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Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.34.3.3  Primitive equivalent of ax-groth
 
Theoremexpandan 42301 Expand conjunction to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ ↔ πœ“)    &   (πœ’ ↔ πœƒ)    β‡’   ((πœ‘ ∧ πœ’) ↔ Β¬ (πœ“ β†’ Β¬ πœƒ))
 
Theoremexpandexn 42302 Expand an existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ ↔ Β¬ πœ“)    β‡’   (βˆƒπ‘₯πœ‘ ↔ Β¬ βˆ€π‘₯πœ“)
 
Theoremexpandral 42303 Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ ↔ πœ“)    β‡’   (βˆ€π‘₯ ∈ 𝐴 πœ‘ ↔ βˆ€π‘₯(π‘₯ ∈ 𝐴 β†’ πœ“))
 
Theoremexpandrexn 42304 Expand a restricted existential quantifier to primitives while contracting a double negation. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ ↔ Β¬ πœ“)    β‡’   (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ Β¬ βˆ€π‘₯(π‘₯ ∈ 𝐴 β†’ πœ“))
 
Theoremexpandrex 42305 Expand a restricted existential quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ ↔ πœ“)    β‡’   (βˆƒπ‘₯ ∈ 𝐴 πœ‘ ↔ Β¬ βˆ€π‘₯(π‘₯ ∈ 𝐴 β†’ Β¬ πœ“))
 
Theoremexpanduniss 42306* Expand βˆͺ 𝐴 βŠ† 𝐡 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(βˆͺ 𝐴 βŠ† 𝐡 ↔ βˆ€π‘₯(π‘₯ ∈ 𝐴 β†’ βˆ€π‘¦(𝑦 ∈ π‘₯ β†’ 𝑦 ∈ 𝐡)))
 
Theoremismnuprim 42307* Express the predicate on π‘ˆ in ismnu 42274 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(βˆ€π‘§ ∈ π‘ˆ (𝒫 𝑧 βŠ† π‘ˆ ∧ βˆ€π‘“βˆƒπ‘€ ∈ π‘ˆ (𝒫 𝑧 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝑧 (βˆƒπ‘£ ∈ π‘ˆ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))) ↔ βˆ€π‘§(𝑧 ∈ π‘ˆ β†’ βˆ€π‘“ Β¬ βˆ€π‘€(𝑀 ∈ π‘ˆ β†’ Β¬ βˆ€π‘£ Β¬ ((βˆ€π‘‘(𝑑 ∈ 𝑣 β†’ 𝑑 ∈ 𝑧) β†’ Β¬ (𝑣 ∈ π‘ˆ β†’ Β¬ 𝑣 ∈ 𝑀)) β†’ Β¬ βˆ€π‘–(𝑖 ∈ 𝑧 β†’ (𝑣 ∈ π‘ˆ β†’ (𝑖 ∈ 𝑣 β†’ (𝑣 ∈ 𝑓 β†’ Β¬ βˆ€π‘’(𝑒 ∈ 𝑓 β†’ (𝑖 ∈ 𝑒 β†’ Β¬ βˆ€π‘œ(π‘œ ∈ 𝑒 β†’ βˆ€π‘ (𝑠 ∈ π‘œ β†’ 𝑠 ∈ 𝑀))))))))))))
 
Theoremrr-grothprimbi 42308* Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim 42313. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(βˆ€π‘₯βˆƒπ‘¦ ∈ Univ π‘₯ ∈ 𝑦 ↔ βˆ€π‘₯ Β¬ βˆ€π‘¦(π‘₯ ∈ 𝑦 β†’ Β¬ βˆ€π‘§(𝑧 ∈ 𝑦 β†’ βˆ€π‘“ Β¬ βˆ€π‘€(𝑀 ∈ 𝑦 β†’ Β¬ βˆ€π‘£ Β¬ ((βˆ€π‘‘(𝑑 ∈ 𝑣 β†’ 𝑑 ∈ 𝑧) β†’ Β¬ (𝑣 ∈ 𝑦 β†’ Β¬ 𝑣 ∈ 𝑀)) β†’ Β¬ βˆ€π‘–(𝑖 ∈ 𝑧 β†’ (𝑣 ∈ 𝑦 β†’ (𝑖 ∈ 𝑣 β†’ (𝑣 ∈ 𝑓 β†’ Β¬ βˆ€π‘’(𝑒 ∈ 𝑓 β†’ (𝑖 ∈ 𝑒 β†’ Β¬ βˆ€π‘œ(π‘œ ∈ 𝑒 β†’ βˆ€π‘ (𝑠 ∈ π‘œ β†’ 𝑠 ∈ 𝑀)))))))))))))
 
Theoreminagrud 42309 Inaccessible levels of the cumulative hierarchy are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(πœ‘ β†’ 𝐼 ∈ Inacc)    β‡’   (πœ‘ β†’ (𝑅1β€˜πΌ) ∈ Univ)
 
Theoreminaex 42310* Assuming the Tarski-Grothendieck axiom, every ordinal is contained in an inaccessible ordinal. (Contributed by Rohan Ridenour, 13-Aug-2023.)
(𝐴 ∈ On β†’ βˆƒπ‘₯ ∈ Inacc 𝐴 ∈ π‘₯)
 
Theoremgruex 42311* Assuming the Tarski-Grothendieck axiom, every set is contained in a Grothendieck universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
βˆƒπ‘¦ ∈ Univ π‘₯ ∈ 𝑦
 
Theoremrr-groth 42312* An equivalent of ax-groth 10693 using only simple defined symbols. (Contributed by Rohan Ridenour, 13-Aug-2023.)
βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑦 ∧ βˆ€π‘“βˆƒπ‘€ ∈ 𝑦 (𝒫 𝑧 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝑧 (βˆƒπ‘£ ∈ 𝑦 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))))
 
Theoremrr-grothprim 42313* An equivalent of ax-groth 10693 using only primitives. This uses only 123 symbols, which is significantly less than the previous record of 163 established by grothprim 10704 (which uses some defined symbols, and requires 229 symbols if expanded to primitives). (Contributed by Rohan Ridenour, 13-Aug-2023.)
Β¬ βˆ€π‘¦(π‘₯ ∈ 𝑦 β†’ Β¬ βˆ€π‘§(𝑧 ∈ 𝑦 β†’ βˆ€π‘“ Β¬ βˆ€π‘€(𝑀 ∈ 𝑦 β†’ Β¬ βˆ€π‘£ Β¬ ((βˆ€π‘‘(𝑑 ∈ 𝑣 β†’ 𝑑 ∈ 𝑧) β†’ Β¬ (𝑣 ∈ 𝑦 β†’ Β¬ 𝑣 ∈ 𝑀)) β†’ Β¬ βˆ€π‘–(𝑖 ∈ 𝑧 β†’ (𝑣 ∈ 𝑦 β†’ (𝑖 ∈ 𝑣 β†’ (𝑣 ∈ 𝑓 β†’ Β¬ βˆ€π‘’(𝑒 ∈ 𝑓 β†’ (𝑖 ∈ 𝑒 β†’ Β¬ βˆ€π‘œ(π‘œ ∈ 𝑒 β†’ βˆ€π‘ (𝑠 ∈ π‘œ β†’ 𝑠 ∈ 𝑀))))))))))))
 
Theoremismnushort 42314* Express the predicate on π‘ˆ and 𝑧 in ismnu 42274 in a shorter form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 10-Oct-2024.)
(βˆ€π‘“ ∈ 𝒫 π‘ˆβˆƒπ‘€ ∈ π‘ˆ (𝒫 𝑧 βŠ† (π‘ˆ ∩ 𝑀) ∧ (𝑧 ∩ βˆͺ 𝑓) βŠ† βˆͺ (𝑓 ∩ 𝒫 𝒫 𝑀)) ↔ (𝒫 𝑧 βŠ† π‘ˆ ∧ βˆ€π‘“βˆƒπ‘€ ∈ π‘ˆ (𝒫 𝑧 βŠ† 𝑀 ∧ βˆ€π‘– ∈ 𝑧 (βˆƒπ‘£ ∈ π‘ˆ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) β†’ βˆƒπ‘’ ∈ 𝑓 (𝑖 ∈ 𝑒 ∧ βˆͺ 𝑒 βŠ† 𝑀)))))
 
Theoremdfuniv2 42315* Alternative definition of Univ using only simple defined symbols. (Contributed by Rohan Ridenour, 10-Oct-2024.)
Univ = {𝑦 ∣ βˆ€π‘§ ∈ 𝑦 βˆ€π‘“ ∈ 𝒫 π‘¦βˆƒπ‘€ ∈ 𝑦 (𝒫 𝑧 βŠ† (𝑦 ∩ 𝑀) ∧ (𝑧 ∩ βˆͺ 𝑓) βŠ† βˆͺ (𝑓 ∩ 𝒫 𝒫 𝑀))}
 
Theoremrr-grothshortbi 42316* Express "every set is contained in a Grothendieck universe" in a short form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 8-Oct-2024.)
(βˆ€π‘₯βˆƒπ‘¦ ∈ Univ π‘₯ ∈ 𝑦 ↔ βˆ€π‘₯βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 βˆ€π‘“ ∈ 𝒫 π‘¦βˆƒπ‘€ ∈ 𝑦 (𝒫 𝑧 βŠ† (𝑦 ∩ 𝑀) ∧ (𝑧 ∩ βˆͺ 𝑓) βŠ† βˆͺ (𝑓 ∩ 𝒫 𝒫 𝑀))))
 
Theoremrr-grothshort 42317* A shorter equivalent of ax-groth 10693 than rr-groth 42312 using a few more simple defined symbols. (Contributed by Rohan Ridenour, 8-Oct-2024.)
βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ βˆ€π‘§ ∈ 𝑦 βˆ€π‘“ ∈ 𝒫 π‘¦βˆƒπ‘€ ∈ 𝑦 (𝒫 𝑧 βŠ† (𝑦 ∩ 𝑀) ∧ (𝑧 ∩ βˆͺ 𝑓) βŠ† βˆͺ (𝑓 ∩ 𝒫 𝒫 𝑀)))
 
21.35  Mathbox for Steve Rodriguez
 
21.35.1  Miscellanea
 
Theoremnanorxor 42318 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
((πœ‘ ⊼ πœ“) ↔ ((πœ‘ ∨ πœ“) ↔ (πœ‘ ⊻ πœ“)))
 
Theoremundisjrab 42319 Union of two disjoint restricted class abstractions; compare unrab 4264. (Contributed by Steve Rodriguez, 28-Feb-2020.)
(({π‘₯ ∈ 𝐴 ∣ πœ‘} ∩ {π‘₯ ∈ 𝐴 ∣ πœ“}) = βˆ… ↔ ({π‘₯ ∈ 𝐴 ∣ πœ‘} βˆͺ {π‘₯ ∈ 𝐴 ∣ πœ“}) = {π‘₯ ∈ 𝐴 ∣ (πœ‘ ⊻ πœ“)})
 
Theoremiso0 42320 The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
βˆ… Isom 𝑅, 𝑆 (βˆ…, βˆ…)
 
Theoremssrecnpr 42321 ℝ is a subset of both ℝ and β„‚. (Contributed by Steve Rodriguez, 22-Nov-2015.)
(𝑆 ∈ {ℝ, β„‚} β†’ ℝ βŠ† 𝑆)
 
Theoremseff 42322 Let set 𝑆 be the real or complex numbers. Then the exponential function restricted to 𝑆 is a mapping from 𝑆 to 𝑆. (Contributed by Steve Rodriguez, 6-Nov-2015.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    β‡’   (πœ‘ β†’ (exp β†Ύ 𝑆):π‘†βŸΆπ‘†)
 
Theoremsblpnf 42323 The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 23672. (Contributed by Steve Rodriguez, 8-Nov-2015.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   π· = ((abs ∘ βˆ’ ) β†Ύ (𝑆 Γ— 𝑆))    β‡’   ((πœ‘ ∧ 𝑃 ∈ 𝑆) β†’ (𝑃(ballβ€˜π·)+∞) = 𝑆)
 
Theoremprmunb2 42324* The primes are unbounded. This generalizes prmunb 16721 to real 𝐴 with arch 12344 and lttrd 11250: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(𝐴 ∈ ℝ β†’ βˆƒπ‘ ∈ β„™ 𝐴 < 𝑝)
 
21.35.2  Ratio test for infinite series convergence and divergence
 
Theoremdvgrat 42325* Ratio test for divergence of a complex infinite series. See e.g. remark "if (absβ€˜((π‘Žβ€˜(𝑛 + 1)) / (π‘Žβ€˜π‘›))) β‰₯ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ π‘Š) β†’ (πΉβ€˜π‘˜) β‰  0)    &   ((πœ‘ ∧ π‘˜ ∈ π‘Š) β†’ (absβ€˜(πΉβ€˜π‘˜)) ≀ (absβ€˜(πΉβ€˜(π‘˜ + 1))))    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) βˆ‰ dom ⇝ )
 
Theoremcvgdvgrat 42326* Ratio test for convergence and divergence of a complex infinite series. If the ratio 𝑅 of the absolute values of successive terms in an infinite sequence 𝐹 converges to less than one, then the infinite sum of the terms of 𝐹 converges to a complex number; and if 𝑅 converges greater then the sum diverges. This combined form of cvgrat 15703 and dvgrat 42325 directly uses the limit of the ratio.

(It also demonstrates how to use climi2 15328 and absltd 15249 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 15249, and how to use r19.29a 3158 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3151 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3151.) (Contributed by Steve Rodriguez, 28-Feb-2020.)

𝑍 = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜π‘)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ π‘Š) β†’ (πΉβ€˜π‘˜) β‰  0)    &   π‘… = (π‘˜ ∈ π‘Š ↦ (absβ€˜((πΉβ€˜(π‘˜ + 1)) / (πΉβ€˜π‘˜))))    &   (πœ‘ β†’ 𝑅 ⇝ 𝐿)    &   (πœ‘ β†’ 𝐿 β‰  1)    β‡’   (πœ‘ β†’ (𝐿 < 1 ↔ seq𝑀( + , 𝐹) ∈ dom ⇝ ))
 
Theoremradcnvrat 42327* Let 𝐿 be the limit, if one exists, of the ratio (absβ€˜((π΄β€˜(π‘˜ + 1)) / (π΄β€˜π‘˜))) (as in the ratio test cvgdvgrat 42326) as π‘˜ increases. Then the radius of convergence of power series Σ𝑛 ∈ β„•0((π΄β€˜π‘›) Β· (π‘₯↑𝑛)) is (1 / 𝐿) if 𝐿 is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence 42326 β€”a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   π· = (π‘˜ ∈ β„•0 ↦ (absβ€˜((π΄β€˜(π‘˜ + 1)) / (π΄β€˜π‘˜))))    &   π‘ = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (π΄β€˜π‘˜) β‰  0)    &   (πœ‘ β†’ 𝐷 ⇝ 𝐿)    &   (πœ‘ β†’ 𝐿 β‰  0)    β‡’   (πœ‘ β†’ 𝑅 = (1 / 𝐿))
 
21.35.3  Multiples
 
Theoremreldvds 42328 The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Rel βˆ₯
 
Theoremnznngen 42329 All positive integers in the set of multiples of n, nβ„€, are the absolute value of n or greater. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (( βˆ₯ β€œ {𝑁}) ∩ β„•) βŠ† (β„€β‰₯β€˜(absβ€˜π‘)))
 
Theoremnzss 42330 The set of multiples of m, mβ„€, is a subset of those of n, nβ„€, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mβ„€ and nβ„€ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑉)    β‡’   (πœ‘ β†’ (( βˆ₯ β€œ {𝑀}) βŠ† ( βˆ₯ β€œ {𝑁}) ↔ 𝑁 βˆ₯ 𝑀))
 
Theoremnzin 42331 The intersection of the set of multiples of m, mβ„€, and those of n, nβ„€, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mβ„€ and nβ„€ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (( βˆ₯ β€œ {𝑀}) ∩ ( βˆ₯ β€œ {𝑁})) = ( βˆ₯ β€œ {(𝑀 lcm 𝑁)}))
 
Theoremnzprmdif 42332 Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑀 ∈ β„™)    &   (πœ‘ β†’ 𝑁 ∈ β„™)    &   (πœ‘ β†’ 𝑀 β‰  𝑁)    β‡’   (πœ‘ β†’ (( βˆ₯ β€œ {𝑀}) βˆ– ( βˆ₯ β€œ {𝑁})) = (( βˆ₯ β€œ {𝑀}) βˆ– ( βˆ₯ β€œ {(𝑀 Β· 𝑁)})))
 
Theoremhashnzfz 42333 Special case of hashdvds 16582: the count of multiples in nβ„€ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜(𝐽 βˆ’ 1)))    β‡’   (πœ‘ β†’ (β™―β€˜(( βˆ₯ β€œ {𝑁}) ∩ (𝐽...𝐾))) = ((βŒŠβ€˜(𝐾 / 𝑁)) βˆ’ (βŒŠβ€˜((𝐽 βˆ’ 1) / 𝑁))))
 
Theoremhashnzfz2 42334 Special case of hashnzfz 42333: the count of multiples in nβ„€, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐾 ∈ β„•)    β‡’   (πœ‘ β†’ (β™―β€˜(( βˆ₯ β€œ {𝑁}) ∩ (2...𝐾))) = (βŒŠβ€˜(𝐾 / 𝑁)))
 
Theoremhashnzfzclim 42335* As the upper bound 𝐾 of the constraint interval (𝐽...𝐾) in hashnzfz 42333 increases, the resulting count of multiples tends to (𝐾 / 𝑀) β€”that is, there are approximately (𝐾 / 𝑀) multiples of 𝑀 in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    β‡’   (πœ‘ β†’ (π‘˜ ∈ (β„€β‰₯β€˜(𝐽 βˆ’ 1)) ↦ ((β™―β€˜(( βˆ₯ β€œ {𝑀}) ∩ (𝐽...π‘˜))) / π‘˜)) ⇝ (1 / 𝑀))
 
21.35.4  Function operations
 
Theoremcaofcan 42336* Transfer a cancellation law like mulcan 11726 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π΄βŸΆπ‘‡)    &   (πœ‘ β†’ 𝐺:π΄βŸΆπ‘†)    &   (πœ‘ β†’ 𝐻:π΄βŸΆπ‘†)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯𝑅𝑦) = (π‘₯𝑅𝑧) ↔ 𝑦 = 𝑧))    β‡’   (πœ‘ β†’ ((𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f 𝑅𝐻) ↔ 𝐺 = 𝐻))
 
Theoremofsubid 42337 Function analogue of subid 11354. (Contributed by Steve Rodriguez, 5-Nov-2015.)
((𝐴 ∈ 𝑉 ∧ 𝐹:π΄βŸΆβ„‚) β†’ (𝐹 ∘f βˆ’ 𝐹) = (𝐴 Γ— {0}))
 
Theoremofmul12 42338 Function analogue of mul12 11254. (Contributed by Steve Rodriguez, 13-Nov-2015.)
(((𝐴 ∈ 𝑉 ∧ 𝐹:π΄βŸΆβ„‚) ∧ (𝐺:π΄βŸΆβ„‚ ∧ 𝐻:π΄βŸΆβ„‚)) β†’ (𝐹 ∘f Β· (𝐺 ∘f Β· 𝐻)) = (𝐺 ∘f Β· (𝐹 ∘f Β· 𝐻)))
 
Theoremofdivrec 42339 Function analogue of divrec 11763, a division analogue of ofnegsub 12085. (Contributed by Steve Rodriguez, 3-Nov-2015.)
((𝐴 ∈ 𝑉 ∧ 𝐹:π΄βŸΆβ„‚ ∧ 𝐺:𝐴⟢(β„‚ βˆ– {0})) β†’ (𝐹 ∘f Β· ((𝐴 Γ— {1}) ∘f / 𝐺)) = (𝐹 ∘f / 𝐺))
 
Theoremofdivcan4 42340 Function analogue of divcan4 11774. (Contributed by Steve Rodriguez, 4-Nov-2015.)
((𝐴 ∈ 𝑉 ∧ 𝐹:π΄βŸΆβ„‚ ∧ 𝐺:𝐴⟢(β„‚ βˆ– {0})) β†’ ((𝐹 ∘f Β· 𝐺) ∘f / 𝐺) = 𝐹)
 
Theoremofdivdiv2 42341 Function analogue of divdiv2 11801. (Contributed by Steve Rodriguez, 23-Nov-2015.)
(((𝐴 ∈ 𝑉 ∧ 𝐹:π΄βŸΆβ„‚) ∧ (𝐺:𝐴⟢(β„‚ βˆ– {0}) ∧ 𝐻:𝐴⟢(β„‚ βˆ– {0}))) β†’ (𝐹 ∘f / (𝐺 ∘f / 𝐻)) = ((𝐹 ∘f Β· 𝐻) ∘f / 𝐺))
 
21.35.5  Calculus
 
Theoremlhe4.4ex1a 42342 Example of the Fundamental Theorem of Calculus, part two (ftc2 25330): ∫(1(,)2)((π‘₯↑2) βˆ’ 3) dπ‘₯ = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 25330 as simply the "Fundamental Theorem of Calculus", then ftc1 25328 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
∫(1(,)2)((π‘₯↑2) βˆ’ 3) dπ‘₯ = -(2 / 3)
 
Theoremdvsconst 42343 Derivative of a constant function on the real or complex numbers. The function may return a complex 𝐴 even if 𝑆 is ℝ. (Contributed by Steve Rodriguez, 11-Nov-2015.)
((𝑆 ∈ {ℝ, β„‚} ∧ 𝐴 ∈ β„‚) β†’ (𝑆 D (𝑆 Γ— {𝐴})) = (𝑆 Γ— {0}))
 
Theoremdvsid 42344 Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.)
(𝑆 ∈ {ℝ, β„‚} β†’ (𝑆 D ( I β†Ύ 𝑆)) = (𝑆 Γ— {1}))
 
Theoremdvsef 42345 Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.)
(𝑆 ∈ {ℝ, β„‚} β†’ (𝑆 D (exp β†Ύ 𝑆)) = (exp β†Ύ 𝑆))
 
Theoremexpgrowthi 42346* Exponential growth and decay model. See expgrowth 42348 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐾 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   π‘Œ = (𝑑 ∈ 𝑆 ↦ (𝐢 Β· (expβ€˜(𝐾 Β· 𝑑))))    β‡’   (πœ‘ β†’ (𝑆 D π‘Œ) = ((𝑆 Γ— {𝐾}) ∘f Β· π‘Œ))
 
Theoremdvconstbi 42347* The derivative of a function on 𝑆 is zero iff it is a constant function. Roughly a biconditional 𝑆 analogue of dvconst 25203 and dveq0 25286. Corresponds to integration formula "∫0 dπ‘₯ = 𝐢 " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ π‘Œ:π‘†βŸΆβ„‚)    &   (πœ‘ β†’ dom (𝑆 D π‘Œ) = 𝑆)    β‡’   (πœ‘ β†’ ((𝑆 D π‘Œ) = (𝑆 Γ— {0}) ↔ βˆƒπ‘ ∈ β„‚ π‘Œ = (𝑆 Γ— {𝑐})))
 
Theoremexpgrowth 42348* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 42346 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model 42346); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant Ξ».

Here y' is given as (𝑆 D π‘Œ), C as 𝑐, and ky as ((𝑆 Γ— {𝐾}) ∘f Β· π‘Œ). (𝑆 Γ— {𝐾}) is the constant function that maps any real or complex input to k and ∘f Β· is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf 42346 pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case 42346.

Statements for this and expgrowthi 42346 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

(πœ‘ β†’ 𝑆 ∈ {ℝ, β„‚})    &   (πœ‘ β†’ 𝐾 ∈ β„‚)    &   (πœ‘ β†’ π‘Œ:π‘†βŸΆβ„‚)    &   (πœ‘ β†’ dom (𝑆 D π‘Œ) = 𝑆)    β‡’   (πœ‘ β†’ ((𝑆 D π‘Œ) = ((𝑆 Γ— {𝐾}) ∘f Β· π‘Œ) ↔ βˆƒπ‘ ∈ β„‚ π‘Œ = (𝑑 ∈ 𝑆 ↦ (𝑐 Β· (expβ€˜(𝐾 Β· 𝑑))))))
 
21.35.6  The generalized binomial coefficient operation
 
Syntaxcbcc 42349 Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
 
Definitiondf-bcc 42350* Define a generalized binomial coefficient operation, which unlike df-bc 14131 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
C𝑐 = (𝑐 ∈ β„‚, π‘˜ ∈ β„•0 ↦ ((𝑐 FallFac π‘˜) / (!β€˜π‘˜)))
 
Theorembccval 42351 Value of the generalized binomial coefficient, 𝐢 choose 𝐾. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐢C𝑐𝐾) = ((𝐢 FallFac 𝐾) / (!β€˜πΎ)))
 
Theorembcccl 42352 Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐢C𝑐𝐾) ∈ β„‚)
 
Theorembcc0 42353 The generalized binomial coefficient 𝐢 choose 𝐾 is zero iff 𝐢 is an integer between zero and (𝐾 βˆ’ 1) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   (πœ‘ β†’ ((𝐢C𝑐𝐾) = 0 ↔ 𝐢 ∈ (0...(𝐾 βˆ’ 1))))
 
Theorembccp1k 42354 Generalized binomial coefficient: 𝐢 choose (𝐾 + 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐢C𝑐(𝐾 + 1)) = ((𝐢C𝑐𝐾) Β· ((𝐢 βˆ’ 𝐾) / (𝐾 + 1))))
 
Theorembccm1k 42355 Generalized binomial coefficient: 𝐢 choose (𝐾 βˆ’ 1), when 𝐢 is not (𝐾 βˆ’ 1). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ (β„‚ βˆ– {(𝐾 βˆ’ 1)}))    &   (πœ‘ β†’ 𝐾 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐢C𝑐(𝐾 βˆ’ 1)) = ((𝐢C𝑐𝐾) / ((𝐢 βˆ’ (𝐾 βˆ’ 1)) / 𝐾)))
 
Theorembccn0 42356 Generalized binomial coefficient: 𝐢 choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐢C𝑐0) = 1)
 
Theorembccn1 42357 Generalized binomial coefficient: 𝐢 choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (𝐢C𝑐1) = 𝐢)
 
Theorembccbc 42358 The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐾 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝑁C𝑐𝐾) = (𝑁C𝐾))
 
21.35.7  Binomial series
 
Theoremuzmptshftfval 42359* When 𝐹 is a maps-to function on some set of upper integers 𝑍 that returns a set 𝐡, (𝐹 shift 𝑁) is another maps-to function on the shifted set of upper integers π‘Š. (Contributed by Steve Rodriguez, 22-Apr-2020.)
𝐹 = (π‘₯ ∈ 𝑍 ↦ 𝐡)    &   π΅ ∈ V    &   (π‘₯ = (𝑦 βˆ’ 𝑁) β†’ 𝐡 = 𝐢)    &   π‘ = (β„€β‰₯β€˜π‘€)    &   π‘Š = (β„€β‰₯β€˜(𝑀 + 𝑁))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝐹 shift 𝑁) = (𝑦 ∈ π‘Š ↦ 𝐢))
 
Theoremdvradcnv2 42360* The radius of convergence of the (formal) derivative 𝐻 of the power series 𝐺 is (at least) as large as the radius of convergence of 𝐺. This version of dvradcnv 25702 uses a shifted version of 𝐻 to match the sum form of (β„‚ D 𝐹) in pserdv2 25711 (and shows how to use uzmptshftfval 42359 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
𝐺 = (π‘₯ ∈ β„‚ ↦ (𝑛 ∈ β„•0 ↦ ((π΄β€˜π‘›) Β· (π‘₯↑𝑛))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (πΊβ€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   π» = (𝑛 ∈ β„• ↦ ((𝑛 Β· (π΄β€˜π‘›)) Β· (𝑋↑(𝑛 βˆ’ 1))))    &   (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π‘‹) < 𝑅)    β‡’   (πœ‘ β†’ seq1( + , 𝐻) ∈ dom ⇝ )
 
Theorembinomcxplemwb 42361 Lemma for binomcxp 42370. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•)    β‡’   (πœ‘ β†’ (((𝐢 βˆ’ 𝐾) Β· (𝐢C𝑐𝐾)) + ((𝐢 βˆ’ (𝐾 βˆ’ 1)) Β· (𝐢C𝑐(𝐾 βˆ’ 1)))) = (𝐢 Β· (𝐢C𝑐𝐾)))
 
Theorembinomcxplemnn0 42362* Lemma for binomcxp 42370. When 𝐢 is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15650 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...𝐢), and when the index set is widened beyond 𝐢 the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   ((πœ‘ ∧ 𝐢 ∈ β„•0) β†’ ((𝐴 + 𝐡)↑𝑐𝐢) = Ξ£π‘˜ ∈ β„•0 ((𝐢Cπ‘π‘˜) Β· ((𝐴↑𝑐(𝐢 βˆ’ π‘˜)) Β· (π΅β†‘π‘˜))))
 
Theorembinomcxplemrat 42363* Lemma for binomcxp 42370. As π‘˜ increases, this ratio's absolute value converges to one. Part of equation "Since continuity of the absolute value..." in the Wikibooks proof (proven for the inverse ratio, which we later show is no problem). (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ (π‘˜ ∈ β„•0 ↦ (absβ€˜((𝐢 βˆ’ π‘˜) / (π‘˜ + 1)))) ⇝ 1)
 
Theorembinomcxplemfrat 42364* Lemma for binomcxp 42370. binomcxplemrat 42363 implies that when 𝐢 is not a nonnegative integer, the absolute value of the ratio ((πΉβ€˜(π‘˜ + 1)) / (πΉβ€˜π‘˜)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    β‡’   ((πœ‘ ∧ Β¬ 𝐢 ∈ β„•0) β†’ (π‘˜ ∈ β„•0 ↦ (absβ€˜((πΉβ€˜(π‘˜ + 1)) / (πΉβ€˜π‘˜)))) ⇝ 1)
 
Theorembinomcxplemradcnv 42365* Lemma for binomcxp 42370. By binomcxplemfrat 42364 and radcnvrat 42327 the radius of convergence of power series Ξ£π‘˜ ∈ β„•0((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    &   π‘† = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„•0 ↦ ((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (π‘†β€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    β‡’   ((πœ‘ ∧ Β¬ 𝐢 ∈ β„•0) β†’ 𝑅 = 1)
 
Theorembinomcxplemdvbinom 42366* Lemma for binomcxp 42370. By the power and chain rules, calculate the derivative of ((1 + 𝑏)↑𝑐-𝐢), with respect to 𝑏 in the disk of convergence 𝐷. We later multiply the derivative in the later binomcxplemdvsum 42368 by this derivative to show that ((1 + 𝑏)↑𝑐𝐢) (with a nonnegated 𝐢) and the later sum, since both at 𝑏 = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    &   π‘† = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„•0 ↦ ((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (π‘†β€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   πΈ = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„• ↦ ((π‘˜ Β· (πΉβ€˜π‘˜)) Β· (𝑏↑(π‘˜ βˆ’ 1)))))    &   π· = (β—‘abs β€œ (0[,)𝑅))    β‡’   ((πœ‘ ∧ Β¬ 𝐢 ∈ β„•0) β†’ (β„‚ D (𝑏 ∈ 𝐷 ↦ ((1 + 𝑏)↑𝑐-𝐢))) = (𝑏 ∈ 𝐷 ↦ (-𝐢 Β· ((1 + 𝑏)↑𝑐(-𝐢 βˆ’ 1)))))
 
Theorembinomcxplemcvg 42367* Lemma for binomcxp 42370. The sum in binomcxplemnn0 42362 and its derivative (see the next theorem, binomcxplemdvsum 42368) converge, as long as their base 𝐽 is within the disk of convergence. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    &   π‘† = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„•0 ↦ ((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (π‘†β€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   πΈ = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„• ↦ ((π‘˜ Β· (πΉβ€˜π‘˜)) Β· (𝑏↑(π‘˜ βˆ’ 1)))))    &   π· = (β—‘abs β€œ (0[,)𝑅))    β‡’   ((πœ‘ ∧ 𝐽 ∈ 𝐷) β†’ (seq0( + , (π‘†β€˜π½)) ∈ dom ⇝ ∧ seq1( + , (πΈβ€˜π½)) ∈ dom ⇝ ))
 
Theorembinomcxplemdvsum 42368* Lemma for binomcxp 42370. The derivative of the generalized sum in binomcxplemnn0 42362. Part of remark "This convergence allows to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    &   π‘† = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„•0 ↦ ((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (π‘†β€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   πΈ = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„• ↦ ((π‘˜ Β· (πΉβ€˜π‘˜)) Β· (𝑏↑(π‘˜ βˆ’ 1)))))    &   π· = (β—‘abs β€œ (0[,)𝑅))    &   π‘ƒ = (𝑏 ∈ 𝐷 ↦ Ξ£π‘˜ ∈ β„•0 ((π‘†β€˜π‘)β€˜π‘˜))    β‡’   (πœ‘ β†’ (β„‚ D 𝑃) = (𝑏 ∈ 𝐷 ↦ Ξ£π‘˜ ∈ β„• ((πΈβ€˜π‘)β€˜π‘˜)))
 
Theorembinomcxplemnotnn0 42369* Lemma for binomcxp 42370. When 𝐢 is not a nonnegative integer, the generalized sum in binomcxplemnn0 42362 β€”which we will call 𝑃 β€”is a convergent power series: its base 𝑏 is always of smaller absolute value than the radius of convergence.

pserdv2 25711 gives the derivative of 𝑃, which by dvradcnv 25702 also converges in that radius. When 𝐴 is fixed at one, (𝐴 + 𝑏) times that derivative equals (𝐢 Β· 𝑃) and fraction (𝑃 / ((𝐴 + 𝑏)↑𝑐𝐢)) is always defined with derivative zero, so the fraction is a constantβ€”specifically one, because ((1 + 0)↑𝑐𝐢) = 1. Thus ((1 + 𝑏)↑𝑐𝐢) = (π‘ƒβ€˜π‘).

Finally, let 𝑏 be (𝐡 / 𝐴), and multiply both the binomial ((1 + (𝐡 / 𝐴))↑𝑐𝐢) and the sum (π‘ƒβ€˜(𝐡 / 𝐴)) by (𝐴↑𝑐𝐢) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)

(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   πΉ = (𝑗 ∈ β„•0 ↦ (𝐢C𝑐𝑗))    &   π‘† = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„•0 ↦ ((πΉβ€˜π‘˜) Β· (π‘β†‘π‘˜))))    &   π‘… = sup({π‘Ÿ ∈ ℝ ∣ seq0( + , (π‘†β€˜π‘Ÿ)) ∈ dom ⇝ }, ℝ*, < )    &   πΈ = (𝑏 ∈ β„‚ ↦ (π‘˜ ∈ β„• ↦ ((π‘˜ Β· (πΉβ€˜π‘˜)) Β· (𝑏↑(π‘˜ βˆ’ 1)))))    &   π· = (β—‘abs β€œ (0[,)𝑅))    &   π‘ƒ = (𝑏 ∈ 𝐷 ↦ Ξ£π‘˜ ∈ β„•0 ((π‘†β€˜π‘)β€˜π‘˜))    β‡’   ((πœ‘ ∧ Β¬ 𝐢 ∈ β„•0) β†’ ((𝐴 + 𝐡)↑𝑐𝐢) = Ξ£π‘˜ ∈ β„•0 ((𝐢Cπ‘π‘˜) Β· ((𝐴↑𝑐(𝐢 βˆ’ π‘˜)) Β· (π΅β†‘π‘˜))))
 
Theorembinomcxp 42370* Generalize the binomial theorem binom 15650 to positive real summand 𝐴, real summand 𝐡, and complex exponent 𝐢. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15650; see also https://en.wikipedia.org/wiki/Binomial_series 15650, https://en.wikipedia.org/wiki/Binomial_theorem 15650 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15650. (Contributed by Steve Rodriguez, 22-Apr-2020.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜π΅) < (absβ€˜π΄))    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐡)↑𝑐𝐢) = Ξ£π‘˜ ∈ β„•0 ((𝐢Cπ‘π‘˜) Β· ((𝐴↑𝑐(𝐢 βˆ’ π‘˜)) Β· (π΅β†‘π‘˜))))
 
21.36  Mathbox for Andrew Salmon
 
21.36.1  Principia Mathematica * 10
 
Theorempm10.12 42371* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
(βˆ€π‘₯(πœ‘ ∨ πœ“) β†’ (πœ‘ ∨ βˆ€π‘₯πœ“))
 
Theorempm10.14 42372 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
((βˆ€π‘₯πœ‘ ∧ βˆ€π‘₯πœ“) β†’ ([𝑦 / π‘₯]πœ‘ ∧ [𝑦 / π‘₯]πœ“))
 
Theorempm10.251 42373 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(βˆ€π‘₯ Β¬ πœ‘ β†’ Β¬ βˆ€π‘₯πœ‘)
 
Theorempm10.252 42374 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
(Β¬ βˆƒπ‘₯πœ‘ ↔ βˆ€π‘₯ Β¬ πœ‘)
 
Theorempm10.253 42375 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
(Β¬ βˆ€π‘₯πœ‘ ↔ βˆƒπ‘₯ Β¬ πœ‘)
 
Theoremalbitr 42376 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
((βˆ€π‘₯(πœ‘ ↔ πœ“) ∧ βˆ€π‘₯(πœ“ ↔ πœ’)) β†’ βˆ€π‘₯(πœ‘ ↔ πœ’))
 
Theorempm10.42 42377 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
((βˆƒπ‘₯πœ‘ ∨ βˆƒπ‘₯πœ“) ↔ βˆƒπ‘₯(πœ‘ ∨ πœ“))
 
Theorempm10.52 42378* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆƒπ‘₯πœ‘ β†’ (βˆ€π‘₯(πœ‘ β†’ πœ“) ↔ πœ“))
 
Theorempm10.53 42379 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(Β¬ βˆƒπ‘₯πœ‘ β†’ βˆ€π‘₯(πœ‘ β†’ πœ“))
 
Theorempm10.541 42380* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯(πœ‘ β†’ (πœ’ ∨ πœ“)) ↔ (πœ’ ∨ βˆ€π‘₯(πœ‘ β†’ πœ“)))
 
Theorempm10.542 42381* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯(πœ‘ β†’ (πœ’ β†’ πœ“)) ↔ (πœ’ β†’ βˆ€π‘₯(πœ‘ β†’ πœ“)))
 
Theorempm10.55 42382 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((βˆƒπ‘₯(πœ‘ ∧ πœ“) ∧ βˆ€π‘₯(πœ‘ β†’ πœ“)) ↔ (βˆƒπ‘₯πœ‘ ∧ βˆ€π‘₯(πœ‘ β†’ πœ“)))
 
Theorempm10.56 42383 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
((βˆ€π‘₯(πœ‘ β†’ πœ“) ∧ βˆƒπ‘₯(πœ‘ ∧ πœ’)) β†’ βˆƒπ‘₯(πœ“ ∧ πœ’))
 
Theorempm10.57 42384 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯(πœ‘ β†’ (πœ“ ∨ πœ’)) β†’ (βˆ€π‘₯(πœ‘ β†’ πœ“) ∨ βˆƒπ‘₯(πœ‘ ∧ πœ’)))
 
21.36.2  Principia Mathematica * 11
 
Theorem2alanimi 42385 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
((πœ‘ ∧ πœ“) β†’ πœ’)    β‡’   ((βˆ€π‘₯βˆ€π‘¦πœ‘ ∧ βˆ€π‘₯βˆ€π‘¦πœ“) β†’ βˆ€π‘₯βˆ€π‘¦πœ’)
 
Theorem2al2imi 42386 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    β‡’   (βˆ€π‘₯βˆ€π‘¦πœ‘ β†’ (βˆ€π‘₯βˆ€π‘¦πœ“ β†’ βˆ€π‘₯βˆ€π‘¦πœ’))
 
Theorempm11.11 42387 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
πœ‘    β‡’   βˆ€π‘§βˆ€π‘€[𝑧 / π‘₯][𝑀 / 𝑦]πœ‘
 
Theorempm11.12 42388* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ ∨ πœ“) β†’ (πœ‘ ∨ βˆ€π‘₯βˆ€π‘¦πœ“))
 
Theorem19.21vv 42389* Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1943. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ“ β†’ πœ‘) ↔ (πœ“ β†’ βˆ€π‘₯βˆ€π‘¦πœ‘))
 
Theorem2alim 42390 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ β†’ πœ“) β†’ (βˆ€π‘₯βˆ€π‘¦πœ‘ β†’ βˆ€π‘₯βˆ€π‘¦πœ“))
 
Theorem2albi 42391 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ ↔ πœ“) β†’ (βˆ€π‘₯βˆ€π‘¦πœ‘ ↔ βˆ€π‘₯βˆ€π‘¦πœ“))
 
Theorem2exim 42392 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ β†’ πœ“) β†’ (βˆƒπ‘₯βˆƒπ‘¦πœ‘ β†’ βˆƒπ‘₯βˆƒπ‘¦πœ“))
 
Theorem2exbi 42393 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ ↔ πœ“) β†’ (βˆƒπ‘₯βˆƒπ‘¦πœ‘ ↔ βˆƒπ‘₯βˆƒπ‘¦πœ“))
 
Theoremspsbce-2 42394 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
([𝑧 / π‘₯][𝑀 / 𝑦]πœ‘ β†’ βˆƒπ‘₯βˆƒπ‘¦πœ‘)
 
Theorem19.33-2 42395 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
((βˆ€π‘₯βˆ€π‘¦πœ‘ ∨ βˆ€π‘₯βˆ€π‘¦πœ“) β†’ βˆ€π‘₯βˆ€π‘¦(πœ‘ ∨ πœ“))
 
Theorem19.36vv 42396* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
(βˆƒπ‘₯βˆƒπ‘¦(πœ‘ β†’ πœ“) ↔ (βˆ€π‘₯βˆ€π‘¦πœ‘ β†’ πœ“))
 
Theorem19.31vv 42397* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ‘ ∨ πœ“) ↔ (βˆ€π‘₯βˆ€π‘¦πœ‘ ∨ πœ“))
 
Theorem19.37vv 42398* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆƒπ‘₯βˆƒπ‘¦(πœ“ β†’ πœ‘) ↔ (πœ“ β†’ βˆƒπ‘₯βˆƒπ‘¦πœ‘))
 
Theorem19.28vv 42399* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆ€π‘₯βˆ€π‘¦(πœ“ ∧ πœ‘) ↔ (πœ“ ∧ βˆ€π‘₯βˆ€π‘¦πœ‘))
 
Theorempm11.52 42400 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(βˆƒπ‘₯βˆƒπ‘¦(πœ‘ ∧ πœ“) ↔ Β¬ βˆ€π‘₯βˆ€π‘¦(πœ‘ β†’ Β¬ πœ“))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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