneousLocalization.mk
      { deg := m * i + m * i,
        num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩,
        den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } =
    ∑ i ∈ range (m + m + 1), g i

case refine_3
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 b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun j =>
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
⊢ ∑ i ∈ range (m + m + 1), g i ∈ 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
l : A⁰_ f :=
  HomogeneousLocalization.mk
    { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ?m.302901⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
⊢ (proj 𝒜 i) b ^ m ∈ 𝒜 (m • i)

Tactic:
mem_tac

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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun j =>
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
⊢ ∀ j ∈ range (m + m + 1),
    Localization.mk ((m + m).choose j • ((proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m + m - j))) ⟨f ^ (i + i), ⋯⟩ =
      (algebraMap (A⁰_ f) (Localization.Away f)) (g j)

Tactic:
intro j hj

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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
g : ℕ → A⁰_ f :=
  fun j =>
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
j : ℕ
hj : j ∈ range (m + m + 1)
⊢ Localization.mk ((m + m).choose j • ((proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m + m - j))) ⟨f ^ (i + i), ⋯⟩ =
    (algebraMap (A⁰_ f) (Localization.Away f)) (g j)

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 b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
⊢ a + b ∈ carrier f_deg q

Tactic:
refine fun i => (q.2.mem_or_mem ?_).elim id id

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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal