Theorem: AlgebraicGeometry.StructureSheaf.normalize_finite_fraction_representation
File: Mathlib/AlgebraicGeometry/StructureSheaf.lean

Proof steps:
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: have fractions_eq :
    ∀ i j : ι,
      IsLocalization.mk' (Localization.Away (h i * h j))
        (a i * h j) ⟨h i * h j, Submonoid.mem_powers _⟩ =
      IsLocalization.mk' _ (h i * a j) ⟨h i * h j, Submonoid.mem_powers _⟩ := by
    intro i j
    let D := PrimeSpectrum.basicOpen (h i * h j)
    let iDi : D ⟶ PrimeSpectrum.basicOpen (h i) := homOfLE (PrimeSpectrum.basicOpen_mul_le_left _ _)
    let iDj : D ⟶ PrimeSpectrum.basicOpen (h j) :=
      homOfLE (PrimeSpectrum.basicOpen_mul_le_right _ _)
    -- Crucially, we need injectivity of `toBasicOpen`
    apply toBasicOpen_injective R (h i * h j)
    rw [toBasicOpen_mk', toBasicOpen_mk']
    simp only []
    -- Here, both sides of the equation are equal to a restriction of `s`
    trans
    on_goal 1 =>
      convert congr_arg ((structureSheaf R).1.map iDj.op) (hs j).symm using 1
      convert congr_arg ((structureSheaf R).1.map iDi.op) (hs i) using 1
    all_goals rw [res_const]; apply const_ext; ring
    -- The remaining two goals were generated during the rewrite of `res_const`
    -- These can be solved immediately
    exacts [PrimeSpectrum.basicOpen_mul_le_left _ _, PrimeSpectrum.basicOpen_mul_le_right _ _]
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: have exists_power :
    ∀ i j : ι, ∃ n : ℕ, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n := by
    intro i j
    obtain ⟨⟨c, n, rfl⟩, hc⟩ := IsLocalization.eq.mp (fractions_eq i j)
    use n + 1
    rw [pow_succ]
    dsimp at hc
    convert hc using 1 <;> ring
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: let n := fun p : ι × ι => (exists_power p.1 p.2).choose
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: have n_spec := fun p : ι × ι => (exists_power p.fst p.snd).choose_spec
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: let N := (t ×ˢ t).sup n
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
tactic: have basic_opens_eq : ∀ i : ι, PrimeSpectrum.basicOpen (h i ^ (N + 1)) =
    PrimeSpectrum.basicOpen (h i) := fun i => PrimeSpectrum.basicOpen_pow _ _ (by omega)
state_before: R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ ∃ a' h' iDh',
    U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h' i) ∧
      (∀ i ∈ t, ∀ j ∈ t, a' i * h' j = h' i * a' j) ∧
        ∀ i ∈ t, ((structureSheaf R).val.map (iDh' i).op) s = const R (a' i) (h' i) (PrimeSpectrum.basicOpen (h' i)) ⋯
state_after: case refine_1
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)

case refine_2
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ ∀ i ∈ t,
    ∀ j ∈ t,
      (fun i => a i * h i ^ N) i * (fun i => h i ^ (N + 1)) j = (fun i => h i ^ (N + 1)) i * (fun i => a i * h i ^ N) j

case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ ∀ i ∈ t,
    ((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s =
      const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
        (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
tactic: refine
    ⟨fun i => a i * h i ^ N, fun i => h i ^ (N + 1), fun i => eqToHom (basic_opens_eq i) ≫ iDh i,
      ?_, ?_, ?_⟩
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
⊢ ∀ i ∈ t,
    ((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s =
      const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
        (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
state_after: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
tactic: intro i _
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
state_after: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map (iDh i).op ≫ (structureSheaf R).val.map (eqToHom ⋯).op) s =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
tactic: rw [op_comp, Functor.map_comp]
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map (iDh i).op ≫ (structureSheaf R).val.map (eqToHom ⋯).op) s =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
state_after: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map (eqToHom ⋯).op) (((structureSheaf R).val.map (iDh i).op) s) =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
tactic: change (structureSheaf R).1.map (eqToHom (basic_opens_eq _)).op
    ((structureSheaf R).1.map (iDh i).op s) = _
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ((structureSheaf R).val.map (eqToHom ⋯).op) (((structureSheaf R).val.map (iDh i).op) s) =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
state_after: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ?refine_3.hv =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯

case refine_3.hv
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ∀ x ∈ PrimeSpectrum.basicOpen (h i ^ (N + 1)), h i ∈ x.asIdeal.primeCompl
tactic: rw [← hs, res_const]
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ?refine_3.hv =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯

case refine_3.hv
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ∀ x ∈ PrimeSpectrum.basicOpen (h i ^ (N + 1)), h i ∈ x.asIdeal.primeCompl
state_after: case refine_3.hv
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ ∀ x ∈ PrimeSpectrum.basicOpen (h i ^ (N + 1)), h i ∈ x.asIdeal.primeCompl

case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ?refine_3.hv =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
tactic: swap
state_before: case refine_3
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ const R (a i) (h i) (PrimeSpectrum.basicOpen (h i ^ (N + 1))) ⋯ =
    const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i)
      (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯
state_after: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * (fun i => h i ^ (N + 1)) i = (fun i => a i * h i ^ N) i * h i
tactic: apply const_ext
state_before: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * (fun i => h i ^ (N + 1)) i = (fun i => a i * h i ^ N) i * h i
state_after: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * h i ^ (N + 1) = a i * h i ^ N * h i
tactic: dsimp
state_before: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * h i ^ (N + 1) = a i * h i ^ N * h i
state_after: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * (h i ^ N * h i) = a i * h i ^ N * h i
tactic: rw [pow_succ]
state_before: case refine_3.h
R : Type u
inst✝ : CommRing R
U : Opens ↑(PrimeSpectrum.Top R)
s : ↑((structureSheaf R).val.obj (op U))
ι : Type u_1
t : Finset ι
a h : ι → R
iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U
h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)
hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s
fractions_eq :
  ∀ (i j : ι),
    IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ =
      IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩
exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n
n : ι × ι → ℕ := fun p => ⋯.choose
n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose
N : ℕ := (t ×ˢ t).sup n
basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)
i : ι
a✝ : i ∈ t
⊢ a i * (h i ^ N * h i) = a i * h i ^ N * h i
state_after: no goals
tactic: ring