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Theorem List for Metamath Proof Explorer - 39801-39900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremliminflelimsuplem 39801* The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Theoremliminflelimsup 39802* The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 39823 for a counterexample). The inequality can be strict, see liminfltlimsupex 39807. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Theoremlimsupgtlem 39803* For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))

Theoremlimsupgt 39804* Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim sup‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((𝐹𝑘) − 𝑋) < (lim sup‘𝐹))

Theoremliminfresre 39805 The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)       (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹))

Theoremliminfresicompt 39806* The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℝ)    &   𝑍 = (𝑀[,)+∞)    &   (𝜑𝐴𝑉)       (𝜑 → (lim inf‘(𝑥 ∈ (𝐴𝑍) ↦ 𝐵)) = (lim inf‘(𝑥𝐴𝐵)))

Theoremliminfltlimsupex 39807 An example where the lim inf is strictly smaller than the lim sup. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1))       (lim inf‘𝐹) < (lim sup‘𝐹)

Theoremliminfgelimsup 39808* The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐹𝑉)    &   (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Theoremliminfvalxr 39809* Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝐹    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒(𝐹𝑥))))

Theoremliminfresuz 39810 If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))

Theoremliminflelimsupuz 39811 The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Theoremliminfvalxrmpt 39812* Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))

Theoremliminfresuz2 39813 If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹𝑉)    &   (𝜑 → dom 𝐹 ⊆ ℤ)       (𝜑 → (lim inf‘(𝐹𝑍)) = (lim inf‘𝐹))

Theoremliminfgelimsupuz 39814 The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Theoremliminfval4 39815* Alternate definition of lim inf when the given function is eventually real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝐵)))

Theoremliminfval3 39816* Alternate definition of lim inf when the given function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑥𝐴𝐵)) = -𝑒(lim sup‘(𝑥𝐴 ↦ -𝑒𝐵)))

Theoremliminfequzmpt2 39817* Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝜑    &   𝑗𝐴    &   𝑗𝐵    &   𝐴 = (ℤ𝑀)    &   𝐵 = (ℤ𝑁)    &   (𝜑𝐾𝐴)    &   (𝜑𝐾𝐵)    &   ((𝜑𝑗 ∈ (ℤ𝐾)) → 𝐶𝑉)       (𝜑 → (lim inf‘(𝑗𝐴𝐶)) = (lim inf‘(𝑗𝐵𝐶)))

Theoremliminfvaluz 39818* Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒𝐵)))

Theoremliminf0 39819 The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim inf‘∅) = +∞

Theoremlimsupval4 39820* Alternate definition of lim inf when the given a function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑥𝐴𝐵)) = -𝑒(lim inf‘(𝑥𝐴 ↦ -𝑒𝐵)))

Theoremliminfvaluz2 39821* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim inf‘(𝑘𝑍𝐵)) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝐵)))

Theoremliminfvaluz3 39822* Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -𝑒(𝐹𝑘))))

Theoremliminflelimsupcex 39823 A counterexample for liminflelimsup 39802, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(lim sup‘∅) < (lim inf‘∅)

Theoremlimsupvaluz3 39824* Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝑒𝐵)))

Theoremliminfvaluz4 39825* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘𝑍 ↦ -(𝐹𝑘))))

Theoremlimsupvaluz4 39826* Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ)       (𝜑 → (lim sup‘(𝑘𝑍𝐵)) = -𝑒(lim inf‘(𝑘𝑍 ↦ -𝐵)))

Theoremclimliminflimsupd 39827 If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹))

Theoremliminfreuzlem 39828* Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))

Theoremliminfreuz 39829* Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))

Theoremliminfltlem 39830* Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))

Theoremliminflt 39831* Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(lim inf‘𝐹) < ((𝐹𝑘) + 𝑋))

Theoremclimliminf 39832 A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹)))

Theoremliminflimsupclim 39833 A sequence of real numbers converges if its inferior limit is real, and it is greater or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 39811). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑 → (lim inf‘𝐹) ∈ ℝ)    &   (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))       (𝜑𝐹 ∈ dom ⇝ )

Theoremclimliminflimsup 39834 A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 39814). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))

Theoremclimliminflimsup2 39835 A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 39814). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))

Theoremclimliminflimsup3 39836 A sequence of real numbers converges if and only if its inferior limit is real and equal to its superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))

Theoremclimliminflimsup4 39837 A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹))))

20.31.8  Trigonometry

Theoremcoseq0 39838 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 39839 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 39840 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 39841 The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(cos‘-π) = -1

Theoremsinaover2ne0 39842 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 39843 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))

20.31.9  Continuous Functions

Theoremmulcncff 39844 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 39845* 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 39846* A continuous complex function restricted to a subset is continuous, using "map to" notation. This theorem generalizes cncfmptss 39619 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 39847* A constant function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))

Theoremidcncfg 39848* The identity function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝑥) ∈ (𝐴cn𝐵))

Theoremaddcncf 39849* 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 39850* 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 39851 sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(sin ↾ ℝ) ∈ (ℝ–cn→ℝ)

Theoremaddccncf2 39852* 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 39853 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ ({∅} Cn {∅})

Theoremfsumcncf 39854* 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 39855* 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 39856 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 39857* 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 39858* 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 39859* Composition of continuous functions. A generalization of cncfmpt1f 22697 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))    &   (𝜑𝐹 ∈ (𝐶cn𝐷))       (𝜑 → (𝑥𝐴 ↦ (𝐹𝐵)) ∈ (𝐴cn𝐷))

Theoremaddcncff 39860 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 39861 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 39862* 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 39863* 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 39864* 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 39865 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 39866* 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 39867 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 39868* 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 39869* 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 39870* 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 39871* 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 39872* 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 39873* 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 39874 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 39875* 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 39876* 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 39877* 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 39878* 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 39879* 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 39880* 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 39881* 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 39882* 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 39883* 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 39884* 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 39885* 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 39886* 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 39887* 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)

20.31.10  Derivatives

Theoremdvsinexp 39888* 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 39889 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))

Theoremdvsinax 39890* 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 39891 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 39892* 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 39893 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 39894* 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 39895 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 39896* 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 39897* 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 39898* 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 39899 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 39900 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)))

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