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Theorem List for Metamath Proof Explorer - 40401-40500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcncfmptssg 40401* A continuous complex function restricted to a subset is continuous, using "map to" notation. This theorem generalizes cncfmptss 40137 because it allows to establish a subset for the codomain also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐸)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑𝐷𝐵)    &   ((𝜑𝑥𝐶) → 𝐸𝐷)       (𝜑 → (𝑥𝐶𝐸) ∈ (𝐶cn𝐷))

Theoremconstcncfg 40402* A constant function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))

Theoremidcncfg 40403* The identity function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝑥) ∈ (𝐴cn𝐵))

Theoremaddcncf 40404* The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝑋cn→ℂ))

Theoremcncfshift 40405* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝑇 ∈ ℂ)    &   𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦𝐴 𝑥 = (𝑦 + 𝑇)}    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   𝐺 = (𝑥𝐵 ↦ (𝐹‘(𝑥𝑇)))       (𝜑𝐺 ∈ (𝐵cn→ℂ))

Theoremresincncf 40406 sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(sin ↾ ℝ) ∈ (ℝ–cn→ℝ)

Theoremaddccncf2 40407* Adding a constant is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴 ↦ (𝐵 + 𝑥))       ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → 𝐹 ∈ (𝐴cn→ℂ))

Theorem0cnf 40408 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ ({∅} Cn {∅})

Theoremfsumcncf 40409* The finite sum of continuous complex function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ⊆ ℂ)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝑋cn→ℂ))

Theoremcncfperiod 40410* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝑇 ∈ ℝ)    &   𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦𝐴 𝑥 = (𝑦 + 𝑇)}    &   (𝜑𝐹:dom 𝐹⟶ℂ)    &   (𝜑𝐵 ⊆ dom 𝐹)    &   ((𝜑𝑥𝐴) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   (𝜑 → (𝐹𝐴) ∈ (𝐴cn→ℂ))       (𝜑 → (𝐹𝐵) ∈ (𝐵cn→ℂ))

Theoremsubcncff 40411 The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (𝑋cn→ℂ))    &   (𝜑𝐺 ∈ (𝑋cn→ℂ))       (𝜑 → (𝐹𝑓𝐺) ∈ (𝑋cn→ℂ))

Theoremnegcncfg 40412* The opposite of a continuous function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn→ℂ))       (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ (𝐴cn→ℂ))

Theoremcnfdmsn 40413* A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝑉𝐵𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵}))

Theoremcncfcompt 40414* Composition of continuous functions. A generalization of cncfmpt1f 22763 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))    &   (𝜑𝐹 ∈ (𝐶cn𝐷))       (𝜑 → (𝑥𝐴 ↦ (𝐹𝐵)) ∈ (𝐴cn𝐷))

Theoremaddcncff 40415 The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (𝑋cn→ℂ))    &   (𝜑𝐺 ∈ (𝑋cn→ℂ))       (𝜑 → (𝐹𝑓 + 𝐺) ∈ (𝑋cn→ℂ))

Theoremioccncflimc 40416 Limit at the upper bound, of a continuous function defined on a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ))       (𝜑 → (𝐹𝐵) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐵))

Theoremcncfuni 40417* A function is continuous if it's domain is the union of sets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 𝐵)    &   ((𝜑𝑏𝐵) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))    &   ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))       (𝜑𝐹 ∈ (𝐴cn→ℂ))

Theoremicccncfext 40418* 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.)
𝑥𝐹    &   𝐽 = (topGen‘ran (,))    &   𝑌 = 𝐾    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴[,]𝐵), (𝐹𝑥), if(𝑥 < 𝐴, (𝐹𝐴), (𝐹𝐵))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐾 ∈ Top)    &   (𝜑𝐹 ∈ ((𝐽t (𝐴[,]𝐵)) Cn 𝐾))       (𝜑 → (𝐺 ∈ (𝐽 Cn (𝐾t ran 𝐹)) ∧ (𝐺 ↾ (𝐴[,]𝐵)) = 𝐹))

Theoremcncficcgt0 40419* 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)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→(ℝ ∖ {0})))       (𝜑 → ∃𝑦 ∈ ℝ+𝑥 ∈ (𝐴[,]𝐵)𝑦 ≤ (abs‘𝐶))

Theoremicocncflimc 40420 Limit at the lower bound, of a continuous function defined on a left closed right open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ))       (𝜑 → (𝐹𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐴))

Theoremcncfdmsn 40421* 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}))

Theoremdivcncff 40422 The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (𝑋cn→ℂ))    &   (𝜑𝐺 ∈ (𝑋cn→(ℂ ∖ {0})))       (𝜑 → (𝐹𝑓 / 𝐺) ∈ (𝑋cn→ℂ))

Theoremcncfshiftioo 40423* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝐶 = (𝐴(,)𝐵)    &   (𝜑𝑇 ∈ ℝ)    &   𝐷 = ((𝐴 + 𝑇)(,)(𝐵 + 𝑇))    &   (𝜑𝐹 ∈ (𝐶cn→ℂ))    &   𝐺 = (𝑥𝐷 ↦ (𝐹‘(𝑥𝑇)))       (𝜑𝐺 ∈ (𝐷cn→ℂ))

Theoremcncfiooicclem1 40424* 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.)
𝑥𝜑    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))       (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))

Theoremcncfiooicc 40425* 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.)
𝑥𝜑    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))       (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))

Theoremcncfiooiccre 40426* 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.)
𝑥𝜑    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))       (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))

Theoremcncfioobdlem 40427* 𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶𝑉)    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))       (𝜑 → (𝐺𝐶) = (𝐹𝐶))

Theoremcncfioobd 40428* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘(𝐹𝑦)) ≤ 𝑥)

Theoremjumpncnp 40429 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)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿𝑅)       (𝜑 → ¬ 𝐹 ∈ ((𝐽 CnP (TopOpen‘ℂfld))‘𝐵))

Theoremcncfcompt2 40430* Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   (𝜑 → (𝑥𝐴𝑅) ∈ (𝐴cn𝐵))    &   (𝜑 → (𝑦𝐶𝑆) ∈ (𝐶cn𝐸))    &   (𝜑𝐵𝐶)    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝑥𝐴𝑇) ∈ (𝐴cn𝐸))

Theoremcxpcncf2 40431* The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝐴 ∈ (ℂ ∖ (-∞(,]0)) → (𝑥 ∈ ℂ ↦ (𝐴𝑐𝑥)) ∈ (ℂ–cn→ℂ))

Theoremfprodcncf 40432* The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ (𝐴cn→ℂ))       (𝜑 → (𝑥𝐴 ↦ ∏𝑘𝐵 𝐶) ∈ (𝐴cn→ℂ))

Theoremadd1cncf 40433* Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremadd2cncf 40434* Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 + 𝑥))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremsub1cncfd 40435* Subtracting a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ (𝑥𝐴))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremsub2cncfd 40436* Subtraction from a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ (𝐴𝑥))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremfprodsub2cncf 40437* 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘𝐴 (𝐵𝑥))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremfprodadd2cncf 40438* 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘𝐴 (𝐵 + 𝑥))       (𝜑𝐹 ∈ (ℂ–cn→ℂ))

Theoremfprodsubrecnncnvlem 40439* 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.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘𝐴 (𝐵 − (1 / 𝑛)))    &   𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘𝐴 (𝐵𝑥))    &   𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))       (𝜑𝑆 ⇝ ∏𝑘𝐴 𝐵)

Theoremfprodsubrecnncnv 40440* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℂ)    &   𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (𝐴 − (1 / 𝑛)))       (𝜑𝑆 ⇝ ∏𝑘𝑋 𝐴)

Theoremfprodaddrecnncnvlem 40441* 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.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘𝐴 (𝐵 + (1 / 𝑛)))    &   𝐹 = (𝑥 ∈ ℂ ↦ ∏𝑘𝐴 (𝐵 + 𝑥))    &   𝐺 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))       (𝜑𝑆 ⇝ ∏𝑘𝐴 𝐵)

Theoremfprodaddrecnncnv 40442* 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.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℂ)    &   𝑆 = (𝑛 ∈ ℕ ↦ ∏𝑘𝑋 (𝐴 + (1 / 𝑛)))       (𝜑𝑆 ⇝ ∏𝑘𝑋 𝐴)

20.32.10  Derivatives

Theoremdvsinexp 40443* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))

Theoremdvcosre 40444 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))

Theoremdvsinax 40445* 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)))))

Theoremdvsubf 40446 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓𝐺)) = ((𝑆 D 𝐹) ∘𝑓 − (𝑆 D 𝐺)))

Theoremdvmptconst 40447* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ 0))

Theoremdvcnre 40448 From compex 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))

Theoremdvmptidg 40449* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))       (𝜑 → (𝑆 D (𝑥𝐴𝑥)) = (𝑥𝐴 ↦ 1))

Theoremdvresntr 40450 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐹:𝑋⟶ℂ)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)       (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹𝑌)))

Theoremfperdvper 40451* The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℂ)    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝐺 = (ℝ D 𝐹)       ((𝜑𝑥 ∈ dom 𝐺) → ((𝑥 + 𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑥 + 𝑇)) = (𝐺𝑥)))

Theoremdvmptresicc 40452* Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥 ∈ ℂ ↦ 𝐴)    &   ((𝜑𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ((𝜑𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))

Theoremdvasinbx 40453* 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)))))

Theoremdvresioo 40454 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)))

Theoremdvdivf 40455 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶(ℂ ∖ {0}))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓 / 𝐺)) = ((((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 − ((𝑆 D 𝐺) ∘𝑓 · 𝐹)) ∘𝑓 / (𝐺𝑓 · 𝐺)))

Theoremdvdivbd 40456* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)    &   (𝜑𝑄 ∈ ℝ)    &   ((𝜑𝑥𝑋) → (abs‘𝐶) ≤ 𝑈)    &   ((𝜑𝑥𝑋) → (abs‘𝐵) ≤ 𝑅)    &   ((𝜑𝑥𝑋) → (abs‘𝐷) ≤ 𝑇)    &   ((𝜑𝑥𝑋) → (abs‘𝐴) ≤ 𝑄)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐵)) = (𝑥𝑋𝐷))    &   ((𝜑𝑥𝑋) → 𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → ∀𝑥𝑋 𝐸 ≤ (abs‘𝐵))    &   𝐹 = (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐵)))       (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥𝑋 (abs‘(𝐹𝑥)) ≤ 𝑏)

Theoremdvsubcncf 40457 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓𝐺)) ∈ (𝑋cn→ℂ))

Theoremdvmulcncf 40458 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓 · 𝐺)) ∈ (𝑋cn→ℂ))

Theoremdvcosax 40459* 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)))))

Theoremdvdivcncf 40460 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶(ℂ ∖ {0}))    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓 / 𝐺)) ∈ (𝑋cn→ℂ))

Theoremdvbdfbdioolem1 40461* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐷 ∈ (𝐶(,)𝐵))       (𝜑 → ((abs‘((𝐹𝐷) − (𝐹𝐶))) ≤ (𝐾 · (𝐷𝐶)) ∧ (abs‘((𝐹𝐷) − (𝐹𝐶))) ≤ (𝐾 · (𝐵𝐴))))

Theoremdvbdfbdioolem2 40462* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵𝐴)))       (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹𝑥)) ≤ 𝑀)

Theoremdvbdfbdioo 40463* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(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‘(𝐹𝑥)) ≤ 𝑏)

Theoremioodvbdlimc1lem1 40464* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑅:(ℤ𝑀)⟶(𝐴(,)𝐵))    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝑅𝑗)))    &   (𝜑𝑅 ∈ dom ⇝ )    &   𝐾 = inf({𝑘 ∈ (ℤ𝑀) ∣ ∀𝑖 ∈ (ℤ𝑘)(abs‘((𝑅𝑖) − (𝑅𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )       (𝜑𝑆 ⇝ (lim sup‘𝑆))

Theoremioodvbdlimc1lem2 40465* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(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 / (𝐵𝐴))) + 1)    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝐴 + (1 / 𝑗))))    &   𝑅 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐴 + (1 / 𝑗)))    &   𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)    &   (𝜒 ↔ (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ𝑁)) ∧ (abs‘((𝑆𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧𝐴)) < (1 / 𝑗)))       (𝜑 → (lim sup‘𝑆) ∈ (𝐹 lim 𝐴))

Theoremioodvbdlimc1 40466* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)       (𝜑 → (𝐹 lim 𝐴) ≠ ∅)

Theoremioodvbdlimc2lem 40467* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(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 / (𝐵𝐴))) + 1)    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝐵 − (1 / 𝑗))))    &   𝑅 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐵 − (1 / 𝑗)))    &   𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)    &   (𝜒 ↔ (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ𝑁)) ∧ (abs‘((𝑆𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧𝐵)) < (1 / 𝑗)))       (𝜑 → (lim sup‘𝑆) ∈ (𝐹 lim 𝐵))

Theoremioodvbdlimc2 40468* 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)       (𝜑 → (𝐹 lim 𝐵) ≠ ∅)

Theoremdvdmsscn 40469 𝑋 is a subset of . This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))       (𝜑𝑋 ⊆ ℂ)

Theoremdvmptmulf 40470* Function-builder for derivative, product rule. A version of dvmptmul 23769 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 · 𝐶))) = (𝑥𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))

Theoremdvnmptdivc 40471* Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝑆)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑𝑀 ∈ ℕ0)       ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))

Theoremdvdsn1add 40472 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) → ((¬ 𝐾𝑀𝐾𝑁) → ¬ 𝐾 ∥ (𝑀 + 𝑁)))

Theoremdvxpaek 40473* Derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))

Theoremdvnmptconst 40474* The 𝑁-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑁) = (𝑥𝑋 ↦ 0))

Theoremdvnxpaek 40475* The 𝑛-th derivative of the polynomial (x+A)^K. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))       ((𝜑𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ if(𝐾 < 𝑁, 0, (((!‘𝐾) / (!‘(𝐾𝑁))) · ((𝑥 + 𝐴)↑(𝐾𝑁))))))

Theoremdvnmul 40476* Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   𝐹 = (𝑥𝑋𝐴)    &   𝐺 = (𝑥𝑋𝐵)    &   ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)    &   ((𝜑𝑘 ∈ (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 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))

Theoremdvmptfprodlem 40477* Induction step for dvmptfprod 40478. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   𝑖𝜑    &   𝑗𝜑    &   𝑖𝐹    &   𝑗𝐺    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝐸 ∈ V)    &   (𝜑 → ¬ 𝐸𝐷)    &   (𝜑 → (𝐷 ∪ {𝐸}) ⊆ 𝐼)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (((𝜑𝑥𝑋) ∧ 𝑗𝐷) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐷 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐷 (𝐶 · ∏𝑖 ∈ (𝐷 ∖ {𝑗})𝐴)))    &   ((𝜑𝑥𝑋) → 𝐺 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐹)) = (𝑥𝑋𝐺))    &   (𝑖 = 𝐸𝐴 = 𝐹)    &   (𝑗 = 𝐸𝐶 = 𝐺)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝐷 ∪ {𝐸})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝐷 ∪ {𝐸})(𝐶 · ∏𝑖 ∈ ((𝐷 ∪ {𝐸}) ∖ {𝑗})𝐴)))

Theoremdvmptfprod 40478* Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑖𝜑    &   𝑗𝜑    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)    &   ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝑖 = 𝑗𝐵 = 𝐶)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))

Theoremdvnprodlem1 40479* 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   (𝜑𝐽 ∈ ℕ0)    &   𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)    &   (𝜑𝑇 ∈ Fin)    &   (𝜑𝑍𝑇)    &   (𝜑 → ¬ 𝑍𝑅)    &   (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)       (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))

Theoremdvnprodlem2 40480* Induction step for dvnprodlem2 40480. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑗):𝑋⟶ℂ)    &   𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   (𝜑𝑅𝑇)    &   (𝜑𝑍 ∈ (𝑇𝑅))    &   (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥𝑋 ↦ ∏𝑡𝑅 ((𝐻𝑡)‘𝑥)))‘𝑘) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝐶𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡𝑅 (!‘(𝑐𝑡))) · ∏𝑡𝑅 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))    &   (𝜑𝐽 ∈ (0...𝑁))    &   𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)       (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻𝑡)‘𝑥)))‘𝐽) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))

Theoremdvnprodlem3 40481* The multinomial formula for the 𝑘-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑗):𝑋⟶ℂ)    &   𝐹 = (𝑥𝑋 ↦ ∏𝑡𝑇 ((𝐻𝑡)‘𝑥))    &   𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡𝑇 (𝑐𝑡) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑡𝑇 (!‘(𝑐𝑡))) · ∏𝑡𝑇 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))

Theoremdvnprod 40482* The multinomial formula for the 𝑁-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑘):𝑋⟶ℂ)    &   𝐹 = (𝑥𝑋 ↦ ∏𝑡𝑇 ((𝐻𝑡)‘𝑥))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑇) ∣ Σ𝑡𝑇 (𝑐𝑡) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑡𝑇 (!‘(𝑐𝑡))) · ∏𝑡𝑇 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))

20.32.11  Integrals

Theoremitgsin0pilem1 40483* 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

Theoremibliccsinexp 40484* 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)

Theoremitgsin0pi 40485 Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Theoremiblioosinexp 40486* 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)

Theoremitgsinexplem1 40487* 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))))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))

Theoremitgsinexp 40488* 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)    &   (𝜑𝑁 ∈ (ℤ‘2))       (𝜑 → (𝐼𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Theoremiblconstmpt 40489* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ∈ 𝐿1)

Theoremitgeq1d 40490* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremmbf0 40491 The empty set is a measurable function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ MblFn

Theoremmbfres2cn 40492 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 23457 but here the theorem is extended to complex valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → (𝐹𝐵) ∈ MblFn)    &   (𝜑 → (𝐹𝐶) ∈ MblFn)    &   (𝜑 → (𝐵𝐶) = 𝐴)       (𝜑𝐹 ∈ MblFn)

Theoremvol0 40493 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(vol‘∅) = 0

Theoremditgeqiooicc 40494* 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, (𝐹𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)       (𝜑 → ⨜[𝐴𝐵](𝐹𝑥) d𝑥 = ⨜[𝐴𝐵](𝐺𝑥) d𝑥)

Theoremvolge0 40495 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))

Theoremcnbdibl 40496* A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑 → (vol‘𝐴) ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥)       (𝜑𝐹 ∈ 𝐿1)

Theoremsnmbl 40497 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ dom vol)

Theoremditgeq3d 40498* Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 = 𝐸)       (𝜑 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)

Theoremiblempty 40499 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ 𝐿1

Theoremiblsplit 40500* The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝑈𝐶) ∈ 𝐿1)

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