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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelprn2 38701 A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐶) → 𝐴 = 𝐵)

Theoremlimcmptdm 38702* The domain of a map-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))       (𝜑𝐴 ⊆ ℂ)

Theoremclim2f 38703* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14073. Similar to clim2 14083, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremlimcicciooub 38704 The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)       (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐵) = (𝐹 lim 𝐵))

Theoremltmod 38705 A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴))       (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵))

Theoremislpcn 38706* A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑃 ∈ ℂ)       (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝑆) ↔ ∀𝑒 ∈ ℝ+𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥𝑃)) < 𝑒))

Theoremlptre2pt 38707* If a set in the real line has a limit point than it contains two distinct points that are closer than a given distance. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → ((limPt‘𝐽)‘𝐴) ≠ ∅)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (abs‘(𝑥𝑦)) < 𝐸))

Theoremlimsupre 38708* If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
(𝜑𝐵 ⊆ ℝ)    &   (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   (𝜑𝐹:𝐵⟶ℝ)    &   (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗 → (abs‘(𝐹𝑗)) ≤ 𝑏))       (𝜑 → (lim sup‘𝐹) ∈ ℝ)

Theoremlimcresiooub 38709 The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐷𝐵)       (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) lim 𝐶))

Theoremlimcresioolb 38710 The right limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 < 𝐶)    &   (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   (𝜑𝐷 ∈ ℝ*)    &   (𝜑𝐶𝐷)       (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) lim 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) lim 𝐵))

Theoremlimcleqr 38711 If the left and the right limits are equal, the limit of the function exits and the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿 = 𝑅)       (𝜑𝐿 ∈ (𝐹 lim 𝐵))

Theoremlptioo2cn 38712 The upper bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Theoremlptioo1cn 38713 The lower bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))

Theoremneglimc 38714* Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ -𝐵)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))       (𝜑 → -𝐶 ∈ (𝐺 lim 𝐷))

Theoremaddlimc 38715* Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 + 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐸 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐼 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 lim 𝐷))

Theorem0ellimcdiv 38716* If the numerator converges to 0 and the denominator converges to non zero then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 / 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   (𝜑 → 0 ∈ (𝐹 lim 𝐸))    &   (𝜑𝐷 ∈ (𝐺 lim 𝐸))    &   (𝜑𝐷 ≠ 0)       (𝜑 → 0 ∈ (𝐻 lim 𝐸))

Theoremclim2cf 38717* Express the predicate 𝐹 converges to 𝐴. Similar to clim2 14083, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹𝐴 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘(𝐵𝐴)) < 𝑥))

Theoremlimclner 38718 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))    &   (𝜑𝐿𝑅)       (𝜑 → (𝐹 lim 𝐵) = ∅)

Theoremsublimc 38719* Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐸 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐼 ∈ (𝐺 lim 𝐷))       (𝜑 → (𝐸𝐼) ∈ (𝐻 lim 𝐷))

Theoremreclimc 38720* Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ (1 / 𝐵))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (ℂ ∖ {0}))    &   (𝜑𝐶 ∈ (𝐹 lim 𝐷))    &   (𝜑𝐶 ≠ 0)       (𝜑 → (1 / 𝐶) ∈ (𝐺 lim 𝐷))

Theoremclim0cf 38721* Express the predicate 𝐹 converges to 0. Similar to clim 14073, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘𝐵) < 𝑥))

Theoremlimclr 38722 For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐽 = (topGen‘ran (,))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (-∞(,)𝐵))))    &   (𝜑𝐵 ∈ ((limPt‘𝐽)‘(𝐴 ∩ (𝐵(,)+∞))))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) lim 𝐵))    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) lim 𝐵))       (𝜑 → (((𝐹 lim 𝐵) ≠ ∅ ↔ 𝐿 = 𝑅) ∧ (𝐿 = 𝑅𝐿 ∈ (𝐹 lim 𝐵))))

Theoremdivlimc 38723* Limit of the quotient of two funcions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ (𝐵 / 𝐶))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   (𝜑𝑋 ∈ (𝐹 lim 𝐷))    &   (𝜑𝑌 ∈ (𝐺 lim 𝐷))    &   (𝜑𝑌 ≠ 0)    &   ((𝜑𝑥𝐴) → 𝐶 ≠ 0)       (𝜑 → (𝑋 / 𝑌) ∈ (𝐻 lim 𝐷))

Theoremexpfac 38724* Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → 𝐹 ⇝ 0)

Theoremclimconstmpt 38725* A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴 ∈ ℂ)       (𝜑 → (𝑥𝑍𝐴) ⇝ 𝐴)

Theoremclimresmpt 38726* A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑍 = (ℤ𝑀)    &   𝐹 = (𝑥𝑍𝐴)    &   (𝜑𝑁𝑍)    &   𝐺 = (𝑥 ∈ (ℤ𝑁) ↦ 𝐴)       (𝜑 → (𝐺𝐵𝐹𝐵))

Theoremclimsubmpt 38727* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐷)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐷))

Theoremclimsubc2mpt 38728* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐴) ⇝ 𝐶)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐶𝐵))

Theoremclimsubc1mpt 38729* Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → (𝑘𝑍𝐵) ⇝ 𝐶)       (𝜑 → (𝑘𝑍 ↦ (𝐴𝐵)) ⇝ (𝐴𝐶))

Theoremfnlimfv 38730* The value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐷    &   𝑥𝐹    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝐺𝑋) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))

Theoremclimreclf 38731* The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑𝐴 ∈ ℝ)

Theoremclimeldmeq 38732* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ ))

Theoremclimf2 38733* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 14073, but without the disjoint var constraint 𝜑𝑘 and 𝐹𝑘. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘 ∈ ℤ) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremfnlimcnv 38734* The sequence of function values converges to the value of the limit function 𝐺 at any point of its domain 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐹    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)) ⇝ (𝐺𝑋))

Theoremclimeldmeqmpt 38735* Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑅)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑𝐶𝑆)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝐶) → 𝐷𝑊)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ((𝑘𝐴𝐵) ∈ dom ⇝ ↔ (𝑘𝐶𝐷) ∈ dom ⇝ ))

Theoremclimfveq 38736* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺))

Theoremclim2f2 38737* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 14073. Similar to clim2 14083, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)       (𝜑 → (𝐹𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑥))))

Theoremclimfveqmpt 38738* Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐴𝑅)    &   (𝜑𝑍𝐴)    &   ((𝜑𝑘𝐴) → 𝐵𝑉)    &   (𝜑𝐶𝑆)    &   (𝜑𝑍𝐶)    &   ((𝜑𝑘𝐶) → 𝐷𝑊)    &   ((𝜑𝑘𝑍) → 𝐵 = 𝐷)       (𝜑 → ( ⇝ ‘(𝑘𝐴𝐵)) = ( ⇝ ‘(𝑘𝐶𝐷)))

Theoremclimd 38739* Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))

Theoremclim2d 38740* The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵𝐴)) < 𝑋))

Theoremfnlimfvre 38741* The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   (𝜑𝑋𝐷)       (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)

Theoremallbutfifvre 38742* Given a sequence of real valued functions, and 𝑋 that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)    &   (𝜑𝑋𝐷)       (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ∈ ℝ)

Theoremclimleltrp 38743* The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑘𝜑    &   𝑘𝐹    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) ∈ ℝ)    &   (𝜑𝐹𝐴)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐶)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) ∈ ℝ ∧ (𝐹𝑘) < (𝐶 + 𝑋)))

Theoremfnlimfvre2 38744* The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)       (𝜑 → (𝐺𝑋) ∈ ℝ)

Theoremfnlimf 38745* The limit function of real functions, is a real valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))       (𝜑𝐺:𝐷⟶ℝ)

Theoremfnlimabslt 38746* A sequence of function values, approximates the corresponding limit function value, all but finitely many times. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑚𝜑    &   𝑚𝐹    &   𝑥𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)    &   𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))    &   (𝜑𝑋𝐷)    &   (𝜑𝑌 ∈ ℝ+)       (𝜑 → ∃𝑛𝑍𝑚 ∈ (ℤ𝑛)(((𝐹𝑚)‘𝑋) ∈ ℝ ∧ (abs‘(((𝐹𝑚)‘𝑋) − (𝐺𝑋))) < 𝑌))

21.31.8  Trigonometry

Theoremcoseq0 38747 A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → ((cos‘𝐴) = 0 ↔ ((𝐴 / π) + (1 / 2)) ∈ ℤ))

Theoremsinmulcos 38748 Multiplication formula for sine and cosine. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((sin‘𝐴) · (cos‘𝐵)) = (((sin‘(𝐴 + 𝐵)) + (sin‘(𝐴𝐵))) / 2))

Theoremcoskpi2 38749 The cosine of an integer multiple of negative π is either 1 or negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ ℤ → (cos‘(𝐾 · π)) = if(2 ∥ 𝐾, 1, -1))

Theoremcosnegpi 38750 The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(cos‘-π) = -1

Theoremsinaover2ne0 38751 If 𝐴 in (0, 2π) then sin(𝐴 / 2) is not 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (0(,)(2 · π)) → (sin‘(𝐴 / 2)) ≠ 0)

Theoremcosknegpi 38752 The cosine of an integer multiple of negative π is either 1 ore negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ ℤ → (cos‘(𝐾 · -π)) = if(2 ∥ 𝐾, 1, -1))

21.31.9  Continuous Functions

Theoremmulcncff 38753 The multiplication 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))

Theoremsubcncf 38754* 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 ↦ (𝐴𝐵)) ∈ (𝑋cn→ℂ))

Theoremcncfmptssg 38755* A continuous complex function restricted to a subset is continuous, using "map to" notation. This theorem generalizes cncfmptss 38654 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 38756* A constant function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))

Theoremidcncfg 38757* The identity function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝑥) ∈ (𝐴cn𝐵))

Theoremaddcncf 38758* 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 38759* 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 38760 sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(sin ↾ ℝ) ∈ (ℝ–cn→ℝ)

Theoremaddccncf2 38761* 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 38762 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ ({∅} Cn {∅})

Theoremfsumcncf 38763* 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 38764* 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 38765 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 38766* 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 38767* 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 38768* Composition of continuous functions. A generalization of cncfmpt1f 22524 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))    &   (𝜑𝐹 ∈ (𝐶cn𝐷))       (𝜑 → (𝑥𝐴 ↦ (𝐹𝐵)) ∈ (𝐴cn𝐷))

Theoremdivcncf 38769* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→(ℂ ∖ {0})))       (𝜑 → (𝑥𝑋 ↦ (𝐴 / 𝐵)) ∈ (𝑋cn→ℂ))

Theoremaddcncff 38770 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 38771 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 38772* 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 38773* 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 38774* 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 38775 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 38776* 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 38777 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 38778* 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 38779* 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 38780* 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 38781* 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 38782* 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 38783* 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 38784 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 38785* 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 38786* 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 38787* 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 38788* 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 38789* 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 38790* 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 38791* 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 38792* 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 38793* 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 38794* 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 38795* 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 38796* 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 38797* 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)

21.31.10  Derivatives

Theoremdvsinexp 38798* 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 38799 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))

Theoremdvrecg 38800* Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ (ℂ ∖ {0}))    &   ((𝜑𝑥𝑋) → 𝐶𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐵)) = (𝑥𝑋𝐶))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐵))) = (𝑥𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))

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