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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdirkercncflem3 38901* The Dirichlet Kernel is continuous at 𝑌 points that are multiples of (2 · π). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
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)))))))    &   𝐴 = (𝑌 − π)    &   𝐵 = (𝑌 + π)    &   𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))    &   𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((2 · π) · (sin‘(𝑦 / 2))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → (𝑌 mod (2 · π)) = 0)       (𝜑 → ((𝐷𝑁)‘𝑌) ∈ ((𝐷𝑁) lim 𝑌))
 
Theoremdirkercncflem4 38902* The Dirichlet Kernel is continuos at points that are not multiple of 2 π . This is the easier condition, for the proof of the continuity of the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
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)))))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → (𝑌 mod (2 · π)) ≠ 0)    &   𝐴 = (⌊‘(𝑌 / (2 · π)))    &   𝐵 = (𝐴 + 1)    &   𝐶 = (𝐴 · (2 · π))    &   𝐸 = (𝐵 · (2 · π))       (𝜑 → (𝐷𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑌))
 
Theoremdirkercncf 38903* For any natural number 𝑁, the Dirichlet Kernel (𝐷𝑁) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
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)))))))       (𝑁 ∈ ℕ → (𝐷𝑁) ∈ (ℝ–cn→ℝ))
 
20.31.16  Fourier Series
 
Theoremfourierdlem1 38904 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑𝐼 ∈ (0..^𝑀))    &   (𝜑𝑋 ∈ ((𝑄𝐼)[,](𝑄‘(𝐼 + 1))))       (𝜑𝑋 ∈ (𝐴[,]𝐵))
 
Theoremfourierdlem2 38905* Membership in a 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)))})       (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
 
Theoremfourierdlem3 38906* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})       (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
 
Theoremfourierdlem4 38907* 𝐸 is a function that maps any point to a periodic corresponding point in (𝐴, 𝐵]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))       (𝜑𝐸:ℝ⟶(𝐴(,]𝐵))
 
Theoremfourierdlem5 38908* 𝑆 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑆 = (𝑥 ∈ (-π[,]π) ↦ (sin‘((𝑋 + (1 / 2)) · 𝑥)))       (𝑋 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem6 38909 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 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝐼 < 𝐽)    &   (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴[,]𝐵))       (𝜑𝐽 = (𝐼 + 1))
 
Theoremfourierdlem7 38910* The difference between a point and it's periodic image in the interval, is decreasing. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)       (𝜑 → ((𝐸𝑌) − 𝑌) ≤ ((𝐸𝑋) − 𝑋))
 
Theoremfourierdlem8 38911 A partition interval is a subset of the partitioned 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))
 
Theoremfourierdlem9 38912* 𝐻 is a complex 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)))       (𝜑𝐻:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem10 38913 Condition on the bounds of a non empty subinterval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))       (𝜑 → (𝐴𝐶𝐷𝐵))
 
Theoremfourierdlem11 38914* 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 38915* 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 38916* 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 38917* 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 38918* 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 38919* 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 38920* 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 38921* 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 38922* 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 38923* 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 38924* 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 38925* 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 38926* 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 38927 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ((-π[,]π) ∖ {0}) → (𝐴 mod (2 · π)) ≠ 0)
 
Theoremfourierdlem25 38928* 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 38929* 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 38930 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 38931* 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 38932* 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 38933* 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 38934* 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)
 
Theoremfourierdlem31OLD 38935* 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.) Obsolete version of fourierdlem31 38934 as of 29-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.)
𝑖𝜑    &   𝑟𝜑    &   𝑖𝑉    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑖𝐴𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)    &   𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}    &   𝑉 = (𝑖𝐴 ↦ sup(𝑀, ℝ, < ))    &   𝑁 = sup(ran 𝑉, ℝ, < )       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖𝐴 𝜒)
 
Theoremfourierdlem32 38936 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 38937 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 38938* 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 38939 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 38940* 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 38941* 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 38942* 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 38943* 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 38944* 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 38945* 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 38946* 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)
 
Theoremfourierdlem42OLD 38947* The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of fourierdlem42 38946 as of 29-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.)
(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 = sup(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 38948 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 38949 A condition for having (sin‘(𝐴 / 2)) non zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)
 
Theoremfourierdlem46 38950* 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 38951* 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 38952* 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 38953* 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 38954* 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 38955* 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 38956* 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 38957* 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 38958* 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 38959* 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 38960* Derivative of the 𝐾 function on an interval non 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 38961* 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 38962* 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 38963* 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 38964* 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 38965* 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 38966 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 38967* 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 38968* 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 38969* 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 38970* 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 38971* 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 38972* 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 38973* 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 38974* 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 38975* 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 38976* 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 38977* 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 38978* 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 38979* 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 38980* 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 38981* 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 38982* 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 38983* 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 38984* 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 38985* 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 38986* 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 38987* 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 38988* 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 38989* 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 38990* 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 38991* 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 38992* 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 38993* 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 38994* 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 38995* 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 38996* 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 38997* 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 38998* 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 38999* 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 39000* 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)))
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