Theorem: hasSum_one_div_pow_mul_fourier_mul_bernoulliFun
File: Mathlib/NumberTheory/ZetaValues.lean

tactic: intro y hy
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
⊢ ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))

tactic: let B : C(𝕌, ℂ) :=
  ContinuousMap.mk ((↑) ∘ periodizedBernoulli k)
    (continuous_ofReal.comp (periodizedBernoulli.continuous (by omega)))
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))

tactic: have step1 : ∀ n : ℤ, fourierCoeff B n = -k ! / (2 * π * I * n) ^ k := by
  rw [ContinuousMap.coe_mk]; exact fourierCoeff_bernoulli_eq (by omega : k ≠ 0)
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))

tactic: have step2 :=
  has_pointwise_sum_fourier_series_of_summable
    ((summable_bernoulli_fourier hk).congr fun n => (step1 n).symm) y
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => fourierCoeff (⇑B) i • (fourier i) ↑y) (B ↑y)
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))

tactic: simp_rw [step1] at step2
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => fourierCoeff (⇑B) i • (fourier i) ↑y) (B ↑y)
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))

tactic: convert step2.mul_left (-(2 * ↑π * I) ^ k / (k ! : ℂ)) using 2 with n
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
state_after: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ 1 / ↑n ^ k * (fourier n) ↑y = -(2 * ↑π * I) ^ k / ↑k ! * (-↑k ! / (2 * ↑π * I * ↑n) ^ k) • (fourier n) ↑y

case h.e'_6.h.e'_6
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ ↑(bernoulliFun k y) = B ↑y

tactic: rw [← Ico_insert_right (zero_le_one' ℝ), mem_insert_iff, or_comm] at hx
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
hx : x ∈ Ico 0 1 ∨ x = 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))

tactic: rcases hx with (hx | rfl)
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
hx : x ∈ Ico 0 1 ∨ x = 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))
state_after: case inl
k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
hx : x ∈ Ico 0 1
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))

case inr
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑1) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1))

tactic: exact this hx
state_before: case inl
k : ℕ
hk : 2 ≤ k
x : ℝ
hx✝ : x ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
hx : x ∈ Ico 0 1
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k x))
state_after: no goals

tactic: convert this (left_mem_Ico.mpr zero_lt_one) using 1
state_before: case inr
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑1) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1))
state_after: case h.e'_5
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ (fun n => 1 / ↑n ^ k * (fourier n) ↑1) = fun n => 1 / ↑n ^ k * (fourier n) ↑0

case h.e'_6
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 0)

tactic: rw [AddCircle.coe_period, QuotientAddGroup.mk_zero]
state_before: case h.e'_5
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ (fun n => 1 / ↑n ^ k * (fourier n) ↑1) = fun n => 1 / ↑n ^ k * (fourier n) ↑0
state_after: no goals

tactic: rw [bernoulliFun_endpoints_eq_of_ne_one (by omega : k ≠ 1)]
state_before: case h.e'_6
k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 1) = -(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k 0)
state_after: no goals

tactic: omega
state_before: k : ℕ
hk : 2 ≤ k
hx : 1 ∈ Icc 0 1
this :
  ∀ {y : ℝ},
    y ∈ Ico 0 1 → HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
⊢ k ≠ 1
state_after: no goals

tactic: omega
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
⊢ k ≠ 1
state_after: no goals

tactic: rw [ContinuousMap.coe_mk]
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
state_after: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ ∀ (n : ℤ), fourierCoeff (ofReal' ∘ periodizedBernoulli k) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k

tactic: exact fourierCoeff_bernoulli_eq (by omega : k ≠ 0)
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ ∀ (n : ℤ), fourierCoeff (ofReal' ∘ periodizedBernoulli k) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
state_after: no goals

tactic: omega
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ k ≠ 0
state_after: no goals

tactic: rw [smul_eq_mul, ← mul_assoc, mul_div, mul_neg, div_mul_cancel₀, neg_neg, mul_pow _ (n : ℂ),
  ← div_div, div_self]
state_before: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ 1 / ↑n ^ k * (fourier n) ↑y = -(2 * ↑π * I) ^ k / ↑k ! * (-↑k ! / (2 * ↑π * I * ↑n) ^ k) • (fourier n) ↑y
state_after: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ (2 * ↑π * I) ^ k ≠ 0

case h.e'_5.h.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ ↑k ! ≠ 0

tactic: rw [Ne, pow_eq_zero_iff', not_and_or]
state_before: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ (2 * ↑π * I) ^ k ≠ 0
state_after: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ ¬2 * ↑π * I = 0 ∨ ¬k ≠ 0

tactic: exact Or.inl two_pi_I_ne_zero
state_before: case h.e'_5.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ ¬2 * ↑π * I = 0 ∨ ¬k ≠ 0
state_after: no goals

tactic: exact Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _)
state_before: case h.e'_5.h.h
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
n : ℤ
⊢ ↑k ! ≠ 0
state_after: no goals

tactic: rw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,
  AddCircle.liftIco_coe_apply (show y ∈ Ico 0 (0 + 1) by rwa [zero_add])]
state_before: case h.e'_6.h.e'_6
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ ↑(bernoulliFun k y) = B ↑y
state_after: no goals

tactic: rwa [zero_add]
state_before: k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal' ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
step1 : ∀ (n : ℤ), fourierCoeff (⇑B) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
step2 : HasSum (fun i => (-↑k ! / (2 * ↑π * I * ↑i) ^ k) • (fourier i) ↑y) (B ↑y)
⊢ y ∈ Ico 0 (0 + 1)
state_after: no goals