Theorem: fourierCoeffOn_of_hasDeriv_right
File: Mathlib/Analysis/Fourier/AddCircle.lean

Proof steps:
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ fourierCoeffOn hab f n =
    1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ fourierCoeffOn hab f n =
    1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
tactic: rw [← ofReal_sub]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ fourierCoeffOn hab f n =
    1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ fourierCoeffOn hab f n =
    1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
tactic: have hT : Fact (0 < b - a) := ⟨by linarith⟩
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ fourierCoeffOn hab f n =
    1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =
    1 / (-2 * ↑π * I * ↑n) *
      ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f' x))
tactic: simp_rw [fourierCoeffOn_eq_integral, smul_eq_mul, real_smul, ofReal_div, ofReal_one]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =
    1 / (-2 * ↑π * I * ↑n) *
      ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f' x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: conv => pattern (occs := 1 2 3) fourier _ _ * _ <;> (rw [mul_comm])
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: rw [integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right hf
    (fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.continuousAt |>.continuousWithinAt) hff'
    (fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.hasDerivWithinAt) hf'
    (((map_continuous (fourier (-n))).comp (AddCircle.continuous_mk' _)).intervalIntegrable _ _)]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: have : ∀ u v w : ℂ, u * ((b - a : ℝ) / v * w) = (b - a : ℝ) / v * (u * w) := by intros; ring
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) / (-2 * ↑π * I * ↑n) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: conv in intervalIntegral _ _ _ _ => congr; ext; rw [this]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) / (-2 * ↑π * I * ↑n) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: rw [(by ring : ((b - a : ℝ) : ℂ) / (-2 * π * I * n) = ((b - a : ℝ) : ℂ) * (1 / (-2 * π * I * n)))]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: have s2 : (b : AddCircle (b - a)) = (a : AddCircle (b - a)) := by
    simpa using coe_add_period (b - a) a
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) *
      (f b * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑b) -
          f a * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a) -
        ∫ (x : ℝ) in a..b, ↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (f' x * (fourier (-n)) ↑x)) =
    1 / (-2 * ↑π * I * ↑n) *
      ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) * ((f b - f a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a)) -
      1 / ↑(b - a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x) =
    1 / (-2 * ↑π * I * ↑n) * ((f b - f a) * (fourier (-n)) ↑a) -
      1 / (-2 * ↑π * I * ↑n) * (↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: rw [s2, integral_const_mul, ← sub_mul, mul_sub, mul_sub]
state_before: T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) * ((f b - f a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a)) -
      1 / ↑(b - a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x) =
    1 / (-2 * ↑π * I * ↑n) * ((f b - f a) * (fourier (-n)) ↑a) -
      1 / (-2 * ↑π * I * ↑n) * (↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
state_after: case e_a
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) * ((f b - f a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * (fourier (-n)) ↑a)) =
    1 / (-2 * ↑π * I * ↑n) * ((f b - f a) * (fourier (-n)) ↑a)

case e_a
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)
this : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)
s2 : ↑b = ↑a
⊢ 1 / ↑(b - a) * (↑(b - a) * (1 / (-2 * ↑π * I * ↑n)) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x) =
    1 / (-2 * ↑π * I * ↑n) * (↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
tactic: congr 1