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

Proof steps:
state_before: case intro.mk.intro
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
⊢ ∃ n, ∀ (m : ℕ), n ≤ m → (((f |_ U₁) ⋯ ^ (m + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m + n₁) * y₂) |_ ↑S) h₂
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
⊢ ∀ (m : ℕ), n ≤ m → (((f |_ U₁) ⋯ ^ (m + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m + n₁) * y₂) |_ ↑S) h₂
tactic: use n
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
⊢ ∀ (m : ℕ), n ≤ m → (((f |_ U₁) ⋯ ^ (m + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m + n₁) * y₂) |_ ↑S) h₂
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
m : ℕ
hm : n ≤ m
⊢ (((f |_ U₁) ⋯ ^ (m + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m + n₁) * y₂) |_ ↑S) h₂
tactic: intros m hm
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
m : ℕ
hm : n ≤ m
⊢ (((f |_ U₁) ⋯ ^ (m + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m + n₁) * y₂) |_ ↑S) h₂
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
m : ℕ
hm : n ≤ m
⊢ (((f |_ U₁) ⋯ ^ (m - n + n + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m - n + n + n₁) * y₂) |_ ↑S) h₂
tactic: rw [← tsub_add_cancel_of_le hm]
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n : ℕ
e :
  ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₁) ⋯ ^ n₂ * y₁) |_ ↑S) h₁ =
    ↑⟨(fun x => (f |_ ↑S) ⋯ ^ x) n, ⋯⟩ * (((f |_ U₂) ⋯ ^ n₁ * y₂) |_ ↑S) h₂
m : ℕ
hm : n ≤ m
⊢ (((f |_ U₁) ⋯ ^ (m - n + n + n₂) * y₁) |_ ↑S) h₁ = (((f |_ U₂) ⋯ ^ (m - n + n + n₁) * y₂) |_ ↑S) h₂
state_after: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n m : ℕ
hm : n ≤ m
e :
  (X.presheaf.map (homOfLE ⋯).op) f ^ n *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n₂ * (X.presheaf.map (homOfLE h₁).op) y₁) =
    (X.presheaf.map (homOfLE ⋯).op) f ^ n *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n₁ * (X.presheaf.map (homOfLE h₂).op) y₂)
⊢ (X.presheaf.map (homOfLE ⋯).op) f ^ (m - n) *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n *
        ((X.presheaf.map (homOfLE ⋯).op) f ^ n₂ * (X.presheaf.map (homOfLE h₁).op) y₁)) =
    (X.presheaf.map (homOfLE ⋯).op) f ^ (m - n) *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n *
        ((X.presheaf.map (homOfLE ⋯).op) f ^ n₁ * (X.presheaf.map (homOfLE h₂).op) y₂))
tactic: simp only [TopCat.Presheaf.restrictOpen, TopCat.Presheaf.restrict,
    pow_add, map_pow, map_mul, ← comp_apply, mul_assoc, ← Functor.map_comp, ← op_comp, homOfLE_comp,
    Subtype.coe_mk] at e ⊢
state_before: case h
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
S : ↑X.affineOpens
U₁ U₂ : X.Opens
n₁ n₂ : ℕ
y₁ : ↑Γ(X, U₁)
y₂ : ↑Γ(X, U₂)
f : ↑Γ(X, U₁ ⊔ U₂)
x : ↑Γ(X, X.basicOpen f)
h₁ : ↑S ≤ U₁
h₂ : ↑S ≤ U₂
e₁ :
  (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =
    ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯
e₂ :
  (y₂ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ =
    ((f |_ U₂) ⋯ |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯ ^ n₂ * (x |_ X.basicOpen ((f |_ U₂) ⋯)) ⋯
n m : ℕ
hm : n ≤ m
e :
  (X.presheaf.map (homOfLE ⋯).op) f ^ n *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n₂ * (X.presheaf.map (homOfLE h₁).op) y₁) =
    (X.presheaf.map (homOfLE ⋯).op) f ^ n *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n₁ * (X.presheaf.map (homOfLE h₂).op) y₂)
⊢ (X.presheaf.map (homOfLE ⋯).op) f ^ (m - n) *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n *
        ((X.presheaf.map (homOfLE ⋯).op) f ^ n₂ * (X.presheaf.map (homOfLE h₁).op) y₁)) =
    (X.presheaf.map (homOfLE ⋯).op) f ^ (m - n) *
      ((X.presheaf.map (homOfLE ⋯).op) f ^ n *
        ((X.presheaf.map (homOfLE ⋯).op) f ^ n₁ * (X.presheaf.map (homOfLE h₂).op) y₂))
state_after: no goals
tactic: rw [e]