P. 468 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 46701-46800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	fourierdlem22 46701*	The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝐶 = (-π(,)π) & ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ 𝐿¹) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫𝐶((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) ⇒ ⊢ (𝜑 → ((𝑛 ∈ ℕ₀ → (𝐴‘𝑛) ∈ ℝ) ∧ (𝑛 ∈ ℕ → (𝐵‘𝑛) ∈ ℝ)))

Theorem	fourierdlem23 46702*	If 𝐹 is continuous and 𝑋 is constant, then (𝐹‘(𝑋 + 𝑠)) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ) & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) & ⊢ (𝜑 → 𝐵 ⊆ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐵) → (𝑋 + 𝑠) ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑠 ∈ 𝐵 ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (𝐵–cn→ℂ))

Theorem	fourierdlem24 46703	A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ((-π[,]π) ∖ {0}) → (𝐴 mod (2 · π)) ≠ 0)

Theorem	fourierdlem25 46704*	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.)
⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ((𝑄‘0)[,](𝑄‘𝑀))) & ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝑄) & ⊢ 𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝐶}, ℝ, < ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄‘𝑗)(,)(𝑄‘(𝑗 + 1))))

Theorem	fourierdlem26 46705*	Periodic image of a point 𝑌 that's in the period that begins with the point 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → (𝐸‘𝑋) = 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝑋(,](𝑋 + 𝑇))) ⇒ ⊢ (𝜑 → (𝐸‘𝑌) = (𝐴 + (𝑌 − 𝑋)))

Theorem	fourierdlem27 46706	A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ^) & ⊢ (𝜑 → 𝐵 ∈ ℝ^) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) ⇒ ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵))

Theorem	fourierdlem28 46707*	Derivative of (𝐹‘(𝑋 + 𝑠)). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐷 = (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) & ⊢ (𝜑 → 𝐷:((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ) ⇒ ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐷‘(𝑋 + 𝑠))))

Theorem	fourierdlem29 46708*	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) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))

Theorem	fourierdlem30 46709*	Sum of three small pieces is less than ε. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿¹) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝑋 = (abs‘𝐴) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ 𝑌 = (abs‘𝐶) & ⊢ 𝑍 = (abs‘∫𝐼(𝐹 · -𝐺) d𝑥) & ⊢ (𝜑 → 𝐸 ∈ ℝ⁺) & ⊢ (𝜑 → 𝑅 ∈ ℝ) & ⊢ (𝜑 → ((((𝑋 + 𝑌) + 𝑍) / 𝐸) + 1) ≤ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐵) ≤ 1) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐷) ≤ 1) ⇒ ⊢ (𝜑 → (abs‘(((𝐴 · -(𝐵 / 𝑅)) − (𝐶 · -(𝐷 / 𝑅))) − ∫𝐼(𝐹 · -(𝐺 / 𝑅)) d𝑥)) < 𝐸)

Theorem	fourierdlem31 46710*	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.)
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑟𝜑 & ⊢ Ⅎ𝑖𝑉 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒) & ⊢ 𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} & ⊢ 𝑉 = (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < )) & ⊢ 𝑁 = sup(ran 𝑉, ℝ, < ) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)

Theorem	fourierdlem32 46711	Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) & ⊢ 𝑌 = if(𝐶 = 𝐴, 𝑅, (𝐹‘𝐶)) & ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐶))

Theorem	fourierdlem33 46712	Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) & ⊢ 𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹‘𝐷)) & ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ⇒ ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim_ℂ 𝐷))

Theorem	fourierdlem34 46713*	A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) ⇒ ⊢ (𝜑 → 𝑄:(0...𝑀)–1-1→ℝ)

Theorem	fourierdlem35 46714	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐼 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) & ⊢ (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴(,]𝐵)) & ⊢ (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐼 = 𝐽)

Theorem	fourierdlem36 46715*	𝐹 is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐹 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) & ⊢ 𝑁 = ((♯‘𝐴) − 1) ⇒ ⊢ (𝜑 → 𝐹 Isom < , < ((0...𝑁), 𝐴))

Theorem	fourierdlem37 46716*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))})))

Theorem	fourierdlem38 46717*	The function 𝐹 is continuous on every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ)) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ 𝐻 = (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹)) & ⊢ (𝜑 → ran 𝑄 = 𝐻) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))

Theorem	fourierdlem39 46718*	Integration by parts of ∫(𝐴(,)𝐵)((𝐹‘𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ (𝜑 → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐺‘𝑥)) ≤ 𝑦) & ⊢ (𝜑 → 𝑅 ∈ ℝ⁺) ⇒ ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝐹‘𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 = ((((𝐹‘𝐵) · -((cos‘(𝑅 · 𝐵)) / 𝑅)) − ((𝐹‘𝐴) · -((cos‘(𝑅 · 𝐴)) / 𝑅))) − ∫(𝐴(,)𝐵)((𝐺‘𝑥) · -((cos‘(𝑅 · 𝑥)) / 𝑅)) d𝑥))

Theorem	fourierdlem40 46719*	𝐻 is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ (-π[,]π)) & ⊢ (𝜑 → 𝐵 ∈ (-π[,]π)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ)) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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))

Theorem	fourierdlem41 46720*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))

Theorem	fourierdlem42 46721*	The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 < 𝐶) & ⊢ 𝑇 = (𝐶 − 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ 𝐷 = (abs ∘ − ) & ⊢ 𝐼 = ((𝐴 × 𝐴) ∖ I ) & ⊢ 𝑅 = ran (𝐷 ↾ 𝐼) & ⊢ 𝐸 = inf(𝑅, ℝ, < ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (𝐽 ↾_t (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))) ⇒ ⊢ (𝜑 → 𝐻 ∈ Fin)

Theorem	fourierdlem43 46722	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

Theorem	fourierdlem44 46723	A condition for having (sin‘(𝐴 / 2)) nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)

Theorem	fourierdlem46 46724*	The function 𝐹 has a limit at the bounds of every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (dom 𝐹–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑄 Isom < , < ((0...𝑀), 𝐻)) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶𝐻) & ⊢ (𝜑 → 𝐼 ∈ (0..^𝑀)) & ⊢ (𝜑 → (𝑄‘𝐼) < (𝑄‘(𝐼 + 1))) & ⊢ (𝜑 → ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐻 = ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹)) & ⊢ (𝜑 → ran 𝑄 = 𝐻) ⇒ ⊢ (𝜑 → (((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘𝐼)) ≠ ∅ ∧ ((𝐹 ↾ ((𝑄‘𝐼)(,)(𝑄‘(𝐼 + 1)))) lim_ℂ (𝑄‘(𝐼 + 1))) ≠ ∅))

Theorem	fourierdlem47 46725*	For 𝑟 large enough, the final expression is less than the given positive 𝐸. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐹) ∈ 𝐿¹) & ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (𝑥 ∈ 𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿¹) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ ℂ) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℂ) → 𝐺 ∈ ℂ) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ) → (abs‘𝐺) ≤ 1) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ 𝑋 = (abs‘𝐴) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ 𝑌 = (abs‘𝐶) & ⊢ 𝑍 = ∫𝐼(abs‘𝐹) d𝑥 & ⊢ (𝜑 → 𝐸 ∈ ℝ⁺) & ⊢ ((𝜑 ∧ 𝑟 ∈ ℂ) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (abs‘𝐵) ≤ 1) & ⊢ ((𝜑 ∧ 𝑟 ∈ ℂ) → 𝐷 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑟 ∈ 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)

Theorem	fourierdlem48 46726*	The given periodic function 𝐹 has a right limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) & ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑄‘𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑦 = (𝑋 + (𝑘 · 𝑇)))) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅)

Theorem	fourierdlem49 46727*	The given periodic function 𝐹 has a left limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐷 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍‘𝑥))) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅)

Theorem	fourierdlem50 46728*	Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π)) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵))) & ⊢ 𝑁 = ((♯‘𝑇) − 1) & ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ 𝑈 = (℩𝑖 ∈ (0..^𝑀)((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) & ⊢ (𝜒 ↔ ((((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ (0..^𝑀)) ∧ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄‘𝑘)(,)(𝑄‘(𝑘 + 1))))) ⇒ ⊢ (𝜑 → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄‘𝑈)(,)(𝑄‘(𝑈 + 1)))))

Theorem	fourierdlem51 46729*	𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) & ⊢ (𝜑 → 0 < 𝐵) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝐶 ∈ Fin) & ⊢ (𝜑 → 𝐶 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → (𝐸‘𝑋) ∈ 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} ⇒ ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)

Theorem	fourierdlem52 46730*	d16:d17,d18:jca \|- ( ph -> ( ( S 0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑇 ∈ Fin) & ⊢ 𝑁 = ((♯‘𝑇) − 1) & ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑇 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑇) ⇒ ⊢ (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆‘𝑁) = 𝐵))

Theorem	fourierdlem53 46731*	The limit of 𝐹(𝑠) at (𝑋 + 𝐷) is the limit of 𝐹(𝑋 + 𝑠) at 𝐷. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝐺 = (𝑠 ∈ 𝐴 ↦ (𝐹‘(𝑋 + 𝑠))) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → (𝑋 + 𝑠) ∈ 𝐵) & ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → 𝑠 ≠ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ ((𝐹 ↾ 𝐵) lim_ℂ (𝑋 + 𝐷))) & ⊢ (𝜑 → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐺 lim_ℂ 𝐷))

Theorem	fourierdlem54 46732*	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 = (𝐵 − 𝐴) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))

Theorem	fourierdlem55 46733*	𝑈 is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) ⇒ ⊢ (𝜑 → 𝑈:(-π[,]π)⟶ℝ)

Theorem	fourierdlem56 46734*	Derivative of the 𝐾 function on an interval not containing ' 0 '. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((-π[,]π) ∖ {0})) & ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0) ⇒ ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾‘𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))

Theorem	fourierdlem57 46735*	The derivative of 𝑂. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐶 ∈ 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))))

Theorem	fourierdlem58 46736*	The derivative of 𝐾 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐾 = (𝑠 ∈ 𝐴 ↦ (𝑠 / (2 · (sin‘(𝑠 / 2))))) & ⊢ (𝜑 → 𝐴 ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (topGen‘ran (,))) ⇒ ⊢ (𝜑 → (ℝ D 𝐾) ∈ (𝐴–cn→ℝ))

Theorem	fourierdlem59 46737*	The derivative of 𝐻 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠)) ⇒ ⊢ (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))

Theorem	fourierdlem60 46738*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐵)) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐵)) & ⊢ 𝐻 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ (((𝐹‘(𝐵 + 𝑠)) − 𝑌) / 𝑠)) & ⊢ 𝑁 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ ((𝐹‘(𝐵 + 𝑠)) − 𝑌)) & ⊢ 𝐷 = (𝑠 ∈ ((𝐴 − 𝐵)(,)0) ↦ 𝑠) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Theorem	fourierdlem61 46739*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) & ⊢ (𝜑 → 𝑌 ∈ (𝐹 lim_ℂ 𝐴)) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ (𝜑 → dom 𝐺 = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝐸 ∈ (𝐺 lim_ℂ 𝐴)) & ⊢ 𝐻 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ (((𝐹‘(𝐴 + 𝑠)) − 𝑌) / 𝑠)) & ⊢ 𝑁 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ ((𝐹‘(𝐴 + 𝑠)) − 𝑌)) & ⊢ 𝐷 = (𝑠 ∈ (0(,)(𝐵 − 𝐴)) ↦ 𝑠) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐻 lim_ℂ 0))

Theorem	fourierdlem62 46740	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)

Theorem	fourierdlem63 46741*	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 = (𝐵 − 𝐴) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ (𝜑 → 𝐾 ∈ (0...𝑀)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ (𝜑 → 𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1)))) & ⊢ (𝜑 → (𝐸‘𝑌) < (𝑄‘𝐾)) & ⊢ 𝑋 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ⇒ ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘𝐾))

Theorem	fourierdlem64 46742*	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 = (𝐵 − 𝐴) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) & ⊢ 𝑁 = ((♯‘𝐻) − 1) & ⊢ 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ 𝐿 = sup({𝑘 ∈ ℤ ∣ ((𝑄‘0) + (𝑘 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) & ⊢ 𝐼 = sup({𝑗 ∈ (0..^𝑀) ∣ ((𝑄‘𝑗) + (𝐿 · 𝑇)) ≤ (𝑉‘𝐽)}, ℝ, < ) ⇒ ⊢ (𝜑 → ((𝐼 ∈ (0..^𝑀) ∧ 𝐿 ∈ ℤ) ∧ ∃𝑖 ∈ (0..^𝑀)∃𝑙 ∈ ℤ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1))) ⊆ (((𝑄‘𝑖) + (𝑙 · 𝑇))(,)((𝑄‘(𝑖 + 1)) + (𝑙 · 𝑇)))))

Theorem	fourierdlem65 46743*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑁 = ((♯‘({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄})) − 1) & ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({𝐶, 𝐷} ∪ {𝑦 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · (𝐵 − 𝐴))) ∈ ran 𝑄}))) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝑍 = ((𝑆‘𝑗) + (𝐵 − (𝐸‘(𝑆‘𝑗)))) ⇒ ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐸‘(𝑆‘(𝑗 + 1))) − (𝐿‘(𝐸‘(𝑆‘𝑗)))) = ((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)))

Theorem	fourierdlem66 46744*	Value of the 𝐺 function when the argument is not zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ 𝐴 = ((-π[,]π) ∖ {0}) ⇒ ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑠 ∈ 𝐴) → (𝐺‘𝑠) = (π · (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) · ((𝐷‘𝑛)‘𝑠))))

Theorem	fourierdlem67 46745*	𝐺 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) ⇒ ⊢ (𝜑 → 𝐺:(-π[,]π)⟶ℝ)

Theorem	fourierdlem68 46746*	The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘(𝐹‘𝑡)) ≤ 𝐷) & ⊢ (𝜑 → 𝐸 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))) → (abs‘((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘𝑡)) ≤ 𝐸) & ⊢ (𝜑 → 𝐶 ∈ 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))

Theorem	fourierdlem69 46747*	A piecewise continuous function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)

Theorem	fourierdlem70 46748*	A piecewise continuous function is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) & ⊢ (𝜑 → (𝑄‘0) = 𝐴) & ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

Theorem	fourierdlem71 46749*	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) & ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ) & ⊢ (𝜑 → (𝑄‘0) = 𝐴) & ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹) & ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) & ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)

Theorem	fourierdlem72 46750*	The derivative of 𝑂 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ (𝜑 → 𝑈 ∈ (0..^𝑀)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((𝑄‘𝑈)(,)(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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) ⇒ ⊢ (𝜑 → (ℝ D 𝑂) ∈ ((𝐴(,)𝐵)–cn→ℂ))

Theorem	fourierdlem73 46751*	A version of the Riemann Lebesgue lemma: as 𝑟 increases, the integral in 𝑆 goes to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵)) & ⊢ (𝜑 → (𝑄‘0) = 𝐴) & ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐺(abs‘(𝐺‘𝑥)) ≤ 𝑦) & ⊢ 𝑆 = (𝑟 ∈ ℝ⁺ ↦ ∫(𝐴(,)𝐵)((𝐹‘𝑥) · (sin‘(𝑟 · 𝑥))) d𝑥) & ⊢ 𝐷 = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))) ⇒ ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)(abs‘∫(𝐴(,)𝐵)((𝐹‘𝑥) · (sin‘(𝑟 · 𝑥))) d𝑥) < 𝑒)

Theorem	fourierdlem74 46752*	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.)
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1)))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ) & ⊢ (𝜑 → 𝐸 ∈ ((𝐺 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ 𝐴 = if((𝑉‘(𝑖 + 1)) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘(𝑖 + 1)) < 𝑋, 𝑊, 𝑌)) / (𝑄‘(𝑖 + 1)))) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))))

Theorem	fourierdlem75 46753*	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.)
⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ ℝ) & ⊢ 𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) & ⊢ (𝜑 → 𝐸 ∈ ((𝐺 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐻 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

Theorem	fourierdlem76 46754*	Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2)))))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 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((𝑆‘𝑗) = (𝑄‘𝑖), 𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) & ⊢ (𝜒 ↔ (((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) ⇒ ⊢ ((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑀)) ∧ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))

Theorem	fourierdlem77 46755*	If 𝐻 is bounded, then 𝑈 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑎) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ⁺ ∀𝑠 ∈ (-π[,]π)(abs‘(𝑈‘𝑠)) ≤ 𝑏)

Theorem	fourierdlem78 46756*	𝐺 is continuous when restricted on an interval not containing 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ (-π[,]π)) & ⊢ (𝜑 → 𝐵 ∈ (-π[,]π)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ)) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) ⇒ ⊢ (𝜑 → (𝐺 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℝ))

Theorem	fourierdlem79 46757*	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 = (𝐵 − 𝐴) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐶 < 𝐷) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝑍 = ((𝑆‘𝑗) + if(((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) < ((𝑄‘1) − 𝐴), (((𝑆‘(𝑗 + 1)) − (𝑆‘𝑗)) / 2), (((𝑄‘1) − 𝐴) / 2))) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐿‘(𝐸‘(𝑆‘𝑗)))(,)(𝐸‘(𝑆‘(𝑗 + 1)))) ⊆ ((𝑄‘(𝐼‘(𝑆‘𝑗)))(,)(𝑄‘((𝐼‘(𝑆‘𝑗)) + 1))))

Theorem	fourierdlem80 46758*	The derivative of 𝑂 is bounded on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) & ⊢ 𝐼 = ((𝑋 + (𝑆‘𝑗))(,)(𝑋 + (𝑆‘(𝑗 + 1)))) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧) & ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵)) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑆‘𝑗) < (𝑆‘(𝑗 + 1))) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑆‘𝑗)[,](𝑆‘(𝑗 + 1))) ⊆ (𝐴[,]𝐵)) & ⊢ (((𝜑 ∧ 𝑟 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑟 ∈ ran 𝑆) → ∃𝑘 ∈ (0..^𝑁)𝑟 ∈ ((𝑆‘𝑘)(,)(𝑆‘(𝑘 + 1)))) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (ℝ D (𝐹 ↾ 𝐼)):𝐼⟶ℝ) & ⊢ 𝑌 = (𝑠 ∈ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2))))) & ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ ℝ) ∧ ∀𝑡 ∈ 𝐼 (abs‘(𝐹‘𝑡)) ≤ 𝑤) ∧ ∀𝑡 ∈ 𝐼 (abs‘((ℝ D (𝐹 ↾ 𝐼))‘𝑡)) ≤ 𝑧)) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ dom (ℝ D 𝑂)(abs‘((ℝ D 𝑂)‘𝑠)) ≤ 𝑏)

Theorem	fourierdlem81 46759*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ ℝ⁺) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ 𝐺 = (𝑥 ∈ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ↦ if(𝑥 = (𝑄‘𝑖), 𝑅, if(𝑥 = (𝑄‘(𝑖 + 1)), 𝐿, (𝐹‘𝑥)))) & ⊢ 𝐻 = (𝑥 ∈ ((𝑆‘𝑖)[,](𝑆‘(𝑖 + 1))) ↦ (𝐺‘(𝑥 − 𝑇))) ⇒ ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem82 46760*	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)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) & ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵)) & ⊢ (𝜑 → 𝑅 ∈ (𝐹 lim_ℂ 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡)

Theorem	fourierdlem83 46761*	The fourier partial sum for 𝐹 rewritten as an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝐶 = (-π(,)π) & ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ 𝐿¹) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫𝐶((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (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))))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑆‘𝑁) = ∫𝐶((𝐹‘𝑥) · ((𝐷‘𝑁)‘(𝑥 − 𝑋))) d𝑥)

Theorem	fourierdlem84 46762*	If 𝐹 is piecewise continuous and 𝐷 is continuous, then 𝐺 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1)))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐷 ∈ (ℝ–cn→ℝ)) & ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝐹‘(𝑋 + 𝑠)) · (𝐷‘𝑠))) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Theorem	fourierdlem85 46763*	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) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐼 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℂ) & ⊢ (𝜑 → 𝐸 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = ((if((𝑉‘𝑖) = 𝑋, 𝐸, ((𝑅 − if((𝑉‘𝑖) < 𝑋, 𝑊, 𝑌)) / (𝑄‘𝑖))) · (𝐾‘(𝑄‘𝑖))) · (𝑆‘(𝑄‘𝑖))) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐴 ∈ ((𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)))

Theorem	fourierdlem86 46764*	Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π)) & ⊢ (𝜑 → ¬ 0 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝑂 = (𝑠 ∈ (𝐴[,]𝐵) ↦ ((((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠) · (𝑠 / (2 · (sin‘(𝑠 / 2)))))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 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 / 𝑖⦌𝐿, (𝐹‘(𝑋 + (𝑆‘(𝑗 + 1))))) − 𝐶) / (𝑆‘(𝑗 + 1))) · ((𝑆‘(𝑗 + 1)) / (2 · (sin‘((𝑆‘(𝑗 + 1)) / 2))))) & ⊢ 𝐸 = (((if((𝑆‘𝑗) = (𝑄‘𝑈), ⦋𝑈 / 𝑖⦌𝑅, (𝐹‘(𝑋 + (𝑆‘𝑗)))) − 𝐶) / (𝑆‘𝑗)) · ((𝑆‘𝑗) / (2 · (sin‘((𝑆‘𝑗) / 2))))) & ⊢ 𝑈 = (℩𝑖 ∈ (0..^𝑀)((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1))) ⊆ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ⇒ ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → ((𝐷 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘(𝑗 + 1))) ∧ 𝐸 ∈ ((𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) lim_ℂ (𝑆‘𝑗))) ∧ (𝑂 ↾ ((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))) ∈ (((𝑆‘𝑗)(,)(𝑆‘(𝑗 + 1)))–cn→ℂ)))

Theorem	fourierdlem87 46765*	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.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑊 ∈ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐻‘𝑠)) ≤ 𝑥) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐺 ∈ 𝐿¹) & ⊢ 𝐷 = ((𝑒 / 3) / 𝑎) & ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑎 ∈ ℝ⁺ ∧ ∀𝑛 ∈ ℕ ∀𝑠 ∈ (-π[,]π)(abs‘(𝐺‘𝑠)) ≤ 𝑎) ∧ 𝑢 ∈ dom vol) ∧ (𝑢 ⊆ (-π[,]π) ∧ (vol‘𝑢) ≤ 𝐷)) ∧ 𝑛 ∈ ℕ)) ⇒ ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑢 ∈ dom vol((𝑢 ⊆ (-π[,]π) ∧ (vol‘𝑢) ≤ 𝑑) → ∀𝑘 ∈ ℕ (abs‘∫𝑢((𝑈‘𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)))

Theorem	fourierdlem88 46766*	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) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ (𝜑 → 𝑁 ∈ ℝ) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘(𝑖 + 1)))) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐼 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ) & ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐷 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐿¹)

Theorem	fourierdlem89 46767*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < )) & ⊢ 𝑉 = (𝑖 ∈ (0..^𝑀) ↦ 𝑅) ⇒ ⊢ (𝜑 → if((𝑍‘(𝐸‘(𝑆‘𝐽))) = (𝑄‘(𝐼‘(𝑆‘𝐽))), (𝑉‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝑍‘(𝐸‘(𝑆‘𝐽))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘𝐽)))

Theorem	fourierdlem90 46768*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) & ⊢ 𝐺 = (𝐹 ↾ ((𝐿‘(𝐸‘(𝑆‘𝐽)))(,)(𝐸‘(𝑆‘(𝐽 + 1))))) & ⊢ 𝑅 = (𝑦 ∈ (((𝐿‘(𝐸‘(𝑆‘𝐽))) + 𝑈)(,)((𝐸‘(𝑆‘(𝐽 + 1))) + 𝑈)) ↦ (𝐺‘(𝑦 − 𝑈))) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐿‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ (𝜑 → (𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) ∈ (((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))–cn→ℂ))

Theorem	fourierdlem91 46769*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ 𝑈 = ((𝑆‘(𝐽 + 1)) − (𝐸‘(𝑆‘(𝐽 + 1)))) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < )) & ⊢ 𝑊 = (𝑖 ∈ (0..^𝑀) ↦ 𝐿) ⇒ ⊢ (𝜑 → if((𝐸‘(𝑆‘(𝐽 + 1))) = (𝑄‘((𝐼‘(𝑆‘𝐽)) + 1)), (𝑊‘(𝐼‘(𝑆‘𝐽))), (𝐹‘(𝐸‘(𝑆‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑆‘𝐽)(,)(𝑆‘(𝐽 + 1)))) lim_ℂ (𝑆‘(𝐽 + 1))))

Theorem	fourierdlem92 46770*	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) + 𝑇)) & ⊢ 𝐻 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑇) ∧ (𝑝‘𝑚) = (𝐵 + 𝑇)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem93 46771*	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 ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐻 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐹:(-π[,]π)⟶ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∫(-π[,]π)(𝐹‘𝑡) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))(𝐹‘(𝑋 + 𝑠)) d𝑠)

Theorem	fourierdlem94 46772*	For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅) ⇒ ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))

Theorem	fourierdlem95 46773*	Algebraic manipulation of integrals, used by other lemmas. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ 𝐼 = (ℝ D 𝐹) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐼 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))):((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ((𝐼 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐶 ∈ ((𝐼 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 𝐴 ⊆ ((-π[,]π) ∖ {0})) & ⊢ 𝐸 = (𝑛 ∈ ℕ ↦ (∫𝐴(𝐺‘𝑠) 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))))))) & ⊢ (𝜑 → 𝑂 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐴) → if(0 < 𝑠, 𝑌, 𝑊) = 𝑂) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∫𝐴((𝐷‘𝑛)‘𝑠) d𝑠 = (1 / 2)) ⇒ ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) + (𝑂 / 2)) = ∫𝐴((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠)

Theorem	fourierdlem96 46774*	limit for 𝐹 at the lower bound of an interval for the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ (𝜑 → 𝐽 ∈ (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) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) ⇒ ⊢ (𝜑 → if(((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽))) = (𝑄‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝐽))), ((𝑖 ∈ (0..^𝑀) ↦ 𝑅)‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝐽))), (𝐹‘((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘𝐽))))) ∈ ((𝐹 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) lim_ℂ (𝑉‘𝐽)))

Theorem	fourierdlem97 46775*	𝐹 is continuous on the intervals induced by the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝐺 = (ℝ D 𝐹) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐺 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ (𝜑 → 𝐽 ∈ (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) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) & ⊢ 𝐻 = (𝑠 ∈ ℝ ↦ if(𝑠 ∈ dom 𝐺, (𝐺‘𝑠), 0)) ⇒ ⊢ (𝜑 → (𝐺 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ))

Theorem	fourierdlem98 46776*	𝐹 is continuous on the intervals induced by the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ (𝜑 → 𝐽 ∈ (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) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) ⇒ ⊢ (𝜑 → (𝐹 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) ∈ (((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))–cn→ℂ))

Theorem	fourierdlem99 46777*	limit for 𝐹 at the upper bound of an interval for the moved partition 𝑉. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ (𝜑 → 𝐽 ∈ (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) ∣ ∃ℎ ∈ ℤ (𝑦 + (ℎ · 𝑇)) ∈ ran 𝑄}))) ⇒ ⊢ (𝜑 → if(((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))) = (𝑄‘(((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝐽)) + 1)), ((𝑖 ∈ (0..^𝑀) ↦ 𝐿)‘((𝑦 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ ((𝑢 ∈ (𝐴(,]𝐵) ↦ if(𝑢 = 𝐵, 𝐴, 𝑢))‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘𝑦))}, ℝ, < ))‘(𝑉‘𝐽))), (𝐹‘((𝑣 ∈ ℝ ↦ (𝑣 + ((⌊‘((𝐵 − 𝑣) / 𝑇)) · 𝑇)))‘(𝑉‘(𝐽 + 1))))) ∈ ((𝐹 ↾ ((𝑉‘𝐽)(,)(𝑉‘(𝐽 + 1)))) lim_ℂ (𝑉‘(𝐽 + 1))))

Theorem	fourierdlem100 46778*	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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (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 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)

Theorem	fourierdlem101 46779*	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 ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐺 = (𝑡 ∈ (-π[,]π) ↦ ((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋)))) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐹:(-π[,]π)⟶ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∫(-π[,]π)((𝐹‘𝑡) · ((𝐷‘𝑁)‘(𝑡 − 𝑋))) d𝑡 = ∫((-π − 𝑋)[,](π − 𝑋))((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑁)‘𝑠)) d𝑠)

Theorem	fourierdlem102 46780*	For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) & ⊢ 𝑀 = ((♯‘𝐻) − 1) & ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻)) ⇒ ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))

Theorem	fourierdlem103 46781*	The half lower part of the integral equal to the fourier partial sum, converges to half the left limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹‘𝑡)) ≤ 𝑤) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ 𝑍 = (𝑚 ∈ ℕ ↦ ∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) & ⊢ 𝐸 = (𝑛 ∈ ℕ ↦ (∫(-π(,)0)(𝐺‘𝑠) d𝑠 / π)) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐴 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐵 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) & ⊢ 𝑂 = (𝑈 ↾ (-π[,]𝑑)) & ⊢ 𝑇 = ({-π, 𝑑} ∪ (ran 𝑄 ∩ (-π(,)𝑑))) & ⊢ 𝑁 = ((♯‘𝑇) − 1) & ⊢ 𝐽 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇)) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝐶 = (℩𝑙 ∈ (0..^𝑀)((𝐽‘𝑘)(,)(𝐽‘(𝑘 + 1))) ⊆ ((𝑄‘𝑙)(,)(𝑄‘(𝑙 + 1)))) & ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ (-π(,)0)) ∧ 𝑘 ∈ ℕ) ∧ (abs‘∫(𝑑(,)0)((𝑈‘𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)) ∧ (abs‘∫(-π(,)𝑑)((𝑈‘𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2))) ⇒ ⊢ (𝜑 → 𝑍 ⇝ (𝑊 / 2))

Theorem	fourierdlem104 46782*	The half upper part of the integral equal to the fourier partial sum, converges to half the right limit of the original function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝‘𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘(𝐹‘𝑡)) ≤ 𝑤) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) ∈ (((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))–cn→ℝ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑧 ∈ ℝ ∀𝑡 ∈ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (𝑉‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑉‘𝑖)(,)(𝑉‘(𝑖 + 1)))) lim_ℂ (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) ↦ ((𝐻‘𝑠) · (𝐾‘𝑠))) & ⊢ 𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑛 + (1 / 2)) · 𝑠))) & ⊢ 𝐺 = (𝑠 ∈ (-π[,]π) ↦ ((𝑈‘𝑠) · (𝑆‘𝑠))) & ⊢ 𝑍 = (𝑚 ∈ ℕ ↦ ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑚)‘𝑠)) d𝑠) & ⊢ 𝐸 = (𝑛 ∈ ℕ ↦ (∫(0(,)π)(𝐺‘𝑠) d𝑠 / π)) & ⊢ (𝜑 → 𝑌 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑊 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐴 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐵 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) & ⊢ 𝑂 = (𝑈 ↾ (𝑑[,]π)) & ⊢ 𝑇 = ({𝑑, π} ∪ (ran 𝑄 ∩ (𝑑(,)π))) & ⊢ 𝑁 = ((♯‘𝑇) − 1) & ⊢ 𝐽 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇)) & ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) & ⊢ 𝐶 = (℩𝑙 ∈ (0..^𝑀)((𝐽‘𝑘)(,)(𝐽‘(𝑘 + 1))) ⊆ ((𝑄‘𝑙)(,)(𝑄‘(𝑙 + 1)))) & ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑑 ∈ (0(,)π)) ∧ 𝑘 ∈ ℕ) ∧ (abs‘∫(0(,)𝑑)((𝑈‘𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2)) ∧ (abs‘∫(𝑑(,)π)((𝑈‘𝑠) · (sin‘((𝑘 + (1 / 2)) · 𝑠))) d𝑠) < (𝑒 / 2))) ⇒ ⊢ (𝜑 → 𝑍 ⇝ (𝑌 / 2))

Theorem	fourierdlem105 46783*	A piecewise continuous function is integrable on any closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)+∞)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐶[,]𝐷) ↦ (𝐹‘𝑥)) ∈ 𝐿¹)

Theorem	fourierdlem106 46784*	For a piecewise smooth function, the left and the right limits exist at any point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → (((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) ≠ ∅ ∧ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) ≠ ∅))

Theorem	fourierdlem107 46785*	The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any positive value 𝑋. This lemma generalizes fourierdlem92 46770 where the integral was shifted by the exact period. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ⁺) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = 𝐴) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐻 = ({(𝐴 − 𝑋), 𝐴} ∪ {𝑦 ∈ ((𝐴 − 𝑋)[,]𝐴) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}) & ⊢ 𝑁 = ((♯‘𝐻) − 1) & ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝑍 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄‘𝑖) ≤ (𝑍‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem108 46786*	The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any positive value 𝑋. This lemma generalizes fourierdlem92 46770 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ⁺) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem109 46787*	The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any value 𝑋. This lemma generalizes fourierdlem92 46770 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 − 𝑋) ∧ (𝑝‘𝑚) = (𝐵 − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐻 = ({(𝐴 − 𝑋), (𝐵 − 𝑋)} ∪ {𝑥 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) & ⊢ 𝑁 = ((♯‘𝐻) − 1) & ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻)) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐽 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦)) & ⊢ 𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) ≤ (𝐽‘(𝐸‘𝑥))}, ℝ, < )) ⇒ ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem110 46788*	The integral of a piecewise continuous periodic function 𝐹 is unchanged if the domain is shifted by any value 𝑋. This lemma generalizes fourierdlem92 46770 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝑇 = (𝐵 − 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝐹:ℝ⟶ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) ⇒ ⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)

Theorem	fourierdlem111 46789*	The fourier partial sum for 𝐹 is the sum of two integrals, with the same integrand involving 𝐹 and the Dirichlet kernel 𝐷, but on two opposite intervals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (cos‘(𝑛 · 𝑡))) d𝑡 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑡) · (sin‘(𝑛 · 𝑡))) d𝑡 / π)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) & ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))) & ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ ((𝐹‘(𝑋 + 𝑥)) · ((𝐷‘𝑛)‘𝑥))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ 𝑇 = (2 · π) & ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = (-π − 𝑋) ∧ (𝑝‘𝑚) = (π − 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝑊 = (𝑖 ∈ (0...𝑀) ↦ ((𝑄‘𝑖) − 𝑋)) ⇒ ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆‘𝑛) = (∫(-π(,)0)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠 + ∫(0(,)π)((𝐹‘(𝑋 + 𝑠)) · ((𝐷‘𝑛)‘𝑠)) d𝑠))

Theorem	fourierdlem112 46790*	Here abbreviations (local definitions) are introduced to prove the fourier 46797 theorem. (𝑍‘𝑚) is the m_th partial sum of the fourier series. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑚) + 1) / (2 · π)), ((sin‘((𝑚 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2))))))) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ 𝑁 = ((♯‘({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄})) − 1) & ⊢ 𝑉 = (℩𝑓𝑓 Isom < , < ((0...𝑁), ({(-π + 𝑋), (π + 𝑋)} ∪ {𝑦 ∈ ((-π + 𝑋)[,](π + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ ran 𝑄}))) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑉) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐶 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑈 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1)))) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ (𝜑 → 𝐸 ∈ (((ℝ D 𝐹) ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐼 ∈ (((ℝ D 𝐹) ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑍 = (𝑚 ∈ ℕ ↦ (((𝐴‘0) / 2) + Σ𝑛 ∈ (1...𝑚)(((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋)))))) & ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) & ⊢ (𝜑 → ∃𝑤 ∈ ℝ ∀𝑡 ∈ ℝ (abs‘(𝐹‘𝑡)) ≤ 𝑤) & ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑡 ∈ dom (ℝ D 𝐹)(abs‘((ℝ D 𝐹)‘𝑡)) ≤ 𝑧) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Theorem	fourierdlem113 46791*	Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘𝑖)) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (((ℝ D 𝐹) ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) lim_ℂ (𝑄‘(𝑖 + 1))) ≠ ∅) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))) & ⊢ (𝜑 → (𝐸‘𝑋) ∈ ran 𝑄) ⇒ ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Theorem	fourierdlem114 46792*	Fourier series convergence for periodic, piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}) & ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((π − 𝑥) / 𝑇)) · 𝑇))) & ⊢ 𝐻 = ({-π, π, (𝐸‘𝑋)} ∪ ((-π[,]π) ∖ dom 𝐺)) & ⊢ 𝑀 = ((♯‘𝐻) − 1) & ⊢ 𝑄 = (℩𝑔𝑔 Isom < , < ((0...𝑀), 𝐻)) ⇒ ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Theorem	fourierdlem115 46793*	Fourier serier convergence, for piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑆 = (𝑘 ∈ ℕ ↦ (((𝐴‘𝑘) · (cos‘(𝑘 · 𝑋))) + ((𝐵‘𝑘) · (sin‘(𝑘 · 𝑋))))) ⇒ ⊢ (𝜑 → (seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)) ∧ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)))

Theorem	fourierd 46794*	Fourier series convergence for periodic, piecewise smooth functions. The series converges to the average value of the left and the right limit of the function. Thus, if the function is continuous at a given point, the series converges exactly to the function value, see fouriercnp 46798. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 46799 for an alternative form of the theorem that makes this fact explicit. When the first derivative is continuous, a simpler version of the theorem can be stated, see fouriercn 46804. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) ⇒ ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2))

Theorem	fourierclimd 46795*	Fourier series convergence, for piecewise smooth functions. See fourierd 46794 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ⇒ ⊢ (𝜑 → seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2)))

Theorem	fourierclim 46796*	Fourier series convergence, for piecewise smooth functions. See fourier 46797 for the analogous Σ equation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹:ℝ⟶ℝ & ⊢ 𝑇 = (2 · π) & ⊢ (𝑥 ∈ ℝ → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ ((-π(,)π) ∖ dom 𝐺) ∈ Fin & ⊢ 𝐺 ∈ (dom 𝐺–cn→ℂ) & ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ 𝑋 ∈ ℝ & ⊢ 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) & ⊢ 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) ⇒ ⊢ seq1( + , 𝑆) ⇝ (((𝐿 + 𝑅) / 2) − ((𝐴‘0) / 2))

Theorem	fourier 46797*	Fourier series convergence for periodic, piecewise smooth functions. The series converges to the average value of the left and the right limit of the function. Thus, if the function is continuous at a given point, the series converges exactly to the function value, see fouriercnp 46798. Notice that for a piecewise smooth function, the left and right limits always exist, see fourier2 46799 for an alternative form of the theorem that makes this fact explicit. When the first derivative is continuous, a simpler version of the theorem can be stated, see fouriercn 46804. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹:ℝ⟶ℝ & ⊢ 𝑇 = (2 · π) & ⊢ (𝑥 ∈ ℝ → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ ((-π(,)π) ∖ dom 𝐺) ∈ Fin & ⊢ 𝐺 ∈ (dom 𝐺–cn→ℂ) & ⊢ (𝑥 ∈ ((-π[,)π) ∖ dom 𝐺) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝑥 ∈ ((-π(,]π) ∖ dom 𝐺) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ 𝑋 ∈ ℝ & ⊢ 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋) & ⊢ 𝑅 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) ⇒ ⊢ (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝐿 + 𝑅) / 2)

Theorem	fouriercnp 46798*	If 𝐹 is continuous at the point 𝑋, then its Fourier series at 𝑋, converges to (𝐹‘𝑋). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 CnP 𝐽)‘𝑋)) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) ⇒ ⊢ (𝜑 → (((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = (𝐹‘𝑋))

Theorem	fourier2 46799*	Fourier series convergence, for a piecewise smooth function. Here it is also proven the existence of the left and right limits of 𝐹 at any given point 𝑋. See fourierd 46794 for a comparison. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) & ⊢ 𝑇 = (2 · π) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥)) & ⊢ 𝐺 = ((ℝ D 𝐹) ↾ (-π(,)π)) & ⊢ (𝜑 → ((-π(,)π) ∖ dom 𝐺) ∈ Fin) & ⊢ (𝜑 → 𝐺 ∈ (dom 𝐺–cn→ℂ)) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π[,)π) ∖ dom 𝐺)) → ((𝐺 ↾ (𝑥(,)+∞)) lim_ℂ 𝑥) ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((-π(,]π) ∖ dom 𝐺)) → ((𝐺 ↾ (-∞(,)𝑥)) lim_ℂ 𝑥) ≠ ∅) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ 𝐴 = (𝑛 ∈ ℕ₀ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π)) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (∫(-π(,)π)((𝐹‘𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π)) ⇒ ⊢ (𝜑 → ∃𝑙 ∈ ((𝐹 ↾ (-∞(,)𝑋)) lim_ℂ 𝑋)∃𝑟 ∈ ((𝐹 ↾ (𝑋(,)+∞)) lim_ℂ 𝑋)(((𝐴‘0) / 2) + Σ𝑛 ∈ ℕ (((𝐴‘𝑛) · (cos‘(𝑛 · 𝑋))) + ((𝐵‘𝑛) · (sin‘(𝑛 · 𝑋))))) = ((𝑙 + 𝑟) / 2))

Theorem	sqwvfoura 46800*	Fourier coefficients for the square wave function. Since the square function is an odd function, there is no contribution from the 𝐴 coefficients. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝑇 = (2 · π) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if((𝑥 mod 𝑇) < π, 1, -1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ⊢ (𝜑 → (∫(-π(,)π)((𝐹‘𝑥) · (cos‘(𝑁 · 𝑥))) d𝑥 / π) = 0)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49400 495 49401-49500 496 49501-49600 497 49601-49700 498 49701-49800 499 49801-49900 500 49901-50000 501 50001-50100 502 50101-50200 503 50201-50300 504 50301-50400 505 50401-50434

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