-- 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 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 := ⋯ }
⊢ Localization.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 := ⋯ }.num
      ⟨↑{ 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 := ⋯ }.den,
        ⋯⟩ =
    (algebraMap (A⁰_ f) (Localization.Away f)) (∑ i ∈ range (m + m + 1), g i)

Tactic:
dsimp only [Subtype.coe_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 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 := ⋯ }
⊢ Localization.mk (∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)))
      ⟨f ^ (i + i), ⋯⟩ =
    (algebraMap (A⁰_ f) (Localization.Away f)) (∑ i ∈ range (m + m + 1), g i)

Proof state:
case neg.e_a.e_x
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)
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m + m - j) =
    (proj 𝒜 i) a ^ m * ((proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j))

Tactic:
dsimp

Result:
case neg.e_a.e_x
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)
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ ↑(((decompose 𝒜) a) i) ^ j * ↑(((decompose 𝒜) b) i) ^ (m + m - j) =
    ↑(((decompose 𝒜) a) i) ^ m * (↑(((decompose 𝒜) a) i) ^ (j - m) * ↑(((decompose 𝒜) b) i) ^ (m + m - j))

Proof state:
case refine_1
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
⊢ (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j) ∈ 𝒜 (m * i)

Tactic:
rw [(_ : m * i = _)]

Result:
case refine_1
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
⊢ (proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j) ∈ 𝒜 ?m.376170

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
⊢ m * i = ?m.376170

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 := ⋯ }
⊢ f ^ i ∈ 𝒜 (i * m)

Tactic:
mem_tac

Result:
no goals