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Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxlimpnfv 43801* 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*+∞ ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)π‘₯ ≀ (πΉβ€˜π‘˜)))
 
Theoremxlimclim2lem 43802* Lemma for xlimclim2 43803. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘— ∈ 𝑍 (𝐹 β†Ύ (β„€β‰₯β€˜π‘—)):(β„€β‰₯β€˜π‘—)βŸΆβ„)    β‡’   (πœ‘ β†’ (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
 
Theoremxlimclim2 43803 Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 43576), 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴))
 
Theoremxlimmnf 43804* 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.)
β„²π‘˜πΉ    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*-∞ ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)(πΉβ€˜π‘˜) ≀ π‘₯))
 
Theoremxlimpnf 43805* 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.)
β„²π‘˜πΉ    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*+∞ ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)π‘₯ ≀ (πΉβ€˜π‘˜)))
 
Theoremxlimmnfmpt 43806* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐡 ∈ ℝ*)    &   πΉ = (π‘˜ ∈ 𝑍 ↦ 𝐡)    β‡’   (πœ‘ β†’ (𝐹~~>*-∞ ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)𝐡 ≀ π‘₯))
 
Theoremxlimpnfmpt 43807* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐡 ∈ ℝ*)    &   πΉ = (π‘˜ ∈ 𝑍 ↦ 𝐡)    β‡’   (πœ‘ β†’ (𝐹~~>*+∞ ↔ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)π‘₯ ≀ 𝐡))
 
Theoremclimxlim2lem 43808 In this lemma for climxlim2 43809 there is the additional assumption that the converging function is complex-valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„‚)    &   (πœ‘ β†’ 𝐹 ⇝ 𝐴)    β‡’   (πœ‘ β†’ 𝐹~~>*𝐴)
 
Theoremclimxlim2 43809 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐹 ⇝ 𝐴)    β‡’   (πœ‘ β†’ 𝐹~~>*𝐴)
 
Theoremdfxlim2v 43810* 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)(πΉβ€˜π‘˜) ≀ π‘₯) ∨ (𝐴 = +∞ ∧ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)π‘₯ ≀ (πΉβ€˜π‘˜)))))
 
Theoremdfxlim2 43811* 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.)
β„²π‘˜πΉ    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)(πΉβ€˜π‘˜) ≀ π‘₯) ∨ (𝐴 = +∞ ∧ βˆ€π‘₯ ∈ ℝ βˆƒπ‘— ∈ 𝑍 βˆ€π‘˜ ∈ (β„€β‰₯β€˜π‘—)π‘₯ ≀ (πΉβ€˜π‘˜)))))
 
Theoremclimresd 43812 A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ (β„€β‰₯β€˜π‘€)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴))
 
Theoremclimresdm 43813 A real function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐹 ∈ dom ⇝ ↔ (𝐹 β†Ύ (β„€β‰₯β€˜π‘€)) ∈ dom ⇝ ))
 
Theoremdmclimxlim 43814 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„)    &   (πœ‘ β†’ 𝐹 ∈ dom ⇝ )    β‡’   (πœ‘ β†’ 𝐹 ∈ dom ~~>*)
 
Theoremxlimmnflimsup2 43815 A sequence of extended reals converges to -∞ if and only if its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*-∞ ↔ (lim supβ€˜πΉ) = -∞))
 
Theoremxlimuni 43816 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.)
(πœ‘ β†’ 𝐹~~>*𝐴)    &   (πœ‘ β†’ 𝐹~~>*𝐡)    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremxlimclimdm 43817 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐹~~>*𝐴)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ 𝐹 ∈ dom ⇝ )
 
Theoremxlimfun 43818 The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Fun ~~>*
 
Theoremxlimmnflimsup 43819 If a sequence of extended reals converges to -∞ then its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐹~~>*-∞)    β‡’   (πœ‘ β†’ (lim supβ€˜πΉ) = -∞)
 
Theoremxlimdm 43820 Two ways to express that a function has a limit. (The expression (~~>*β€˜πΉ) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(𝐹 ∈ dom ~~>* ↔ 𝐹~~>*(~~>*β€˜πΉ))
 
Theoremxlimpnfxnegmnf2 43821* A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Ⅎ𝑗𝐹    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*+∞ ↔ (𝑗 ∈ 𝑍 ↦ -𝑒(πΉβ€˜π‘—))~~>*-∞))
 
Theoremxlimresdm 43822 A function converges in the extended reals iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝐹 ∈ (ℝ* ↑pm β„‚))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    β‡’   (πœ‘ β†’ (𝐹 ∈ dom ~~>* ↔ (𝐹 β†Ύ (β„€β‰₯β€˜π‘€)) ∈ dom ~~>*))
 
Theoremxlimpnfliminf 43823 If a sequence of extended reals converges to +∞ then its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    &   (πœ‘ β†’ 𝐹~~>*+∞)    β‡’   (πœ‘ β†’ (lim infβ€˜πΉ) = +∞)
 
Theoremxlimpnfliminf2 43824 A sequence of extended reals converges to +∞ if and only if its superior limit is also +∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹~~>*+∞ ↔ (lim infβ€˜πΉ) = +∞))
 
Theoremxlimliminflimsup 43825 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹 ∈ dom ~~>* ↔ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)))
 
Theoremxlimlimsupleliminf 43826 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)    β‡’   (πœ‘ β†’ (𝐹 ∈ dom ~~>* ↔ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)))
 
21.38.8  Trigonometry
 
Theoremcoseq0 43827 A complex number whose cosine is zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ β„‚ β†’ ((cosβ€˜π΄) = 0 ↔ ((𝐴 / Ο€) + (1 / 2)) ∈ β„€))
 
Theoremsinmulcos 43828 Multiplication formula for sine and cosine. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((sinβ€˜π΄) Β· (cosβ€˜π΅)) = (((sinβ€˜(𝐴 + 𝐡)) + (sinβ€˜(𝐴 βˆ’ 𝐡))) / 2))
 
Theoremcoskpi2 43829 The cosine of an integer multiple of negative Ο€ is either 1 or negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ β„€ β†’ (cosβ€˜(𝐾 Β· Ο€)) = if(2 βˆ₯ 𝐾, 1, -1))
 
Theoremcosnegpi 43830 The cosine of negative Ο€ is negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(cosβ€˜-Ο€) = -1
 
Theoremsinaover2ne0 43831 If 𝐴 in (0, 2Ο€) then sin(𝐴 / 2) is not 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (0(,)(2 Β· Ο€)) β†’ (sinβ€˜(𝐴 / 2)) β‰  0)
 
Theoremcosknegpi 43832 The cosine of an integer multiple of negative Ο€ is either 1 or negative 1. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ β„€ β†’ (cosβ€˜(𝐾 Β· -Ο€)) = if(2 βˆ₯ 𝐾, 1, -1))
 
21.38.9  Continuous Functions
 
Theoremmulcncff 43833 The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹 ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐺 ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ (𝑋–cnβ†’β„‚))
 
Theoremcncfmptssg 43834* A continuous complex function restricted to a subset is continuous, using maps-to notation. This theorem generalizes cncfmptss 43550 because it allows to establish a subset for the codomain also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐸)    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cn→𝐡))    &   (πœ‘ β†’ 𝐢 βŠ† 𝐴)    &   (πœ‘ β†’ 𝐷 βŠ† 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐸 ∈ 𝐷)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↦ 𝐸) ∈ (𝐢–cn→𝐷))
 
Theoremconstcncfg 43835* A constant function is a continuous function on β„‚. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ 𝐢)    &   (πœ‘ β†’ 𝐢 βŠ† β„‚)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (𝐴–cn→𝐢))
 
Theoremidcncfg 43836* The identity function is a continuous function on β„‚. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐡 βŠ† β„‚)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ π‘₯) ∈ (𝐴–cn→𝐡))
 
Theoremcncfshift 43837* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝑇 ∈ β„‚)    &   π΅ = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (𝑦 + 𝑇)}    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))    &   πΊ = (π‘₯ ∈ 𝐡 ↦ (πΉβ€˜(π‘₯ βˆ’ 𝑇)))    β‡’   (πœ‘ β†’ 𝐺 ∈ (𝐡–cnβ†’β„‚))
 
Theoremresincncf 43838 sin restricted to reals is continuous from reals to reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(sin β†Ύ ℝ) ∈ (ℝ–cn→ℝ)
 
Theoremaddccncf2 43839* Adding a constant is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ (𝐡 + π‘₯))    β‡’   ((𝐴 βŠ† β„‚ ∧ 𝐡 ∈ β„‚) β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))
 
Theorem0cnf 43840 The empty set is a continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
βˆ… ∈ ({βˆ…} Cn {βˆ…})
 
Theoremfsumcncf 43841* The finite sum of continuous complex function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑋 βŠ† β„‚)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ Ξ£π‘˜ ∈ 𝐴 𝐡) ∈ (𝑋–cnβ†’β„‚))
 
Theoremcncfperiod 43842* A periodic continuous function stays continuous if the domain is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   π΅ = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (𝑦 + 𝑇)}    &   (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)    &   (πœ‘ β†’ 𝐡 βŠ† dom 𝐹)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πΉβ€˜(π‘₯ + 𝑇)) = (πΉβ€˜π‘₯))    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐴) ∈ (𝐴–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ (𝐡–cnβ†’β„‚))
 
Theoremsubcncff 43843 The subtraction of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹 ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐺 ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝐹 ∘f βˆ’ 𝐺) ∈ (𝑋–cnβ†’β„‚))
 
Theoremnegcncfg 43844* The opposite of a continuous function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (𝐴–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ -𝐡) ∈ (𝐴–cnβ†’β„‚))
 
Theoremcnfdmsn 43845* A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (π‘₯ ∈ {𝐴} ↦ 𝐡) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐡}))
 
Theoremcncfcompt 43846* Composition of continuous functions. A generalization of cncfmpt1f 24199 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐡) ∈ (𝐴–cn→𝐢))    &   (πœ‘ β†’ 𝐹 ∈ (𝐢–cn→𝐷))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ (πΉβ€˜π΅)) ∈ (𝐴–cn→𝐷))
 
Theoremaddcncff 43847 The sum of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹 ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐺 ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ (𝑋–cnβ†’β„‚))
 
Theoremioccncflimc 43848 Limit at the upper bound of a continuous function defined on a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,]𝐡)–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (πΉβ€˜π΅) ∈ ((𝐹 β†Ύ (𝐴(,)𝐡)) limβ„‚ 𝐡))
 
Theoremcncfuni 43849* 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.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 βŠ† βˆͺ 𝐡)    &   ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝐴 ∩ 𝑏) ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝐴))    &   ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (𝐹 β†Ύ 𝑏) ∈ ((𝐴 ∩ 𝑏)–cnβ†’β„‚))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))
 
Theoremicccncfext 43850* 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.)
β„²π‘₯𝐹    &   π½ = (topGenβ€˜ran (,))    &   π‘Œ = βˆͺ 𝐾    &   πΊ = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ (𝐴[,]𝐡), (πΉβ€˜π‘₯), if(π‘₯ < 𝐴, (πΉβ€˜π΄), (πΉβ€˜π΅))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐾 ∈ Top)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐽 β†Ύt (𝐴[,]𝐡)) Cn 𝐾))    β‡’   (πœ‘ β†’ (𝐺 ∈ (𝐽 Cn (𝐾 β†Ύt ran 𝐹)) ∧ (𝐺 β†Ύ (𝐴[,]𝐡)) = 𝐹))
 
Theoremcncficcgt0 43851* 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 = (π‘₯ ∈ (𝐴[,]𝐡) ↦ 𝐢)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’(ℝ βˆ– {0})))    β‡’   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ+ βˆ€π‘₯ ∈ (𝐴[,]𝐡)𝑦 ≀ (absβ€˜πΆ))
 
Theoremicocncflimc 43852 Limit at the lower bound, of a continuous function defined on a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,)𝐡)–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (πΉβ€˜π΄) ∈ ((𝐹 β†Ύ (𝐴(,)𝐡)) limβ„‚ 𝐴))
 
Theoremcncfdmsn 43853* A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (π‘₯ ∈ {𝐴} ↦ 𝐡) ∈ ({𝐴}–cnβ†’{𝐡}))
 
Theoremdivcncff 43854 The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹 ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐺 ∈ (𝑋–cnβ†’(β„‚ βˆ– {0})))    β‡’   (πœ‘ β†’ (𝐹 ∘f / 𝐺) ∈ (𝑋–cnβ†’β„‚))
 
Theoremcncfshiftioo 43855* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   πΆ = (𝐴(,)𝐡)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   π· = ((𝐴 + 𝑇)(,)(𝐡 + 𝑇))    &   (πœ‘ β†’ 𝐹 ∈ (𝐢–cnβ†’β„‚))    &   πΊ = (π‘₯ ∈ 𝐷 ↦ (πΉβ€˜(π‘₯ βˆ’ 𝑇)))    β‡’   (πœ‘ β†’ 𝐺 ∈ (𝐷–cnβ†’β„‚))
 
Theoremcncfiooicclem1 43856* A continuous function 𝐹 on an open interval (𝐴(,)𝐡) 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 𝐡. 𝐹 can be complex-valued. This lemma assumes 𝐴 < 𝐡, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
β„²π‘₯πœ‘    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    β‡’   (πœ‘ β†’ 𝐺 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
 
Theoremcncfiooicc 43857* A continuous function 𝐹 on an open interval (𝐴(,)𝐡) 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 𝐡. 𝐹 can be complex-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
β„²π‘₯πœ‘    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    β‡’   (πœ‘ β†’ 𝐺 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))
 
Theoremcncfiooiccre 43858* A continuous function 𝐹 on an open interval (𝐴(,)𝐡) 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 𝐡. 𝐹 is assumed to be real-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
β„²π‘₯πœ‘    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cn→ℝ))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    β‡’   (πœ‘ β†’ 𝐺 ∈ ((𝐴[,]𝐡)–cn→ℝ))
 
Theoremcncfioobdlem 43859* 𝐺 actually extends 𝐹. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆπ‘‰)    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ if(π‘₯ = 𝐴, 𝑅, if(π‘₯ = 𝐡, 𝐿, (πΉβ€˜π‘₯))))    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ (πΊβ€˜πΆ) = (πΉβ€˜πΆ))
 
Theoremcncfioobd 43860* A continuous function 𝐹 on an open interval (𝐴(,)𝐡) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐡 is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ (𝐴(,)𝐡)(absβ€˜(πΉβ€˜π‘¦)) ≀ π‘₯)
 
Theoremjumpncnp 43861 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 = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π½ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐹:π΄βŸΆβ„‚)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ((limPtβ€˜π½)β€˜(𝐴 ∩ (-∞(,)𝐡))))    &   (πœ‘ β†’ 𝐡 ∈ ((limPtβ€˜π½)β€˜(𝐴 ∩ (𝐡(,)+∞))))    &   (πœ‘ β†’ 𝐿 ∈ ((𝐹 β†Ύ (-∞(,)𝐡)) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝑅 ∈ ((𝐹 β†Ύ (𝐡(,)+∞)) limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐿 β‰  𝑅)    β‡’   (πœ‘ β†’ Β¬ 𝐹 ∈ ((𝐽 CnP (TopOpenβ€˜β„‚fld))β€˜π΅))
 
Theoremcxpcncf2 43862* The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝐴 ∈ (β„‚ βˆ– (-∞(,]0)) β†’ (π‘₯ ∈ β„‚ ↦ (𝐴↑𝑐π‘₯)) ∈ (ℂ–cnβ†’β„‚))
 
Theoremfprodcncf 43863* The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴 ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ (𝐴–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ βˆπ‘˜ ∈ 𝐡 𝐢) ∈ (𝐴–cnβ†’β„‚))
 
Theoremadd1cncf 43864* Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ (π‘₯ + 𝐴))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremadd2cncf 43865* Addition to a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ (𝐴 + π‘₯))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremsub1cncfd 43866* Subtracting a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ (π‘₯ βˆ’ 𝐴))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremsub2cncfd 43867* Subtraction from a constant is a continuous function. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ (𝐴 βˆ’ π‘₯))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremfprodsub2cncf 43868* 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 βˆ’ π‘₯))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremfprodadd2cncf 43869* 𝐹 is continuous. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 + π‘₯))    β‡’   (πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremfprodsubrecnncnvlem 43870* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 βˆ’ (1 / 𝑛)))    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 βˆ’ π‘₯))    &   πΊ = (𝑛 ∈ β„• ↦ (1 / 𝑛))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodsubrecnncnv 43871* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝑋 (𝐴 βˆ’ (1 / 𝑛)))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 𝐴)
 
Theoremfprodaddrecnncnvlem 43872* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 + (1 / 𝑛)))    &   πΉ = (π‘₯ ∈ β„‚ ↦ βˆπ‘˜ ∈ 𝐴 (𝐡 + π‘₯))    &   πΊ = (𝑛 ∈ β„• ↦ (1 / 𝑛))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodaddrecnncnv 43873* 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.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑋 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   π‘† = (𝑛 ∈ β„• ↦ βˆπ‘˜ ∈ 𝑋 (𝐴 + (1 / 𝑛)))    β‡’   (πœ‘ β†’ 𝑆 ⇝ βˆπ‘˜ ∈ 𝑋 𝐴)
 
21.38.10  Derivatives
 
Theoremdvsinexp 43874* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ ((sinβ€˜π‘₯)↑𝑁))) = (π‘₯ ∈ β„‚ ↦ ((𝑁 Β· ((sinβ€˜π‘₯)↑(𝑁 βˆ’ 1))) Β· (cosβ€˜π‘₯))))
 
Theoremdvcosre 43875 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(ℝ D (π‘₯ ∈ ℝ ↦ (cosβ€˜π‘₯))) = (π‘₯ ∈ ℝ ↦ -(sinβ€˜π‘₯))
 
Theoremdvsinax 43876* Derivative exercise: the derivative with respect to y of sin(Ay), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ β„‚ β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (sinβ€˜(𝐴 Β· 𝑦)))) = (𝑦 ∈ β„‚ ↦ (𝐴 Β· (cosβ€˜(𝐴 Β· 𝑦)))))
 
Theoremdvsubf 43877 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ dom (𝑆 D 𝐹) = 𝑋)    &   (πœ‘ β†’ dom (𝑆 D 𝐺) = 𝑋)    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f βˆ’ 𝐺)) = ((𝑆 D 𝐹) ∘f βˆ’ (𝑆 D 𝐺)))
 
Theoremdvmptconst 43878* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐴 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝐴 ↦ 𝐡)) = (π‘₯ ∈ 𝐴 ↦ 0))
 
Theoremdvcnre 43879 From complex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:β„‚βŸΆβ„‚ ∧ ℝ βŠ† dom (β„‚ D 𝐹)) β†’ (ℝ D (𝐹 β†Ύ ℝ)) = ((β„‚ D 𝐹) β†Ύ ℝ))
 
Theoremdvmptidg 43880* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐴 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝐴 ↦ π‘₯)) = (π‘₯ ∈ 𝐴 ↦ 1))
 
Theoremdvresntr 43881 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   π½ = (𝐾 β†Ύt 𝑆)    &   πΎ = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ ((intβ€˜π½)β€˜π‘‹) = π‘Œ)    β‡’   (πœ‘ β†’ (𝑆 D 𝐹) = (𝑆 D (𝐹 β†Ύ π‘Œ)))
 
Theoremfperdvper 43882* The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„‚)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (πΉβ€˜(π‘₯ + 𝑇)) = (πΉβ€˜π‘₯))    &   πΊ = (ℝ D 𝐹)    β‡’   ((πœ‘ ∧ π‘₯ ∈ dom 𝐺) β†’ ((π‘₯ + 𝑇) ∈ dom 𝐺 ∧ (πΊβ€˜(π‘₯ + 𝑇)) = (πΊβ€˜π‘₯)))
 
Theoremdvasinbx 43883* Derivative exercise: the derivative with respect to y of A x sin(By), given two constants 𝐴 and 𝐡. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (β„‚ D (𝑦 ∈ β„‚ ↦ (𝐴 Β· (sinβ€˜(𝐡 Β· 𝑦))))) = (𝑦 ∈ β„‚ ↦ ((𝐴 Β· 𝐡) Β· (cosβ€˜(𝐡 Β· 𝑦)))))
 
Theoremdvresioo 43884 Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 βŠ† ℝ ∧ 𝐹:π΄βŸΆβ„‚) β†’ (ℝ D (𝐹 β†Ύ (𝐡(,)𝐢))) = ((ℝ D 𝐹) β†Ύ (𝐡(,)𝐢)))
 
Theoremdvdivf 43885 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆ(β„‚ βˆ– {0}))    &   (πœ‘ β†’ dom (𝑆 D 𝐹) = 𝑋)    &   (πœ‘ β†’ dom (𝑆 D 𝐺) = 𝑋)    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f / 𝐺)) = ((((𝑆 D 𝐹) ∘f Β· 𝐺) ∘f βˆ’ ((𝑆 D 𝐺) ∘f Β· 𝐹)) ∘f / (𝐺 ∘f Β· 𝐺)))
 
Theoremdvdivbd 43886* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐢))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ π‘ˆ ∈ ℝ)    &   (πœ‘ β†’ 𝑅 ∈ ℝ)    &   (πœ‘ β†’ 𝑇 ∈ ℝ)    &   (πœ‘ β†’ 𝑄 ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜πΆ) ≀ π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π΅) ≀ 𝑅)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π·) ≀ 𝑇)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (absβ€˜π΄) ≀ 𝑄)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐡)) = (π‘₯ ∈ 𝑋 ↦ 𝐷))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 𝐸 ≀ (absβ€˜π΅))    &   πΉ = (𝑆 D (π‘₯ ∈ 𝑋 ↦ (𝐴 / 𝐡)))    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘₯ ∈ 𝑋 (absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑏)
 
Theoremdvsubcncf 43887 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f βˆ’ 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvmulcncf 43888 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f Β· 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvcosax 43889* Derivative exercise: the derivative with respect to x of cos(Ax), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ β„‚ β†’ (β„‚ D (π‘₯ ∈ β„‚ ↦ (cosβ€˜(𝐴 Β· π‘₯)))) = (π‘₯ ∈ β„‚ ↦ (𝐴 Β· -(sinβ€˜(𝐴 Β· π‘₯)))))
 
Theoremdvdivcncf 43890 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐺:π‘‹βŸΆ(β„‚ βˆ– {0}))    &   (πœ‘ β†’ (𝑆 D 𝐹) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (𝑆 D 𝐺) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (𝑆 D (𝐹 ∘f / 𝐺)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremdvbdfbdioolem1 43891* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝐾)    &   (πœ‘ β†’ 𝐢 ∈ (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐷 ∈ (𝐢(,)𝐡))    β‡’   (πœ‘ β†’ ((absβ€˜((πΉβ€˜π·) βˆ’ (πΉβ€˜πΆ))) ≀ (𝐾 Β· (𝐷 βˆ’ 𝐢)) ∧ (absβ€˜((πΉβ€˜π·) βˆ’ (πΉβ€˜πΆ))) ≀ (𝐾 Β· (𝐡 βˆ’ 𝐴))))
 
Theoremdvbdfbdioolem2 43892* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ 𝐾 ∈ ℝ)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝐾)    &   π‘€ = ((absβ€˜(πΉβ€˜((𝐴 + 𝐡) / 2))) + (𝐾 Β· (𝐡 βˆ’ 𝐴)))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑀)
 
Theoremdvbdfbdioo 43893* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ π‘Ž)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜(πΉβ€˜π‘₯)) ≀ 𝑏)
 
Theoremioodvbdlimc1lem1 43894* If 𝐹 has bounded derivative on (𝐴(,)𝐡) 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cn→ℝ))    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑅:(β„€β‰₯β€˜π‘€)⟢(𝐴(,)𝐡))    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(π‘…β€˜π‘—)))    &   (πœ‘ β†’ 𝑅 ∈ dom ⇝ )    &   πΎ = inf({π‘˜ ∈ (β„€β‰₯β€˜π‘€) ∣ βˆ€π‘– ∈ (β„€β‰₯β€˜π‘˜)(absβ€˜((π‘…β€˜π‘–) βˆ’ (π‘…β€˜π‘˜))) < (π‘₯ / (sup(ran (𝑧 ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘§))), ℝ, < ) + 1))}, ℝ, < )    β‡’   (πœ‘ β†’ 𝑆 ⇝ (lim supβ€˜π‘†))
 
Theoremioodvbdlimc1lem2 43895* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   π‘Œ = sup(ran (π‘₯ ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘₯))), ℝ, < )    &   π‘€ = ((βŒŠβ€˜(1 / (𝐡 βˆ’ 𝐴))) + 1)    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(𝐴 + (1 / 𝑗))))    &   π‘… = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (𝐴 + (1 / 𝑗)))    &   π‘ = if(𝑀 ≀ ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), 𝑀)    &   (πœ’ ↔ (((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘)) ∧ (absβ€˜((π‘†β€˜π‘—) βˆ’ (lim supβ€˜π‘†))) < (π‘₯ / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐡)) ∧ (absβ€˜(𝑧 βˆ’ 𝐴)) < (1 / 𝑗)))    β‡’   (πœ‘ β†’ (lim supβ€˜π‘†) ∈ (𝐹 limβ„‚ 𝐴))
 
Theoremioodvbdlimc1 43896* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐴) β‰  βˆ…)
 
Theoremioodvbdlimc2lem 43897* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    &   π‘Œ = sup(ran (π‘₯ ∈ (𝐴(,)𝐡) ↦ (absβ€˜((ℝ D 𝐹)β€˜π‘₯))), ℝ, < )    &   π‘€ = ((βŒŠβ€˜(1 / (𝐡 βˆ’ 𝐴))) + 1)    &   π‘† = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (πΉβ€˜(𝐡 βˆ’ (1 / 𝑗))))    &   π‘… = (𝑗 ∈ (β„€β‰₯β€˜π‘€) ↦ (𝐡 βˆ’ (1 / 𝑗)))    &   π‘ = if(𝑀 ≀ ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), ((βŒŠβ€˜(π‘Œ / (π‘₯ / 2))) + 1), 𝑀)    &   (πœ’ ↔ (((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘)) ∧ (absβ€˜((π‘†β€˜π‘—) βˆ’ (lim supβ€˜π‘†))) < (π‘₯ / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐡)) ∧ (absβ€˜(𝑧 βˆ’ 𝐡)) < (1 / 𝑗)))    β‡’   (πœ‘ β†’ (lim supβ€˜π‘†) ∈ (𝐹 limβ„‚ 𝐡))
 
Theoremioodvbdlimc2 43898* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐹:(𝐴(,)𝐡)βŸΆβ„)    &   (πœ‘ β†’ dom (ℝ D 𝐹) = (𝐴(,)𝐡))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜((ℝ D 𝐹)β€˜π‘₯)) ≀ 𝑦)    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐡) β‰  βˆ…)
 
Theoremdvdmsscn 43899 𝑋 is a subset of β„‚. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝑋 ∈ ((TopOpenβ€˜β„‚fld) β†Ύt 𝑆))    β‡’   (πœ‘ β†’ 𝑋 βŠ† β„‚)
 
Theoremdvmptmulf 43900* Function-builder for derivative, product rule. A version of dvmptmul 25247 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐴)) = (π‘₯ ∈ 𝑋 ↦ 𝐡))    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ 𝐷 ∈ π‘Š)    &   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ 𝐢)) = (π‘₯ ∈ 𝑋 ↦ 𝐷))    β‡’   (πœ‘ β†’ (𝑆 D (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐢))) = (π‘₯ ∈ 𝑋 ↦ ((𝐡 Β· 𝐢) + (𝐷 Β· 𝐴))))
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