Theorem: _private.Mathlib.NumberTheory.NumberField.House.0.NumberField.house.ξ_mulVec_eq_0
File: Mathlib/NumberTheory/NumberField/House.lean

Proof steps:
state_before: K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
⊢ a *ᵥ NumberField.house.ξ K x = 0
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ (a *ᵥ NumberField.house.ξ K x) k = 0 k
tactic: funext k
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ (a *ᵥ NumberField.house.ξ K x) k = 0 k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ (a *ᵥ NumberField.house.ξ K x) k = 0
tactic: simp only [Pi.zero_apply]
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ (a *ᵥ NumberField.house.ξ K x) k = 0
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
tactic: rw [eq_comm]
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
tactic: have lin_0 : ∀ u, ∑ r, ∑ l, (a' K a k l r u * x (l, r) : 𝓞 K) = 0 := by
    intros u
    have hξ := ξ_ne_0 K x hxl
    rw [Ne, Function.funext_iff, not_forall] at hξ
    rcases hξ with ⟨l, hξ⟩
    rw [Function.funext_iff] at hmulvec0
    specialize hmulvec0 ⟨k, u⟩
    simp only [Fintype.sum_prod_type, mulVec, dotProduct, asiegel] at hmulvec0
    rw [sum_comm] at hmulvec0
    exact mod_cast hmulvec0
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
tactic: have : 0 = ∑ u, (∑ r, ∑ l, a' K a k l r u * x (l, r) : 𝓞 K) * (newBasis K) u := by
    simp only [lin_0, zero_mul, sum_const_zero]
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ r : K →+* ℂ,
      ∑ l : β, ↑(x (l, r)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l r u) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
tactic: have : 0 = ∑ r, ∑ l, x (l, r) * ∑ u, a' K a k l r u * (newBasis K) u := by
    conv at this => enter [2, 2, u]; rw [sum_mul]
    rw [sum_comm] at this
    rw [this]; congr 1; ext1 r
    conv => enter [1, 2, l]; rw [sum_mul]
    rw [sum_comm]; congr 1; ext1 r
    rw [mul_sum]; congr 1; ext1 r
    ring
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ r : K →+* ℂ,
      ∑ l : β, ↑(x (l, r)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l r u) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
tactic: rw [sum_comm] at this
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ 0 = (a *ᵥ NumberField.house.ξ K x) k
state_after: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u =
    (a *ᵥ NumberField.house.ξ K x) k
tactic: rw [this]
state_before: case h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u =
    (a *ᵥ NumberField.house.ξ K x) k
state_after: case h.e_f
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ (fun y =>
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u) =
    fun i => (fun j => a k j) i * NumberField.house.ξ K x i
tactic: congr 1
state_before: case h.e_f
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
⊢ (fun y =>
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u) =
    fun i => (fun j => a k j) i * NumberField.house.ξ K x i
state_after: case h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ ∑ x_1 : K →+* ℂ,
      ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u =
    (fun j => a k j) l * NumberField.house.ξ K x l
tactic: ext1 l
state_before: case h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ ∑ x_1 : K →+* ℂ,
      ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u =
    (fun j => a k j) l * NumberField.house.ξ K x l
state_after: case h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ ∑ x_1 : K →+* ℂ,
      ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u =
    ∑ i : K →+* ℂ, (fun j => a k j) l * (↑(x (l, i)) * (NumberField.house.newBasis K) i)
tactic: rw [ξ, mul_sum]
state_before: case h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ ∑ x_1 : K →+* ℂ,
      ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u =
    ∑ i : K →+* ℂ, (fun j => a k j) l * (↑(x (l, i)) * (NumberField.house.newBasis K) i)
state_after: case h.e_f.h.e_f
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ (fun x_1 => ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u) =
    fun i => (fun j => a k j) l * (↑(x (l, i)) * (NumberField.house.newBasis K) i)
tactic: congr 1
state_before: case h.e_f.h.e_f
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l : β
⊢ (fun x_1 => ↑(x (l, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l x_1 u) * (NumberField.house.newBasis K) u) =
    fun i => (fun j => a k j) l * (↑(x (l, i)) * (NumberField.house.newBasis K) i)
state_after: case h.e_f.h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l✝ : β
l : K →+* ℂ
⊢ ↑(x (l✝, l)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l✝ l u) * (NumberField.house.newBasis K) u =
    (fun j => a k j) l✝ * (↑(x (l✝, l)) * (NumberField.house.newBasis K) l)
tactic: ext1 l
state_before: case h.e_f.h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l✝ : β
l : K →+* ℂ
⊢ ↑(x (l✝, l)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l✝ l u) * (NumberField.house.newBasis K) u =
    (fun j => a k j) l✝ * (↑(x (l✝, l)) * (NumberField.house.newBasis K) l)
state_after: case h.e_f.h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l✝ : β
l : K →+* ℂ
⊢ ↑(x (l✝, l)) * (a k l✝ * (NumberField.house.newBasis K) l) =
    (fun j => a k j) l✝ * (↑(x (l✝, l)) * (NumberField.house.newBasis K) l)
tactic: rw [← lin_1]
state_before: case h.e_f.h.e_f.h
K : Type u_1
inst✝⁴ : Field K
inst✝³ : NumberField K
inst✝² : DecidableEq (K →+* ℂ)
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
a : Matrix α β (𝓞 K)
ha : a ≠ 0
p q : ℕ
cardα : Fintype.card α = p
cardβ : Fintype.card β = q
h0p : 0 < p
hpq : p < q
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0
this✝ :
  0 =
    ∑ u : K →+* ℂ,
      (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u
this :
  0 =
    ∑ y : β,
      ∑ x_1 : K →+* ℂ,
        ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u
l✝ : β
l : K →+* ℂ
⊢ ↑(x (l✝, l)) * (a k l✝ * (NumberField.house.newBasis K) l) =
    (fun j => a k j) l✝ * (↑(x (l✝, l)) * (NumberField.house.newBasis K) l)
state_after: no goals
tactic: ring