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Theorem List for Metamath Proof Explorer - 42001-42100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxlimclim2 42001 Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 41774), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐹~~>*𝐴𝐹𝐴))
 
Theoremxlimmnf 42002* A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥))
 
Theoremxlimpnf 42003* A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))
 
Theoremxlimmnfmpt 42004* A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝐵𝑥))
 
Theoremxlimpnfmpt 42005* A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℝ*)    &   𝐹 = (𝑘𝑍𝐵)       (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥𝐵))
 
Theoremclimxlim2lem 42006 In this lemma for climxlim2 42007 there is the additional assumption that the converging function is complex-valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹:𝑍⟶ℂ)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremclimxlim2 42007 A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹𝐴)       (𝜑𝐹~~>*𝐴)
 
Theoremdfxlim2v 42008* An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))
 
Theoremdfxlim2 42009* An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑘𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐹𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝑥 ≤ (𝐹𝑘)))))
 
Theoremclimresd 42010 A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)       (𝜑 → ((𝐹 ↾ (ℤ𝑀)) ⇝ 𝐴𝐹𝐴))
 
Theoremclimresdm 42011 A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹𝑉)       (𝜑 → (𝐹 ∈ dom ⇝ ↔ (𝐹 ↾ (ℤ𝑀)) ∈ dom ⇝ ))
 
Theoremdmclimxlim 42012 A real valued sequence that converges w.r.t. the topology on the complex numbers, converges w.r.t. the topology on the extended reals (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ)    &   (𝜑𝐹 ∈ dom ⇝ )       (𝜑𝐹 ∈ dom ~~>*)
 
Theoremxlimmnflimsup2 42013 A sequence of extended reals converges to -∞ if and only if its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
 
Theoremxlimuni 42014 An infinite sequence converges to at most one limit (w.r.t. to the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐹~~>*𝐴)    &   (𝜑𝐹~~>*𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremxlimclimdm 42015 A sequence of extended reals that converges to a real w.r.t. the standard topology on the extended reals, also converges w.r.t. to the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*𝐴)    &   (𝜑𝐴 ∈ ℝ)       (𝜑𝐹 ∈ dom ⇝ )
 
Theoremxlimfun 42016 The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Fun ~~>*
 
Theoremxlimmnflimsup 42017 If a sequence of extended reals converges to -∞ then its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*-∞)       (𝜑 → (lim sup‘𝐹) = -∞)
 
Theoremxlimdm 42018 Two ways to express that a function has a limit. (The expression (~~>*‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*‘𝐹))
 
Theoremxlimpnfxnegmnf2 42019* A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
𝑗𝐹    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ (𝑗𝑍 ↦ -𝑒(𝐹𝑗))~~>*-∞))
 
Theoremxlimresdm 42020 A function converges in the extended reals iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝐹 ∈ (ℝ*pm ℂ))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ𝑀)) ∈ dom ~~>*))
 
Theoremxlimpnfliminf 42021 If a sequence of extended reals converges to +∞ then its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)    &   (𝜑𝐹~~>*+∞)       (𝜑 → (lim inf‘𝐹) = +∞)
 
Theoremxlimpnfliminf2 42022 A sequence of extended reals converges to +∞ if and only if its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
 
Theoremxlimliminflimsup 42023 A sequence of extended reals converges if and only if its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
 
Theoremxlimlimsupleliminf 42024 A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
 
20.36.8  Trigonometry
 
Theoremcoseq0 42025 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 42026 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 42027 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 42028 The cosine of negative π is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(cos‘-π) = -1
 
Theoremsinaover2ne0 42029 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 42030 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))
 
20.36.9  Continuous Functions
 
Theoremmulcncff 42031 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 42032* 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 42033* A continuous complex function restricted to a subset is continuous, using maps-to notation. This theorem generalizes cncfmptss 41748 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 42034* A constant function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))
 
Theoremidcncfg 42035* The identity function is a continuous function on . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℂ)       (𝜑 → (𝑥𝐴𝑥) ∈ (𝐴cn𝐵))
 
Theoremaddcncf 42036* 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 42037* 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 42038 sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(sin ↾ ℝ) ∈ (ℝ–cn→ℝ)
 
Theoremaddccncf2 42039* 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 42040 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ ({∅} Cn {∅})
 
Theoremfsumcncf 42041* 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 42042* 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 42043 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 42044* 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 42045* 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 42046* Composition of continuous functions. A generalization of cncfmpt1f 23450 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn𝐶))    &   (𝜑𝐹 ∈ (𝐶cn𝐷))       (𝜑 → (𝑥𝐴 ↦ (𝐹𝐵)) ∈ (𝐴cn𝐷))
 
Theoremaddcncff 42047 The sum 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 42048 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 42049* 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.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 𝐵)    &   ((𝜑𝑏𝐵) → (𝐴𝑏) ∈ ((TopOpen‘ℂfld) ↾t 𝐴))    &   ((𝜑𝑏𝐵) → (𝐹𝑏) ∈ ((𝐴𝑏)–cn→ℂ))       (𝜑𝐹 ∈ (𝐴cn→ℂ))
 
Theoremicccncfext 42050* 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 42051* 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 42052 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 42053* 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 42054 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 42055* 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 42056* 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 42057* 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 42058* 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 42059* 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 42060* 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 42061 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 42062* 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 42063* 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 42064* 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 42065* 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 42066* 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 42067* 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 42068* 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 42069* 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 42070* 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 42071* 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 42072* 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 42073* 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 42074* 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)
 
20.36.10  Derivatives
 
Theoremdvsinexp 42075* 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 42076 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))
 
Theoremdvsinax 42077* 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 42078 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 42079* 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 42080 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 42081* 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 42082 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 42083* 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 42084* 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 42085* 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 42086 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 42087 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 42088* 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 42089 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 42090 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 42091* 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 42092 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 42093* 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 42094* 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 42095* 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 42096* 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 42097* 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 42098* 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 42099* 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 42100* 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) ≠ ∅)
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