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Theorem List for Metamath Proof Explorer - 43201-43300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfourierdlem11 43201* If there is a partition, than the lower bound is strictly less than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
 
Theoremfourierdlem12 43202* A point of a partition is not an element of any open interval determined by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ran 𝑄)       ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
 
Theoremfourierdlem13 43203* Value of 𝑉 in terms of value of 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝐼 ∈ (0...𝑀))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))       (𝜑 → ((𝑄𝐼) = ((𝑉𝐼) − 𝑋) ∧ (𝑉𝐼) = (𝑋 + (𝑄𝐼))))
 
Theoremfourierdlem14 43204* Given the partition 𝑉, 𝑄 is the partition shifted to the left by 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))       (𝜑𝑄 ∈ (𝑂𝑀))
 
Theoremfourierdlem15 43205* The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
 
Theoremfourierdlem16 43206* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐴 = (𝑛 ∈ ℕ0 ↦ (∫𝐶((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (((𝐴𝑁) ∈ ℝ ∧ (𝑥𝐶 ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫𝐶((𝐹𝑥) · (cos‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ))
 
Theoremfourierdlem17 43207* The defined 𝐿 is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))       (𝜑𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
 
Theoremfourierdlem18 43208* The function 𝑆 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))       (𝜑𝑆 ∈ ((-π[,]π)–cn→ℝ))
 
Theoremfourierdlem19 43209* If two elements of 𝐷 have the same periodic image in (𝐴(,]𝐵) then they are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝑋 ∈ ℝ)    &   𝐷 = {𝑦 ∈ ((𝐴 + 𝑋)(,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶}    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑊𝐷)    &   (𝜑𝑍𝐷)    &   (𝜑 → (𝐸𝑍) = (𝐸𝑊))       (𝜑 → ¬ 𝑊 < 𝑍)
 
Theoremfourierdlem20 43210* Every interval in the partition 𝑆 is included in an interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑 → (𝑄‘0) ≤ 𝐴)    &   (𝜑𝐵 ≤ (𝑄𝑀))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))    &   𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) ≤ (𝑆𝐽)}, ℝ, < )       (𝜑 → ∃𝑖 ∈ (0..^𝑀)((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
 
Theoremfourierdlem21 43211* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (((𝐵𝑁) ∈ ℝ ∧ (𝑥𝐶 ↦ ((𝐹𝑥) · (sin‘(𝑁 · 𝑥)))) ∈ 𝐿1) ∧ ∫𝐶((𝐹𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ))
 
Theoremfourierdlem22 43212* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐴 = (𝑛 ∈ ℕ0 ↦ (∫𝐶((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))       (𝜑 → ((𝑛 ∈ ℕ0 → (𝐴𝑛) ∈ ℝ) ∧ (𝑛 ∈ ℕ → (𝐵𝑛) ∈ ℝ)))
 
Theoremfourierdlem23 43213* If 𝐹 is continuous and 𝑋 is constant, then (𝐹‘(𝑋 + 𝑠)) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝐵 ⊆ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑠𝐵) → (𝑋 + 𝑠) ∈ 𝐴)       (𝜑 → (𝑠𝐵 ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (𝐵cn→ℂ))
 
Theoremfourierdlem24 43214 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ((-π[,]π) ∖ {0}) → (𝐴 mod (2 · π)) ≠ 0)
 
Theoremfourierdlem25 43215* If 𝐶 is not in the range of the partition, then it is in an open interval induced by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑𝐶 ∈ ((𝑄‘0)[,](𝑄𝑀)))    &   (𝜑 → ¬ 𝐶 ∈ ran 𝑄)    &   𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝐶}, ℝ, < )       (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
 
Theoremfourierdlem26 43216* Periodic image of a point 𝑌 that's in the period that begins with the point 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → (𝐸𝑋) = 𝐵)    &   (𝜑𝑌 ∈ (𝑋(,](𝑋 + 𝑇)))       (𝜑 → (𝐸𝑌) = (𝐴 + (𝑌𝑋)))
 
Theoremfourierdlem27 43217 A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑𝐼 ∈ (0..^𝑀))       (𝜑 → ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵))
 
Theoremfourierdlem28 43218* Derivative of (𝐹‘(𝑋 + 𝑠)). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝐷 = (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))    &   (𝜑𝐷:((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)       (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐷‘(𝑋 + 𝑠))))
 
Theoremfourierdlem29 43219* Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))       (𝐴 ∈ (-π[,]π) → (𝐾𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
 
Theoremfourierdlem30 43220* Sum of three small pieces is less than ε. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿1)    &   ((𝜑𝑥𝐼) → 𝐹 ∈ ℂ)    &   ((𝜑𝑥𝐼) → 𝐺 ∈ ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   𝑋 = (abs‘𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   𝑌 = (abs‘𝐶)    &   𝑍 = (abs‘∫𝐼(𝐹 · -𝐺) d𝑥)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((((𝑋 + 𝑌) + 𝑍) / 𝐸) + 1) ≤ 𝑅)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (abs‘𝐵) ≤ 1)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (abs‘𝐷) ≤ 1)       (𝜑 → (abs‘(((𝐴 · -(𝐵 / 𝑅)) − (𝐶 · -(𝐷 / 𝑅))) − ∫𝐼(𝐹 · -(𝐺 / 𝑅)) d𝑥)) < 𝐸)
 
Theoremfourierdlem31 43221* If 𝐴 is finite and for any element in 𝐴 there is a number 𝑚 such that a property holds for all numbers larger than 𝑚, then there is a number 𝑛 such that the property holds for all numbers larger than 𝑛 and for all elements in 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
𝑖𝜑    &   𝑟𝜑    &   𝑖𝑉    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑖𝐴𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)    &   𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}    &   𝑉 = (𝑖𝐴 ↦ inf(𝑀, ℝ, < ))    &   𝑁 = sup(ran 𝑉, ℝ, < )       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖𝐴 𝜒)
 
Theoremfourierdlem32 43222 Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))    &   𝑌 = if(𝐶 = 𝐴, 𝑅, (𝐹𝐶))    &   𝐽 = ((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵))       (𝜑𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim 𝐶))
 
Theoremfourierdlem33 43223 Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))    &   𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹𝐷))    &   𝐽 = ((TopOpen‘ℂfld) ↾t ((𝐴(,)𝐵) ∪ {𝐵}))       (𝜑𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim 𝐷))
 
Theoremfourierdlem34 43224* A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑𝑄:(0...𝑀)–1-1→ℝ)
 
Theoremfourierdlem35 43225 There is a single point in (𝐴(,]𝐵) that's distant from 𝑋 a multiple integer of 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴(,]𝐵))    &   (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴(,]𝐵))       (𝜑𝐼 = 𝐽)
 
Theoremfourierdlem36 43226* 𝐹 is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐹 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴))    &   𝑁 = ((♯‘𝐴) − 1)       (𝜑𝐹 Isom < , < ((0...𝑁), 𝐴))
 
Theoremfourierdlem37 43227* 𝐼 is a function that maps any real point to the point that in the partition that immediately precedes the corresponding periodic point in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))})))
 
Theoremfourierdlem38 43228* The function 𝐹 is continuous on every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (dom 𝐹cn→ℂ))    &   𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   𝐻 = (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹))    &   (𝜑 → ran 𝑄 = 𝐻)       ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
 
Theoremfourierdlem39 43229* Integration by parts of ∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   𝐺 = (ℝ D 𝐹)    &   (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐺𝑥)) ≤ 𝑦)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 = ((((𝐹𝐵) · -((cos‘(𝑅 · 𝐵)) / 𝑅)) − ((𝐹𝐴) · -((cos‘(𝑅 · 𝐴)) / 𝑅))) − ∫(𝐴(,)𝐵)((𝐺𝑥) · -((cos‘(𝑅 · 𝑥)) / 𝑅)) d𝑥))
 
Theoremfourierdlem40 43230* 𝐻 is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐴 ∈ (-π[,]π))    &   (𝜑𝐵 ∈ (-π[,]π))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))       (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
 
Theoremfourierdlem41 43231* Lemma used to prove that every real is a limit point for the domain of the derivative of the periodic function to be approximated. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)       (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))
 
Theoremfourierdlem42 43232* The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   𝑇 = (𝐶𝐵)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   𝐷 = (abs ∘ − )    &   𝐼 = ((𝐴 × 𝐴) ∖ I )    &   𝑅 = ran (𝐷𝐼)    &   𝐸 = inf(𝑅, ℝ, < )    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (𝐽t (𝑋[,]𝑌))    &   𝐻 = {𝑥 ∈ (𝑋[,]𝑌) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ 𝐴}    &   (𝜓 ↔ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏)) ∧ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑎 + (𝑗 · 𝑇)) ∈ 𝐴 ∧ (𝑏 + (𝑘 · 𝑇)) ∈ 𝐴)))       (𝜑𝐻 ∈ Fin)
 
Theoremfourierdlem43 43233 𝐾 is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))       𝐾:(-π[,]π)⟶ℝ
 
Theoremfourierdlem44 43234 A condition for having (sin‘(𝐴 / 2)) nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)
 
Theoremfourierdlem46 43235* The function 𝐹 has a limit at the bounds of every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (dom 𝐹cn→ℂ))    &   ((𝜑𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim 𝑥) ≠ ∅)    &   ((𝜑𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim 𝑥) ≠ ∅)    &   (𝜑𝑄 Isom < , < ((0...𝑀), 𝐻))    &   (𝜑𝑄:(0...𝑀)⟶𝐻)    &   (𝜑𝐼 ∈ (0..^𝑀))    &   (𝜑 → (𝑄𝐼) < (𝑄‘(𝐼 + 1)))    &   (𝜑 → ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π))    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))    &   (𝜑 → ran 𝑄 = 𝐻)       (𝜑 → (((𝐹 ↾ ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1)))) lim (𝑄𝐼)) ≠ ∅ ∧ ((𝐹 ↾ ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1)))) lim (𝑄‘(𝐼 + 1))) ≠ ∅))
 
Theoremfourierdlem47 43236* For 𝑟 large enough, the final expression is less than the given positive 𝐸. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐼𝐹) ∈ 𝐿1)    &   ((𝜑𝑟 ∈ ℝ) → (𝑥𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿1)    &   ((𝜑𝑥𝐼) → 𝐹 ∈ ℂ)    &   (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℂ) → 𝐺 ∈ ℂ)    &   (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ) → (abs‘𝐺) ≤ 1)    &   (𝜑𝐴 ∈ ℂ)    &   𝑋 = (abs‘𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   𝑌 = (abs‘𝐶)    &   𝑍 = ∫𝐼(abs‘𝐹) d𝑥    &   (𝜑𝐸 ∈ ℝ+)    &   ((𝜑𝑟 ∈ ℂ) → 𝐵 ∈ ℂ)    &   ((𝜑𝑟 ∈ ℝ) → (abs‘𝐵) ≤ 1)    &   ((𝜑𝑟 ∈ ℂ) → 𝐷 ∈ ℂ)    &   ((𝜑𝑟 ∈ ℝ) → (abs‘𝐷) ≤ 1)    &   𝑀 = ((⌊‘((((𝑋 + 𝑌) + 𝑍) / 𝐸) + 1)) + 1)       (𝜑 → ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)(abs‘(((𝐴 · -(𝐵 / 𝑟)) − (𝐶 · -(𝐷 / 𝑟))) − ∫𝐼(𝐹 · -(𝐺 / 𝑟)) d𝑥)) < 𝐸)
 
Theoremfourierdlem48 43237* The given periodic function 𝐹 has a right limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))    &   (𝜒 ↔ ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑄𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑦 = (𝑋 + (𝑘 · 𝑇))))       (𝜑 → ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅)
 
Theoremfourierdlem49 43238* The given periodic function 𝐹 has a left limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))       (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅)
 
Theoremfourierdlem50 43239* Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   𝑁 = ((♯‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑈 = (𝑖 ∈ (0..^𝑀)((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))    &   (𝜒 ↔ ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ (0..^𝑀)) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑘)(,)(𝑄‘(𝑘 + 1)))))       (𝜑 → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
 
Theoremfourierdlem51 43240* 𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐶 ⊆ (𝐴[,]𝐵))    &   (𝜑𝐵𝐶)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → (𝐸𝑋) ∈ 𝐶)    &   𝐷 = ({(𝐴 + 𝑋), (𝐵 + 𝑋)} ∪ {𝑦 ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶})    &   𝐹 = (℩𝑓𝑓 Isom < , < ((0...((♯‘𝐷) − 1)), 𝐷))    &   𝐻 = {𝑦 ∈ ((𝐴 + 𝑋)(,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶}       (𝜑𝑋 ∈ ran 𝐹)
 
Theoremfourierdlem52 43241* d16:d17,d18:jca |- ( ph -> ( ( S 0) ≤ 𝐴𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑇 ∈ Fin)    &   𝑁 = ((♯‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ⊆ (𝐴[,]𝐵))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑇)       (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
 
Theoremfourierdlem53 43242* The limit of 𝐹(𝑠) at (𝑋 + 𝐷) is the limit of 𝐹(𝑋 + 𝑠) at 𝐷. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐺 = (𝑠𝐴 ↦ (𝐹‘(𝑋 + 𝑠)))    &   ((𝜑𝑠𝐴) → (𝑋 + 𝑠) ∈ 𝐵)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑠𝐴) → 𝑠𝐷)    &   (𝜑𝐶 ∈ ((𝐹𝐵) lim (𝑋 + 𝐷)))    &   (𝜑𝐷 ∈ ℂ)       (𝜑𝐶 ∈ (𝐺 lim 𝐷))
 
Theoremfourierdlem54 43243* Given a partition 𝑄 and an arbitrary interval [𝐶, 𝐷], a partition 𝑆 on [𝐶, 𝐷] is built such that it preserves any periodic function piecewise continuous on 𝑄 will be piecewise continuous on 𝑆, with the same limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑇 = (𝐵𝐴)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))       (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
 
Theoremfourierdlem55 43244* 𝑈 is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))       (𝜑𝑈:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem56 43245* Derivative of the 𝐾 function on an interval not containing ' 0 '. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   (𝜑 → (𝐴(,)𝐵) ⊆ ((-π[,]π) ∖ {0}))    &   ((𝜑𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)       (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))
 
Theoremfourierdlem57 43246* The derivative of 𝑂. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)    &   (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))       ((𝜑 → ((ℝ D 𝑂):(𝐴(,)𝐵)⟶ℝ ∧ (ℝ D 𝑂) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2))))) ∧ (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑠 / 2))))
 
Theoremfourierdlem58 43247* The derivative of 𝐾 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠𝐴 ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))    &   (𝜑𝐴 ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ 𝐴)    &   (𝜑𝐴 ∈ (topGen‘ran (,)))       (𝜑 → (ℝ D 𝐾) ∈ (𝐴cn→ℝ))
 
Theoremfourierdlem59 43248* The derivative of 𝐻 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))       (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
 
Theoremfourierdlem60 43249* Given a differentiable function 𝐹, with finite limit of the derivative at 𝐴 the derived function 𝐻 has a limit at 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑𝑌 ∈ (𝐹 lim 𝐵))    &   𝐺 = (ℝ D 𝐹)    &   (𝜑 → dom 𝐺 = (𝐴(,)𝐵))    &   (𝜑𝐸 ∈ (𝐺 lim 𝐵))    &   𝐻 = (𝑠 ∈ ((𝐴𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠))    &   𝑁 = (𝑠 ∈ ((𝐴𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌))    &   𝐷 = (𝑠 ∈ ((𝐴𝐵)(,)0) ↦ 𝑠)       (𝜑𝐸 ∈ (𝐻 lim 0))
 
Theoremfourierdlem61 43250* Given a differentiable function 𝐹, with finite limit of the derivative at 𝐴 the derived function 𝐻 has a limit at 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑𝑌 ∈ (𝐹 lim 𝐴))    &   𝐺 = (ℝ D 𝐹)    &   (𝜑 → dom 𝐺 = (𝐴(,)𝐵))    &   (𝜑𝐸 ∈ (𝐺 lim 𝐴))    &   𝐻 = (𝑠 ∈ (0(,)(𝐵𝐴)) ↦ (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠))    &   𝑁 = (𝑠 ∈ (0(,)(𝐵𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌))    &   𝐷 = (𝑠 ∈ (0(,)(𝐵𝐴)) ↦ 𝑠)       (𝜑𝐸 ∈ (𝐻 lim 0))
 
Theoremfourierdlem62 43251 The function 𝐾 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑦 ∈ (-π[,]π) ↦ if(𝑦 = 0, 1, (𝑦 / (2 · (sin‘(𝑦 / 2))))))       𝐾 ∈ ((-π[,]π)–cn→ℝ)
 
Theoremfourierdlem63 43252* The upper bound of intervals in the moved partition are mapped to points that are not greater than the corresponding upper bounds in the original partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑇 = (𝐵𝐴)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝐾 ∈ (0...𝑀))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   (𝜑𝑌 ∈ ((𝑆𝐽)[,)(𝑆‘(𝐽 + 1))))    &   (𝜑 → (𝐸𝑌) < (𝑄𝐾))    &   𝑋 = ((𝑄𝐾) − ((𝐸𝑌) − 𝑌))       (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄𝐾))
 
Theoremfourierdlem64 43253* The partition 𝑉 is finer than 𝑄, when 𝑄 is moved on the same interval where 𝑉 lies. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑇 = (𝐵𝐴)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝐿 = sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉𝐽)}, ℝ, < )    &   𝐼 = sup({𝑗 ∈ (0..^𝑀) ∣ ((𝑄𝑗) + (𝐿 · 𝑇)) ≤ (𝑉𝐽)}, ℝ, < )       (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐿 ∈ ℤ) ∧ ∃𝑖 ∈ (0..^𝑀)∃𝑙 ∈ ℤ ((𝑉𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))))
 
Theoremfourierdlem65 43254* The distance of two adjacent points in the moved partition is preserved in the original partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵𝐴))) ∈ ran 𝑄})) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵𝐴))) ∈ ran 𝑄})))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝑍 = ((𝑆𝑗) + (𝐵 − (𝐸‘(𝑆𝑗))))       ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆𝑗)))
 
Theoremfourierdlem66 43255* Value of the 𝐺 function when the argument is not zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   𝐴 = ((-π[,]π) ∖ {0})       (((𝜑𝑛 ∈ ℕ) ∧ 𝑠𝐴) → (𝐺𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷𝑛)‘𝑠))))
 
Theoremfourierdlem67 43256* 𝐺 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   (𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))       (𝜑𝐺:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem68 43257* The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))    &   (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   ((𝜑𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹𝑡)) ≤ 𝐷)    &   (𝜑𝐸 ∈ ℝ)    &   ((𝜑𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸)    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))       (𝜑 → (dom (ℝ D 𝑂) = (𝐴(,)𝐵) ∧ ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏))
 
Theoremfourierdlem69 43258* A piecewise continuous function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑𝐹 ∈ 𝐿1)
 
Theoremfourierdlem70 43259* A piecewise continuous function is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑 → (𝑄‘0) = 𝐴)    &   (𝜑 → (𝑄𝑀) = 𝐵)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹𝑠)) ≤ 𝑥)
 
Theoremfourierdlem71 43260* A periodic piecewise continuous function, possibly undefined on a finite set in each periodic interval, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → dom 𝐹 ⊆ ℝ)    &   (𝜑𝐹:dom 𝐹⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑 → (𝑄‘0) = 𝐴)    &   (𝜑 → (𝑄𝑀) = 𝐵)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)    &   (((𝜑𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))    &   𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))       (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹𝑥)) ≤ 𝑦)
 
Theoremfourierdlem72 43261* The derivative of 𝑂 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   (𝜑𝑈 ∈ (0..^𝑀))    &   (𝜑 → (𝐴(,)𝐵) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1))))    &   𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))    &   𝐾 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))    &   𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((𝐻𝑠) · (𝐾𝑠)))       (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))
 
Theoremfourierdlem73 43262* A version of the Riemann Lebesgue lemma: as 𝑟 increases, the integral in 𝑆 goes to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑 → (𝑄‘0) = 𝐴)    &   (𝜑 → (𝑄𝑀) = 𝐵)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   𝐺 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐺(abs‘(𝐺𝑥)) ≤ 𝑦)    &   𝑆 = (𝑟 ∈ ℝ+ ↦ ∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑟 · 𝑥))) d𝑥)    &   𝐷 = (𝑥 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹𝑥))))       (𝜑 → ∀𝑒 ∈ ℝ+𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)(abs‘∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑟 · 𝑥))) d𝑥) < 𝑒)
 
Theoremfourierdlem74 43263* Given a piecewise smooth function 𝐹, the derived function 𝐻 has a limit at the upper bound of each interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ran 𝑉)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐺 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)    &   (𝜑𝐸 ∈ ((𝐺 ↾ (-∞(,)𝑋)) lim 𝑋))    &   𝐴 = if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1))))       ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))
 
Theoremfourierdlem75 43264* Given a piecewise smooth function 𝐹, the derived function 𝐻 has a limit at the lower bound of each interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ran 𝑉)    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐺 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)    &   (𝜑𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim 𝑋))    &   𝐴 = if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖)))       ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
 
Theoremfourierdlem76 43265* Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   𝑁 = ((♯‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   𝐷 = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))    &   𝐸 = (((if((𝑆𝑗) = (𝑄𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))    &   (𝜒 ↔ (((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))))       ((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
 
Theoremfourierdlem77 43266* If 𝐻 is bounded, then 𝑈 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   (𝜑 → ∃𝑎 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻𝑠)) ≤ 𝑎)       (𝜑 → ∃𝑏 ∈ ℝ+𝑠 ∈ (-π[,]π)(abs‘(𝑈𝑠)) ≤ 𝑏)
 
Theoremfourierdlem78 43267* 𝐺 is continuous when restricted on an interval not containing 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐴 ∈ (-π[,]π))    &   (𝜑𝐵 ∈ (-π[,]π))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   (𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))       (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
 
Theoremfourierdlem79 43268* 𝐸 projects every interval of the partition induced by 𝑆 on 𝐻 into a corresponding interval of the partition induced by 𝑄 on [𝐴, 𝐵]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑇 = (𝐵𝐴)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝑍 = ((𝑆𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2)))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ))       ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆𝑗)))(,)(𝑄‘((𝐼‘(𝑆𝑗)) + 1))))
 
Theoremfourierdlem80 43269* The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))    &   𝐼 = ((𝑋 + (𝑆𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1))))    &   ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤)    &   ((𝜑𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧)    &   (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))    &   ((𝜑𝑗 ∈ (0..^𝑁)) → (𝑆𝑗) < (𝑆‘(𝑗 + 1)))    &   ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝑆𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵))    &   (((𝜑𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆𝑘)(,)(𝑆‘(𝑘 + 1))))    &   ((𝜑𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹𝐼)):𝐼⟶ℝ)    &   𝑌 = (𝑠 ∈ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))    &   (𝜒 ↔ (((((𝜑𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡𝐼 (abs‘(𝐹𝑡)) ≤ 𝑤) ∧ ∀𝑡𝐼 (abs‘((ℝ D (𝐹𝐼))‘𝑡)) ≤ 𝑧))       (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)
 
Theoremfourierdlem81 43270* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by its period 𝑇. In this lemma, 𝑇 is assumed to be strictly positive. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) + 𝑇))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   𝐺 = (𝑥 ∈ ((𝑄𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹𝑥))))    &   𝐻 = (𝑥 ∈ ((𝑆𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐺‘(𝑥𝑇)))       (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem82 43271* Integral by substitution, adding a constant to the function's argument, for a function on an open interval with finite limits ad boundary points. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)    &   (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → ∫(𝐴[,]𝐵)(𝐹𝑡) d𝑡 = ∫((𝐴𝑋)[,](𝐵𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)
 
Theoremfourierdlem83 43272* The fourier partial sum for 𝐹 rewritten as an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐴 = (𝑛 ∈ ℕ0 ↦ (∫𝐶((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))    &   (𝜑𝑋 ∈ ℝ)    &   𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑆𝑁) = ∫𝐶((𝐹𝑥) · ((𝐷𝑁)‘(𝑥𝑋))) d𝑥)
 
Theoremfourierdlem84 43273* If 𝐹 is piecewise coninuous and 𝐷 is continuous, then 𝐺 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐷 ∈ (ℝ–cn→ℝ))    &   𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷𝑠)))       (𝜑𝐺 ∈ 𝐿1)
 
Theoremfourierdlem85 43274* Limit of the function 𝐺 at the lower bounds of the partition intervals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ran 𝑉)    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   (𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐼 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ)    &   (𝜑𝐸 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim 𝑋))    &   𝐴 = ((if((𝑉𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄𝑖))) · (𝐾‘(𝑄𝑖))) · (𝑆‘(𝑄𝑖)))       ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐺 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))
 
Theoremfourierdlem86 43275* Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   𝑁 = ((♯‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   𝐷 = (((if((𝑆‘(𝑗 + 1)) = (𝑄‘(𝑈 + 1)), 𝑈 / 𝑖𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2)))))    &   𝐸 = (((if((𝑆𝑗) = (𝑄𝑈), 𝑈 / 𝑖𝑅, (𝐹‘(𝑋 + (𝑆𝑗)))) − 𝐶) / (𝑆𝑗)) · ((𝑆𝑗) / (2 · (sin‘((𝑆𝑗) / 2)))))    &   𝑈 = (𝑖 ∈ (0..^𝑀)((𝑆𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))       ((𝜑𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) lim (𝑆𝑗))) ∧ (𝑂 ↾ ((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))
 
Theoremfourierdlem87 43276* The integral of 𝐺 goes uniformly ( with respect to 𝑛) to zero if the measure of the domain of integration goes to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻𝑠)) ≤ 𝑥)    &   ((𝜑𝑛 ∈ ℕ) → 𝐺 ∈ 𝐿1)    &   𝐷 = ((𝑒 / 3) / 𝑎)    &   (𝜒 ↔ (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ+ ∧ ∀𝑛 ∈ ℕ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐺𝑠)) ≤ 𝑎) ∧ 𝑢 ∈ dom vol) ∧ (𝑢 ⊆ (-π[,]π) ∧ (vol‘𝑢) ≤ 𝐷)) ∧ 𝑛 ∈ ℕ))       ((𝜑𝑒 ∈ ℝ+) → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((𝑢 ⊆ (-π[,]π) ∧ (vol‘𝑢) ≤ 𝑑) → ∀𝑘 ∈ ℕ (abs‘∫𝑢((𝑈𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)))
 
Theoremfourierdlem88 43277* Given a piecewise continuous function 𝐹, a continuous function 𝐾 and a continuous function 𝑆, the function 𝐺 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ran 𝑉)    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   (𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐼 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)    &   (𝜑𝐶 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐷 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim 𝑋))       (𝜑𝐺 ∈ 𝐿1)
 
Theoremfourierdlem89 43278* Given a piecewise continuous function and changing the interval and the partition, the limit at the lower bound of each interval of the moved partition is still finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝑍‘(𝐸𝑥))}, ℝ, < ))    &   𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅)       (𝜑 → if((𝑍‘(𝐸‘(𝑆𝐽))) = (𝑄‘(𝐼‘(𝑆𝐽))), (𝑉‘(𝐼‘(𝑆𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆𝐽))))) ∈ ((𝐹 ↾ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1)))) lim (𝑆𝐽)))
 
Theoremfourierdlem90 43279* Given a piecewise continuous function, it is still continuous with respect to an open interval of the moved partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))    &   𝐺 = (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1)))))    &   𝑅 = (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦𝑈)))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → (𝐹 ↾ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1)))) ∈ (((𝑆𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))
 
Theoremfourierdlem91 43280* Given a piecewise continuous function and changing the interval and the partition, the limit at the upper bound of each interval of the moved partition is still finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1))))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝑍‘(𝐸𝑥))}, ℝ, < ))    &   𝑊 = (𝑖 ∈ (0..^𝑀) ↦ 𝐿)       (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆𝐽))), (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1)))) lim (𝑆‘(𝐽 + 1))))
 
Theoremfourierdlem92 43281* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by its period 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑇 ∈ ℝ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) + 𝑇))    &   𝐻 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem93 43282* Integral by substitution (the domain is shifted by 𝑋) for a piecewise continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐹:(-π[,]π)⟶ℂ)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑 → ∫(-π[,]π)(𝐹𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)
 
Theoremfourierdlem94 43283* For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑇 = (2 · π)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)) ≠ ∅)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))) ≠ ∅)       (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅))
 
Theoremfourierdlem95 43284* Algebraic manipulation of integrals, used by other lemmas. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ran 𝑉)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   𝐼 = (ℝ D 𝐹)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ)    &   (𝜑𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐴 ∈ dom vol)    &   (𝜑𝐴 ⊆ ((-π[,]π) ∖ {0}))    &   𝐸 = (𝑛 ∈ ℕ ↦ (∫𝐴(𝐺𝑠) d𝑠 / π))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   (𝜑𝑂 ∈ ℝ)    &   ((𝜑𝑠𝐴) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑂)    &   ((𝜑𝑛 ∈ ℕ) → ∫𝐴((𝐷𝑛)‘𝑠) d𝑠 = (1 / 2))       ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) + (𝑂 / 2)) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠)
 
Theoremfourierdlem96 43285* limit for 𝐹 at the lower bound of an interval for the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   (𝜑𝐽 ∈ (0..^((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))    &   𝑉 = (℩𝑔𝑔 Isom < , < ((0...((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))       (𝜑 → if(((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘(𝑉𝐽))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝐽))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝐽))), (𝐹‘((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘(𝑉𝐽))))) ∈ ((𝐹 ↾ ((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))) lim (𝑉𝐽)))
 
Theoremfourierdlem97 43286* 𝐹 is continuous on the intervals induced by the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐺 = (ℝ D 𝐹)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   (𝜑𝐽 ∈ (0..^((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))    &   𝑉 = (℩𝑔𝑔 Isom < , < ((0...((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))    &   𝐻 = (𝑠 ∈ ℝ ↦ if(𝑠 ∈ dom 𝐺, (𝐺𝑠), 0))       (𝜑 → (𝐺 ↾ ((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ))
 
Theoremfourierdlem98 43287* 𝐹 is continuous on the intervals induced by the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   (𝜑𝐽 ∈ (0..^((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))    &   𝑉 = (℩𝑔𝑔 Isom < , < ((0...((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))       (𝜑 → (𝐹 ↾ ((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ))
 
Theoremfourierdlem99 43288* limit for 𝐹 at the upper bound of an interval for the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   (𝜑𝐽 ∈ (0..^((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)))    &   𝑉 = (℩𝑔𝑔 Isom < , < ((0...((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1)), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃ ∈ ℤ (𝑦 + ( · 𝑇)) ∈ ran 𝑄})))       (𝜑 → if(((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝐽)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉𝐽))), (𝐹‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑉𝐽)(,)(𝑉‘(𝐽 + 1)))) lim (𝑉‘(𝐽 + 1))))
 
Theoremfourierdlem100 43289* A piecewise continuous function is integrable on any closed interval. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐽‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹𝑥)) ∈ 𝐿1)
 
Theoremfourierdlem101 43290* Integral by substitution for a piecewise continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹𝑡) · ((𝐷𝑁)‘(𝑡𝑋))))    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐹:(-π[,]π)⟶ℂ)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑 → ∫(-π[,]π)((𝐹𝑡) · ((𝐷𝑁)‘(𝑡𝑋))) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑁)‘𝑠)) d𝑠)
 
Theoremfourierdlem102 43291* For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑇 = (2 · π)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))    &   (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)    &   (𝜑𝐺 ∈ (dom 𝐺cn→ℂ))    &   ((𝜑𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim 𝑥) ≠ ∅)    &   ((𝜑𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim 𝑥) ≠ ∅)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇)))    &   𝐻 = ({-π, π, (𝐸𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺))    &   𝑀 = ((♯‘𝐻) − 1)    &   𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻))       (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅))
 
Theoremfourierdlem103 43292* The half lower part of the integral equal to the fourier partial sum, converges to half the left limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ran 𝑉)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   𝑍 = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)    &   𝐸 = (𝑛 ∈ ℕ ↦ (∫(-π(,)0)(𝐺𝑠) d𝑠 / π))    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐴 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐵 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   𝑂 = (𝑈 ↾ (-π[,]𝑑))    &   𝑇 = ({-π, 𝑑} ∪ (ran 𝑄 ∩ (-π(,)𝑑)))    &   𝑁 = ((♯‘𝑇) − 1)    &   𝐽 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝐶 = (𝑙 ∈ (0..^𝑀)((𝐽𝑘)(,)(𝐽‘(𝑘 + 1))) ⊆ ((𝑄𝑙)(,)(𝑄‘(𝑙 + 1))))    &   (𝜒 ↔ (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ (-π(,)0)) ∧ 𝑘 ∈ ℕ) ∧ (abs‘∫(𝑑(,)0)((𝑈𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)) ∧ (abs‘∫(-π(,)𝑑)((𝑈𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)))       (𝜑𝑍 ⇝ (𝑊 / 2))
 
Theoremfourierdlem104 43293* The half upper part of the integral equal to the fourier partial sum, converges to half the right limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ran 𝑉)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹𝑡)) ≤ 𝑤)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧)    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉𝑖)(,)(𝑉‘(𝑖 + 1)))) lim (𝑉‘(𝑖 + 1))))    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠)))    &   𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈𝑠) · (𝑆𝑠)))    &   𝑍 = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑚)‘𝑠)) d𝑠)    &   𝐸 = (𝑛 ∈ ℕ ↦ (∫(0(,)π)(𝐺𝑠) d𝑠 / π))    &   (𝜑𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋))    &   (𝜑𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐴 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim 𝑋))    &   (𝜑𝐵 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim 𝑋))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   𝑂 = (𝑈 ↾ (𝑑[,]π))    &   𝑇 = ({𝑑, π} ∪ (ran 𝑄 ∩ (𝑑(,)π)))    &   𝑁 = ((♯‘𝑇) − 1)    &   𝐽 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝐶 = (𝑙 ∈ (0..^𝑀)((𝐽𝑘)(,)(𝐽‘(𝑘 + 1))) ⊆ ((𝑄𝑙)(,)(𝑄‘(𝑙 + 1))))    &   (𝜒 ↔ (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ (0(,)π)) ∧ 𝑘 ∈ ℕ) ∧ (abs‘∫(0(,)𝑑)((𝑈𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)) ∧ (abs‘∫(𝑑(,)π)((𝑈𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)))       (𝜑𝑍 ⇝ (𝑌 / 2))
 
Theoremfourierdlem105 43294* A piecewise continuous function is integrable on any closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ (𝐶(,)+∞))       (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹𝑥)) ∈ 𝐿1)
 
Theoremfourierdlem106 43295* For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝑇 = (2 · π)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π))    &   (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin)    &   (𝜑𝐺 ∈ (dom 𝐺cn→ℂ))    &   ((𝜑𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim 𝑥) ≠ ∅)    &   ((𝜑𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim 𝑥) ≠ ∅)    &   (𝜑𝑋 ∈ ℝ)       (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅))
 
Theoremfourierdlem107 43296* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any positive value 𝑋. This lemma generalizes fourierdlem92 43281 where the integral was shifted by the exact period. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ+)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴𝑋) ∧ (𝑝𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({(𝐴𝑋), 𝐴} ∪ {𝑦 ∈ ((𝐴𝑋)[,]𝐴) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝑍‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → ∫((𝐴𝑋)[,](𝐵𝑋))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem108 43297* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any positive value 𝑋. This lemma generalizes fourierdlem92 43281 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ+)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑 → ∫((𝐴𝑋)[,](𝐵𝑋))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem109 43298* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any value 𝑋. This lemma generalizes fourierdlem92 43281 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴𝑋) ∧ (𝑝𝑚) = (𝐵𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({(𝐴𝑋), (𝐵𝑋)} ∪ {𝑥 ∈ ((𝐴𝑋)[,](𝐵𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((♯‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄𝑗) ≤ (𝐽‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → ∫((𝐴𝑋)[,](𝐵𝑋))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem110 43299* The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any value 𝑋. This lemma generalizes fourierdlem92 43281 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))       (𝜑 → ∫((𝐴𝑋)[,](𝐵𝑋))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremfourierdlem111 43300* The fourier partial sum for 𝐹 is the sum of two integrals, with the same integrand involving 𝐹 and the Dirichlet Kernel 𝐷, but on two opposite intervals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑛 ∈ ℕ0 ↦ (∫(-π(,)π)((𝐹𝑡) · (cos‘(𝑛 · 𝑡))) d𝑡 / π))    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹𝑡) · (sin‘(𝑛 · 𝑡))) d𝑡 / π))    &   𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵𝑛) · (sin‘(𝑛 · 𝑋))))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐹:ℝ⟶ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝐺 = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷𝑛)‘𝑥)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   𝑇 = (2 · π)    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝𝑚) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑊 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄𝑖) − 𝑋))       ((𝜑𝑛 ∈ ℕ) → (𝑆𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷𝑛)‘𝑠)) d𝑠))
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