Theorem: AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen
File: Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean

tactic: have := (hU.isLocalization_basicOpen f).2
state_before: X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
state_after: X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x

tactic: obtain ⟨⟨y, _, n, rfl⟩, d⟩ := this x
state_before: X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
state_after: case intro.mk.mk.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x

tactic: use n, y
state_before: case intro.mk.mk.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x

tactic: delta TopCat.Presheaf.restrictOpen TopCat.Presheaf.restrict
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ (X.presheaf.map (homOfLE ⋯).op) y = (X.presheaf.map (homOfLE ⋯).op) f ^ n * x

tactic: simpa [mul_comm x] using d.symm
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : Opens ↑↑X.toPresheafedSpace
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
  ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
y : ↑Γ(X, U)
n : ℕ
d :
  x * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x => f ^ x) n, ⋯⟩).2 =
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) (y, ⟨(fun x => f ^ x) n, ⋯⟩).1
⊢ (X.presheaf.map (homOfLE ⋯).op) y = (X.presheaf.map (homOfLE ⋯).op) f ^ n * x
state_after: no goals