P. 447 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44601-44700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ioccncflimc 44601	Limit at the upper bound of a continuous function defined on a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵))
Theorem	cncfuni 44602*	A complex function on a subset of the complex numbers is continuous if its domain is the union of relatively open subsets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐴 ∩ 𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))    &   ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝐹 ↾ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cn→ℂ))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	icccncfext 44603*	A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴[,]𝐵), (𝐹‘𝑥), if(𝑥 < 𝐴, (𝐹‘𝐴), (𝐹‘𝐵))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ Top)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t (𝐴[,]𝐵)) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐺 ∈ (𝐽 Cn (𝐾 ↾t ran 𝐹)) ∧ (𝐺 ↾ (𝐴[,]𝐵)) = 𝐹))
Theorem	cncficcgt0 44604*	A the absolute value of a continuous function on a closed interval, that is never 0, has a strictly positive lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ+ ∀𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶))
Theorem	icocncflimc 44605	Limit at the lower bound, of a continuous function defined on a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴))
Theorem	cncfdmsn 44606*	A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ ({𝐴}–cn→{𝐵}))
Theorem	divcncff 44607	The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→(ℂ ∖ {0})))    ⇒   ⊢ (𝜑 → (𝐹 ∘f / 𝐺) ∈ (𝑋–cn→ℂ))
Theorem	cncfshiftioo 44608*	A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ 𝐶 = (𝐴(,)𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ 𝐷 = ((𝐴 + 𝑇)(,)(𝐵 + 𝑇))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ))    &   ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ))
Theorem	cncfiooicclem1 44609*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. This lemma assumes 𝐴 < 𝐵, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	cncfiooicc 44610*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 can be complex-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	cncfiooiccre 44611*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) can be extended to a continuous function 𝐺 on the corresponding closed interval, if it has a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵. 𝐹 is assumed to be real-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))
Theorem	cncfioobdlem 44612*	𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶𝑉)    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → (𝐺‘𝐶) = (𝐹‘𝐶))
Theorem	cncfioobd 44613*	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑦)) ≤ 𝑥)
Theorem	jumpncnp 44614	Jump discontinuity or discontinuity of the first kind: if the left and the right limit don't match, the function is discontinuous at the point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐿 ≠ 𝑅)    ⇒   ⊢ (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵))
Theorem	cxpcncf2 44615*	The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → (𝑥 ∈ ℂ ↦ (𝐴↑𝑐𝑥)) ∈ (ℂ–cn→ℂ))
Theorem	fprodcncf 44616*	The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (𝐴–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝐵 𝐶) ∈ (𝐴–cn→ℂ))
Theorem	add1cncf 44617*	Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	add2cncf 44618*	Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 + 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	sub1cncfd 44619*	Subtracting a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	sub2cncfd 44620*	Subtraction from a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	fprodsub2cncf 44621*	𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	fprodadd2cncf 44622*	𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	fprodsubrecnncnvlem 44623*	The sequence 𝑆 of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − (1 / 𝑛)))    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 − 𝑥))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodsubrecnncnv 44624*	The sequence 𝑆 of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (𝐴 − (1 / 𝑛)))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 𝐴)
Theorem	fprodaddrecnncnvlem 44625*	The sequence 𝑆 of finite products, where every factor is added an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + (1 / 𝑛)))    &   ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘 ∈ 𝐴 (𝐵 + 𝑥))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝐴 𝐵)
Theorem	fprodaddrecnncnv 44626*	The sequence 𝑆 of finite products, where every factor is added an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (𝐴 + (1 / 𝑛)))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 𝐴)
21.40.10  Derivatives
Theorem	dvsinexp 44627*	The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
Theorem	dvcosre 44628	The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))
Theorem	dvsinax 44629*	Derivative exercise: the derivative with respect to y of sin(Ay), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
Theorem	dvsubf 44630	The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 D (𝐹 ∘f − 𝐺)) = ((𝑆 D 𝐹) ∘f − (𝑆 D 𝐺)))
Theorem	dvmptconst 44631*	Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ 0))
Theorem	dvcnre 44632	From complex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐹:ℂ⟶ℂ ∧ ℝ ⊆ dom (ℂ D 𝐹)) → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
Theorem	dvmptidg 44633*	Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝐴 ↦ 𝑥)) = (𝑥 ∈ 𝐴 ↦ 1))
Theorem	dvresntr 44634	Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)    ⇒   ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌)))
Theorem	fperdvper 44635*	The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℝ⟶ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))    &   ⊢ 𝐺 = (ℝ D 𝐹)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐺) → ((𝑥 + 𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑥 + 𝑇)) = (𝐺‘𝑥)))
Theorem	dvasinbx 44636*	Derivative exercise: the derivative with respect to y of A x sin(By), given two constants 𝐴 and 𝐵. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · (sin‘(𝐵 · 𝑦))))) = (𝑦 ∈ ℂ ↦ ((𝐴 · 𝐵) · (cos‘(𝐵 · 𝑦)))))
Theorem	dvresioo 44637	Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ (𝐵(,)𝐶)))
Theorem	dvdivf 44638	The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶(ℂ ∖ {0}))    &   ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 D (𝐹 ∘f / 𝐺)) = ((((𝑆 D 𝐹) ∘f · 𝐺) ∘f − ((𝑆 D 𝐺) ∘f · 𝐹)) ∘f / (𝐺 ∘f · 𝐺)))
Theorem	dvdivbd 44639*	A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑈 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑄 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐶) ≤ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐵) ≤ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐷) ≤ 𝑇)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (abs‘𝐴) ≤ 𝑄)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐸 ≤ (abs‘𝐵))    &   ⊢ 𝐹 = (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
Theorem	dvsubcncf 44640	A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑆 D (𝐹 ∘f − 𝐺)) ∈ (𝑋–cn→ℂ))
Theorem	dvmulcncf 44641	A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)    &   ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) ∈ (𝑋–cn→ℂ))
Theorem	dvcosax 44642*	Derivative exercise: the derivative with respect to x of cos(Ax), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (𝑥 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑥)))))
Theorem	dvdivcncf 44643	A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶(ℂ ∖ {0}))    &   ⊢ (𝜑 → (𝑆 D 𝐹) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑆 D 𝐺) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑆 D (𝐹 ∘f / 𝐺)) ∈ (𝑋–cn→ℂ))
Theorem	dvbdfbdioolem1 44644*	Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐶(,)𝐵))    ⇒   ⊢ (𝜑 → ((abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐷 − 𝐶)) ∧ (abs‘((𝐹‘𝐷) − (𝐹‘𝐶))) ≤ (𝐾 · (𝐵 − 𝐴))))
Theorem	dvbdfbdioolem2 44645*	A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   ⊢ 𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵 − 𝐴)))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑀)
Theorem	dvbdfbdioo 44646*	A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎)    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏)
Theorem	ioodvbdlimc1lem1 44647*	If 𝐹 has bounded derivative on (𝐴(,)𝐵) then a sequence of points in its image converges to its lim sup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑅:(ℤ≥‘𝑀)⟶(𝐴(,)𝐵))    &   ⊢ 𝑆 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))    &   ⊢ (𝜑 → 𝑅 ∈ dom ⇝ )    &   ⊢ 𝐾 = inf({𝑘 ∈ (ℤ≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )    ⇒   ⊢ (𝜑 → 𝑆 ⇝ (lim sup‘𝑆))
Theorem	ioodvbdlimc1lem2 44648*	Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   ⊢ 𝑌 = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )    &   ⊢ 𝑀 = ((⌊‘(1 / (𝐵 − 𝐴))) + 1)    &   ⊢ 𝑆 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝐴 + (1 / 𝑗))))    &   ⊢ 𝑅 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐴 + (1 / 𝑗)))    &   ⊢ 𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)    &   ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)))    ⇒   ⊢ (𝜑 → (lim sup‘𝑆) ∈ (𝐹 limℂ 𝐴))
Theorem	ioodvbdlimc1 44649*	A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    ⇒   ⊢ (𝜑 → (𝐹 limℂ 𝐴) ≠ ∅)
Theorem	ioodvbdlimc2lem 44650*	Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   ⊢ 𝑌 = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )    &   ⊢ 𝑀 = ((⌊‘(1 / (𝐵 − 𝐴))) + 1)    &   ⊢ 𝑆 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝐵 − (1 / 𝑗))))    &   ⊢ 𝑅 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐵 − (1 / 𝑗)))    &   ⊢ 𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)    &   ⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐵)) < (1 / 𝑗)))    ⇒   ⊢ (𝜑 → (lim sup‘𝑆) ∈ (𝐹 limℂ 𝐵))
Theorem	ioodvbdlimc2 44651*	A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    &   ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    ⇒   ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅)
Theorem	dvdmsscn 44652	𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    ⇒   ⊢ (𝜑 → 𝑋 ⊆ ℂ)
Theorem	dvmptmulf 44653*	Function-builder for derivative, product rule. A version of dvmptmul 25478 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷))    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
Theorem	dvnmptdivc 44654*	Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶)))
Theorem	dvdsn1add 44655	If 𝐾 divides 𝑁 but 𝐾 does not divide 𝑀, then 𝐾 does not divide (𝑀 + 𝑁). (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((¬ 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁) → ¬ 𝐾 ∥ (𝑀 + 𝑁)))
Theorem	dvxpaek 44656*	Derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥 ∈ 𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
Theorem	dvnmptconst 44657*	The 𝑁-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))
Theorem	dvnxpaek 44658*	The 𝑛-th derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ if(𝐾 < 𝑁, 0, (((!‘𝐾) / (!‘(𝐾 − 𝑁))) · ((𝑥 + 𝐴)↑(𝐾 − 𝑁))))))
Theorem	dvnmul 44659*	Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ 𝐴)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)    &   ⊢ 𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))    &   ⊢ 𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶‘𝑘)‘𝑥) · ((𝐷‘(𝑁 − 𝑘))‘𝑥)))))
Theorem	dvmptfprodlem 44660*	Induction step for dvmptfprod 44661. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ Ⅎ𝑖𝐹    &   ⊢ Ⅎ𝑗𝐺    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    &   ⊢ (𝜑 → 𝐸 ∈ V)    &   ⊢ (𝜑 → ¬ 𝐸 ∈ 𝐷)    &   ⊢ (𝜑 → (𝐷 ∪ {𝐸}) ⊆ 𝐼)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝐷) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐷 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐷 (𝐶 · ∏𝑖 ∈ (𝐷 ∖ {𝑗})𝐴)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ ℂ)    &   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐹)) = (𝑥 ∈ 𝑋 ↦ 𝐺))    &   ⊢ (𝑖 = 𝐸 → 𝐴 = 𝐹)    &   ⊢ (𝑗 = 𝐸 → 𝐶 = 𝐺)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝐷 ∪ {𝐸})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝐷 ∪ {𝐸})(𝐶 · ∏𝑖 ∈ ((𝐷 ∪ {𝐸}) ∖ {𝑗})𝐴)))
Theorem	dvmptfprod 44661*	Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝜑    &   ⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))    &   ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
Theorem	dvnprodlem1 44662*	𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑇)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)    &   ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)    ⇒   ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
Theorem	dvnprodlem2 44663*	Induction step for dvnprodlem2 44663. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)    &   ⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))    &   ⊢ (𝜑 → 𝑅 ⊆ 𝑇)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅))    &   ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))    &   ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))    &   ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
Theorem	dvnprodlem3 44664*	The multinomial formula for the 𝑘-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))    &   ⊢ 𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
Theorem	dvnprod 44665*	The multinomial formula for the 𝑁-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑇 ((𝐻‘𝑡)‘𝑥))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑡 ∈ 𝑇 (𝑐‘𝑡) = 𝑛})    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ (𝐶‘𝑁)(((!‘𝑁) / ∏𝑡 ∈ 𝑇 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑇 (((𝑆 D𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
21.40.11  Integrals
Theorem	itgsin0pilem1 44666*	Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝐶 = (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡))    ⇒   ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
Theorem	ibliccsinexp 44667*	sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1)
Theorem	itgsin0pi 44668	Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
Theorem	iblioosinexp 44669*	sin^n on an open integral is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1)
Theorem	itgsinexplem1 44670*	Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))    &   ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))    &   ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))    &   ⊢ 𝐼 = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))    &   ⊢ 𝐿 = (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)))    &   ⊢ 𝑀 = (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
Theorem	itgsinexp 44671*	A recursive formula for the integral of sin^N on the interval (0,π) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    ⇒   ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
Theorem	iblconstmpt 44672*	A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)
Theorem	itgeq1d 44673*	Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Theorem	mbfres2cn 44674	Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐵 and 𝐶 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. Similar to mbfres2 25162 but here the theorem is extended to complex-valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn)    &   ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	vol0 44675	The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (vol‘∅) = 0
Theorem	ditgeqiooicc 44676*	A function 𝐹 on an open interval, has the same directed integral as its extension 𝐺 on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐹‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵](𝐺‘𝑥) d𝑥)
Theorem	volge0 44677	The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))
Theorem	cnbdibl 44678*	A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ (𝜑 → (vol‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)
Theorem	snmbl 44679	A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → {𝐴} ∈ dom vol)
Theorem	ditgeq3d 44680*	Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 = 𝐸)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐷 d𝑥 = ⨜[𝐴 → 𝐵]𝐸 d𝑥)
Theorem	iblempty 44681	The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ∅ ∈ 𝐿1
Theorem	iblsplit 44682*	The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1)
Theorem	volsn 44683	A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → (vol‘{𝐴}) = 0)
Theorem	itgvol0 44684*	If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0))
Theorem	itgcoscmulx 44685*	Exercise: the integral of 𝑥 ↦ cos𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴))
Theorem	iblsplitf 44686*	A version of iblsplit 44682 using bound-variable hypotheses instead of distinct variable conditions". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1)
Theorem	ibliooicc 44687*	If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1)
Theorem	volioc 44688	The measure of a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,]𝐵)) = (𝐵 − 𝐴))
Theorem	iblspltprt 44689*	If a function is integrable on any interval of a partition, then it is integrable on the whole interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑡𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑃‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))    &   ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁))) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁)) ↦ 𝐴) ∈ 𝐿1)
Theorem	itgsincmulx 44690*	Exercise: the integral of 𝑥 ↦ sin𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
Theorem	itgsubsticclem 44691*	lemma for itgsubsticc 44692. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)    &   ⊢ 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ)    &   ⊢ (𝜑 → 𝐾 ≤ 𝐿)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgsubsticc 44692*	Integration by u-substitution. The main difference with respect to itgsubst 25566 is that here we consider the range of 𝐴(𝑥) to be in the closed interval (𝐾[,]𝐿). If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶) ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ)    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgioocnicc 44693*	The integral of a piecewise continuous function 𝐹 on an open interval is equal to the integral of the continuous function 𝐺, in the corresponding closed interval. 𝐺 is equal to 𝐹 on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴))    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐿1 ∧ ∫(𝐴[,]𝐵)(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
Theorem	iblcncfioo 44694	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)
Theorem	itgspltprt 44695*	The ∫ integral splits on a given partition 𝑃. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑃:(𝑀...𝑁)⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))    &   ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁))) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫((𝑃‘𝑀)[,](𝑃‘𝑁))𝐴 d𝑡 = Σ𝑖 ∈ (𝑀..^𝑁)∫((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1)))𝐴 d𝑡)
Theorem	itgiccshift 44696*	The integral of a function, 𝐹 stays the same if its closed interval domain is shifted by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇)))    ⇒   ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
Theorem	itgperiod 44697*	The integral of a periodic function, with period 𝑇 stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
Theorem	itgsbtaddcnst 44698*	Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
Theorem	volico 44699	The measure of left-closed, right-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
Theorem	sublevolico 44700	The Lebesgue measure of a left-closed, right-open interval is greater than or equal to the difference of the two bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))