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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement

21.31.16  Fourier Series

Theoremfourierdlem1 39001 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 39002* 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 39003* 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 39004* 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 39005* 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 39006 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 39007* 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 39008 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 39009* 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 39010 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 39011* 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 39012* 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 39013* 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 39014* 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 39015* 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 39016* 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 39017* 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 39018* 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 39019* 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 39020* 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 39021* 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 39022* 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 39023* 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 39024 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ((-π[,]π) ∖ {0}) → (𝐴 mod (2 · π)) ≠ 0)

Theoremfourierdlem25 39025* 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 39026* 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 39027 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 39028* 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 39029* 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 39030* 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 39031* 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 39032 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 39033 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 39034* 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 39035 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 39036* 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 39037* 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 39038* 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 39039* 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 39040* 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 39041* 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 39042* 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 39043 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 39044 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 39045* 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 39046* 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 39047* 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 39048* 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 39049* 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 39050* 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 39051* 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 39052* 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 39053* 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 39054* 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 39055* 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 39056* 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 39057* 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 39058* 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 39059* 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 39060* 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 39061 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 39062* 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 39063* 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 39064* 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 39065* 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 39066* 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 39067* 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 39068* 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 39069* 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 39070* 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 39071* 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 39072* 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 39073* 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 39074* 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 39075* 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 39076* 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 39077* 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 39078* 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 39079* 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 39080* 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 39081* 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 39082* 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 39083* 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 39084* 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 39085* 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 39086* 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 39087* 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 39088* 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 39089* 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 39090* 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 39091* 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 39092* 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 39093* 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 39094* 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 39095* 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 39096* 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 39097* 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 39098* 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 39099* 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 39100* 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)

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