![]() |
Metamath
Proof Explorer Theorem List (p. 439 of 484) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30767) |
![]() (30768-32290) |
![]() (32291-48346) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rr-grothshortbi 43801* | Express "every set is contained in a Grothendieck universe" in a short form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 8-Oct-2024.) |
β’ (βπ₯βπ¦ β Univ π₯ β π¦ β βπ₯βπ¦(π₯ β π¦ β§ βπ§ β π¦ βπ β π« π¦βπ€ β π¦ (π« π§ β (π¦ β© π€) β§ (π§ β© βͺ π) β βͺ (π β© π« π« π€)))) | ||
Theorem | rr-grothshort 43802* | A shorter equivalent of ax-groth 10841 than rr-groth 43797 using a few more simple defined symbols. (Contributed by Rohan Ridenour, 8-Oct-2024.) |
β’ βπ¦(π₯ β π¦ β§ βπ§ β π¦ βπ β π« π¦βπ€ β π¦ (π« π§ β (π¦ β© π€) β§ (π§ β© βͺ π) β βͺ (π β© π« π« π€))) | ||
Theorem | nanorxor 43803 | 'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.) |
β’ ((π βΌ π) β ((π β¨ π) β (π β» π))) | ||
Theorem | undisjrab 43804 | Union of two disjoint restricted class abstractions; compare unrab 4301. (Contributed by Steve Rodriguez, 28-Feb-2020.) |
β’ (({π₯ β π΄ β£ π} β© {π₯ β π΄ β£ π}) = β β ({π₯ β π΄ β£ π} βͺ {π₯ β π΄ β£ π}) = {π₯ β π΄ β£ (π β» π)}) | ||
Theorem | iso0 43805 | The empty set is an π , π isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
β’ β Isom π , π (β , β ) | ||
Theorem | ssrecnpr 43806 | β is a subset of both β and β. (Contributed by Steve Rodriguez, 22-Nov-2015.) |
β’ (π β {β, β} β β β π) | ||
Theorem | seff 43807 | Let set π be the real or complex numbers. Then the exponential function restricted to π is a mapping from π to π. (Contributed by Steve Rodriguez, 6-Nov-2015.) |
β’ (π β π β {β, β}) β β’ (π β (exp βΎ π):πβΆπ) | ||
Theorem | sblpnf 43808 | The infinity ball in the absolute value metric is just the whole space. π analogue of blpnf 24316. (Contributed by Steve Rodriguez, 8-Nov-2015.) |
β’ (π β π β {β, β}) & β’ π· = ((abs β β ) βΎ (π Γ π)) β β’ ((π β§ π β π) β (π(ballβπ·)+β) = π) | ||
Theorem | prmunb2 43809* | The primes are unbounded. This generalizes prmunb 16877 to real π΄ with arch 12494 and lttrd 11400: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ (π΄ β β β βπ β β π΄ < π) | ||
Theorem | dvgrat 43810* | Ratio test for divergence of a complex infinite series. See e.g. remark "if (absβ((πβ(π + 1)) / (πβπ))) β₯ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.) |
β’ π = (β€β₯βπ) & β’ π = (β€β₯βπ) & β’ (π β π β π) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΉβπ) β 0) & β’ ((π β§ π β π) β (absβ(πΉβπ)) β€ (absβ(πΉβ(π + 1)))) β β’ (π β seqπ( + , πΉ) β dom β ) | ||
Theorem | cvgdvgrat 43811* |
Ratio test for convergence and divergence of a complex infinite series.
If the ratio π
of the absolute values of successive
terms in an
infinite sequence πΉ converges to less than one, then the
infinite
sum of the terms of πΉ converges to a complex number; and
if π
converges greater then the sum diverges. This combined form of
cvgrat 15856 and dvgrat 43810 directly uses the limit of the ratio.
(It also demonstrates how to use climi2 15482 and absltd 15403 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191 15403, and how to use r19.29a 3152 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3145 at https://groups.google.com/g/metamath/c/2RPikOiXLMo 3145.) (Contributed by Steve Rodriguez, 28-Feb-2020.) |
β’ π = (β€β₯βπ) & β’ π = (β€β₯βπ) & β’ (π β π β π) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΉβπ) β 0) & β’ π = (π β π β¦ (absβ((πΉβ(π + 1)) / (πΉβπ)))) & β’ (π β π β πΏ) & β’ (π β πΏ β 1) β β’ (π β (πΏ < 1 β seqπ( + , πΉ) β dom β )) | ||
Theorem | radcnvrat 43812* | Let πΏ be the limit, if one exists, of the ratio (absβ((π΄β(π + 1)) / (π΄βπ))) (as in the ratio test cvgdvgrat 43811) as π increases. Then the radius of convergence of power series Ξ£π β β0((π΄βπ) Β· (π₯βπ)) is (1 / πΏ) if πΏ is nonzero. Proof "The limit involved in the ratio test..." in https://en.wikipedia.org/wiki/Radius_of_convergence 43811 βa few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.) |
β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) & β’ (π β π΄:β0βΆβ) & β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) & β’ π· = (π β β0 β¦ (absβ((π΄β(π + 1)) / (π΄βπ)))) & β’ π = (β€β₯βπ) & β’ (π β π β β0) & β’ ((π β§ π β π) β (π΄βπ) β 0) & β’ (π β π· β πΏ) & β’ (π β πΏ β 0) β β’ (π β π = (1 / πΏ)) | ||
Theorem | reldvds 43813 | The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ Rel β₯ | ||
Theorem | nznngen 43814 | 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.) |
β’ (π β π β β€) β β’ (π β (( β₯ β {π}) β© β) β (β€β₯β(absβπ))) | ||
Theorem | nzss 43815 | 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.) |
β’ (π β π β β€) & β’ (π β π β π) β β’ (π β (( β₯ β {π}) β ( β₯ β {π}) β π β₯ π)) | ||
Theorem | nzin 43816 | 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.) |
β’ (π β π β β€) & β’ (π β π β β€) β β’ (π β (( β₯ β {π}) β© ( β₯ β {π})) = ( β₯ β {(π lcm π)})) | ||
Theorem | nzprmdif 43817 | Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ (π β π β β) & β’ (π β π β β) & β’ (π β π β π) β β’ (π β (( β₯ β {π}) β ( β₯ β {π})) = (( β₯ β {π}) β ( β₯ β {(π Β· π)}))) | ||
Theorem | hashnzfz 43818 | Special case of hashdvds 16738: the count of multiples in nβ€ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ (π β π β β) & β’ (π β π½ β β€) & β’ (π β πΎ β (β€β₯β(π½ β 1))) β β’ (π β (β―β(( β₯ β {π}) β© (π½...πΎ))) = ((ββ(πΎ / π)) β (ββ((π½ β 1) / π)))) | ||
Theorem | hashnzfz2 43819 | Special case of hashnzfz 43818: the count of multiples in nβ€, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ (π β π β (β€β₯β2)) & β’ (π β πΎ β β) β β’ (π β (β―β(( β₯ β {π}) β© (2...πΎ))) = (ββ(πΎ / π))) | ||
Theorem | hashnzfzclim 43820* | As the upper bound πΎ of the constraint interval (π½...πΎ) in hashnzfz 43818 increases, the resulting count of multiples tends to (πΎ / π) βthat is, there are approximately (πΎ / π) multiples of π in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
β’ (π β π β β) & β’ (π β π½ β β€) β β’ (π β (π β (β€β₯β(π½ β 1)) β¦ ((β―β(( β₯ β {π}) β© (π½...π))) / π)) β (1 / π)) | ||
Theorem | caofcan 43821* | Transfer a cancellation law like mulcan 11876 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.) |
β’ (π β π΄ β π) & β’ (π β πΉ:π΄βΆπ) & β’ (π β πΊ:π΄βΆπ) & β’ (π β π»:π΄βΆπ) & β’ ((π β§ (π₯ β π β§ π¦ β π β§ π§ β π)) β ((π₯π π¦) = (π₯π π§) β π¦ = π§)) β β’ (π β ((πΉ βf π πΊ) = (πΉ βf π π») β πΊ = π»)) | ||
Theorem | ofsubid 43822 | Function analogue of subid 11504. (Contributed by Steve Rodriguez, 5-Nov-2015.) |
β’ ((π΄ β π β§ πΉ:π΄βΆβ) β (πΉ βf β πΉ) = (π΄ Γ {0})) | ||
Theorem | ofmul12 43823 | Function analogue of mul12 11404. (Contributed by Steve Rodriguez, 13-Nov-2015.) |
β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ (πΊ:π΄βΆβ β§ π»:π΄βΆβ)) β (πΉ βf Β· (πΊ βf Β· π»)) = (πΊ βf Β· (πΉ βf Β· π»))) | ||
Theorem | ofdivrec 43824 | Function analogue of divrec 11913, a division analogue of ofnegsub 12235. (Contributed by Steve Rodriguez, 3-Nov-2015.) |
β’ ((π΄ β π β§ πΉ:π΄βΆβ β§ πΊ:π΄βΆ(β β {0})) β (πΉ βf Β· ((π΄ Γ {1}) βf / πΊ)) = (πΉ βf / πΊ)) | ||
Theorem | ofdivcan4 43825 | Function analogue of divcan4 11924. (Contributed by Steve Rodriguez, 4-Nov-2015.) |
β’ ((π΄ β π β§ πΉ:π΄βΆβ β§ πΊ:π΄βΆ(β β {0})) β ((πΉ βf Β· πΊ) βf / πΊ) = πΉ) | ||
Theorem | ofdivdiv2 43826 | Function analogue of divdiv2 11951. (Contributed by Steve Rodriguez, 23-Nov-2015.) |
β’ (((π΄ β π β§ πΉ:π΄βΆβ) β§ (πΊ:π΄βΆ(β β {0}) β§ π»:π΄βΆ(β β {0}))) β (πΉ βf / (πΊ βf / π»)) = ((πΉ βf Β· π») βf / πΊ)) | ||
Theorem | lhe4.4ex1a 43827 | Example of the Fundamental Theorem of Calculus, part two (ftc2 25992): β«(1(,)2)((π₯β2) β 3) dπ₯ = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 25992 as simply the "Fundamental Theorem of Calculus", then ftc1 25990 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.) |
β’ β«(1(,)2)((π₯β2) β 3) dπ₯ = -(2 / 3) | ||
Theorem | dvsconst 43828 | Derivative of a constant function on the real or complex numbers. The function may return a complex π΄ even if π is β. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
β’ ((π β {β, β} β§ π΄ β β) β (π D (π Γ {π΄})) = (π Γ {0})) | ||
Theorem | dvsid 43829 | Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
β’ (π β {β, β} β (π D ( I βΎ π)) = (π Γ {1})) | ||
Theorem | dvsef 43830 | Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.) |
β’ (π β {β, β} β (π D (exp βΎ π)) = (exp βΎ π)) | ||
Theorem | expgrowthi 43831* | Exponential growth and decay model. See expgrowth 43833 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.) |
β’ (π β π β {β, β}) & β’ (π β πΎ β β) & β’ (π β πΆ β β) & β’ π = (π‘ β π β¦ (πΆ Β· (expβ(πΎ Β· π‘)))) β β’ (π β (π D π) = ((π Γ {πΎ}) βf Β· π)) | ||
Theorem | dvconstbi 43832* | The derivative of a function on π is zero iff it is a constant function. Roughly a biconditional π analogue of dvconst 25859 and dveq0 25946. Corresponds to integration formula "β«0 dπ₯ = πΆ " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
β’ (π β π β {β, β}) & β’ (π β π:πβΆβ) & β’ (π β dom (π D π) = π) β β’ (π β ((π D π) = (π Γ {0}) β βπ β β π = (π Γ {π}))) | ||
Theorem | expgrowth 43833* |
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 43831 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 43831); 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 (π D π), C as π, and ky as ((π Γ {πΎ}) βf Β· π). (π Γ {πΎ}) is the constant function that maps any real or complex input to k and βf 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 43831 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 43831. Statements for this and expgrowthi 43831 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.) |
β’ (π β π β {β, β}) & β’ (π β πΎ β β) & β’ (π β π:πβΆβ) & β’ (π β dom (π D π) = π) β β’ (π β ((π D π) = ((π Γ {πΎ}) βf Β· π) β βπ β β π = (π‘ β π β¦ (π Β· (expβ(πΎ Β· π‘)))))) | ||
Syntax | cbcc 43834 | Extend class notation to include the generalized binomial coefficient operation. |
class Cπ | ||
Definition | df-bcc 43835* | Define a generalized binomial coefficient operation, which unlike df-bc 14289 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ Cπ = (π β β, π β β0 β¦ ((π FallFac π) / (!βπ))) | ||
Theorem | bccval 43836 | Value of the generalized binomial coefficient, πΆ choose πΎ. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β0) β β’ (π β (πΆCππΎ) = ((πΆ FallFac πΎ) / (!βπΎ))) | ||
Theorem | bcccl 43837 | Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β0) β β’ (π β (πΆCππΎ) β β) | ||
Theorem | bcc0 43838 | The generalized binomial coefficient πΆ choose πΎ is zero iff πΆ is an integer between zero and (πΎ β 1) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β0) β β’ (π β ((πΆCππΎ) = 0 β πΆ β (0...(πΎ β 1)))) | ||
Theorem | bccp1k 43839 | Generalized binomial coefficient: πΆ choose (πΎ + 1). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β0) β β’ (π β (πΆCπ(πΎ + 1)) = ((πΆCππΎ) Β· ((πΆ β πΎ) / (πΎ + 1)))) | ||
Theorem | bccm1k 43840 | Generalized binomial coefficient: πΆ choose (πΎ β 1), when πΆ is not (πΎ β 1). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β (β β {(πΎ β 1)})) & β’ (π β πΎ β β) β β’ (π β (πΆCπ(πΎ β 1)) = ((πΆCππΎ) / ((πΆ β (πΎ β 1)) / πΎ))) | ||
Theorem | bccn0 43841 | Generalized binomial coefficient: πΆ choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) β β’ (π β (πΆCπ0) = 1) | ||
Theorem | bccn1 43842 | Generalized binomial coefficient: πΆ choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) β β’ (π β (πΆCπ1) = πΆ) | ||
Theorem | bccbc 43843 | The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π β β0) & β’ (π β πΎ β β0) β β’ (π β (πCππΎ) = (πCπΎ)) | ||
Theorem | uzmptshftfval 43844* | When πΉ is a maps-to function on some set of upper integers π that returns a set π΅, (πΉ shift π) is another maps-to function on the shifted set of upper integers π. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ πΉ = (π₯ β π β¦ π΅) & β’ π΅ β V & β’ (π₯ = (π¦ β π) β π΅ = πΆ) & β’ π = (β€β₯βπ) & β’ π = (β€β₯β(π + π)) & β’ (π β π β β€) & β’ (π β π β β€) β β’ (π β (πΉ shift π) = (π¦ β π β¦ πΆ)) | ||
Theorem | dvradcnv2 43845* | The radius of convergence of the (formal) derivative π» of the power series πΊ is (at least) as large as the radius of convergence of πΊ. This version of dvradcnv 26370 uses a shifted version of π» to match the sum form of (β D πΉ) in pserdv2 26380 (and shows how to use uzmptshftfval 43844 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) & β’ π = sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*, < ) & β’ π» = (π β β β¦ ((π Β· (π΄βπ)) Β· (πβ(π β 1)))) & β’ (π β π΄:β0βΆβ) & β’ (π β π β β) & β’ (π β (absβπ) < π ) β β’ (π β seq1( + , π») β dom β ) | ||
Theorem | binomcxplemwb 43846 | Lemma for binomcxp 43855. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β πΆ β β) & β’ (π β πΎ β β) β β’ (π β (((πΆ β πΎ) Β· (πΆCππΎ)) + ((πΆ β (πΎ β 1)) Β· (πΆCπ(πΎ β 1)))) = (πΆ Β· (πΆCππΎ))) | ||
Theorem | binomcxplemnn0 43847* | Lemma for binomcxp 43855. When πΆ is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 15803 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set (0...πΆ), and when the index set is widened beyond πΆ the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ ((π β§ πΆ β β0) β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | binomcxplemrat 43848* | Lemma for binomcxp 43855. 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.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ (π β (π β β0 β¦ (absβ((πΆ β π) / (π + 1)))) β 1) | ||
Theorem | binomcxplemfrat 43849* | Lemma for binomcxp 43855. binomcxplemrat 43848 implies that when πΆ is not a nonnegative integer, the absolute value of the ratio ((πΉβ(π + 1)) / (πΉβπ)) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) β β’ ((π β§ Β¬ πΆ β β0) β (π β β0 β¦ (absβ((πΉβ(π + 1)) / (πΉβπ)))) β 1) | ||
Theorem | binomcxplemradcnv 43850* | Lemma for binomcxp 43855. By binomcxplemfrat 43849 and radcnvrat 43812 the radius of convergence of power series Ξ£π β β0((πΉβπ) Β· (πβπ)) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) β β’ ((π β§ Β¬ πΆ β β0) β π = 1) | ||
Theorem | binomcxplemdvbinom 43851* | Lemma for binomcxp 43855. By the power and chain rules, calculate the derivative of ((1 + π)βπ-πΆ), with respect to π in the disk of convergence π·. We later multiply the derivative in the later binomcxplemdvsum 43853 by this derivative to show that ((1 + π)βππΆ) (with a nonnegated πΆ) and the later sum, since both at π = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) β β’ ((π β§ Β¬ πΆ β β0) β (β D (π β π· β¦ ((1 + π)βπ-πΆ))) = (π β π· β¦ (-πΆ Β· ((1 + π)βπ(-πΆ β 1))))) | ||
Theorem | binomcxplemcvg 43852* | Lemma for binomcxp 43855. The sum in binomcxplemnn0 43847 and its derivative (see the next theorem, binomcxplemdvsum 43853) converge, as long as their base π½ 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.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) β β’ ((π β§ π½ β π·) β (seq0( + , (πβπ½)) β dom β β§ seq1( + , (πΈβπ½)) β dom β )) | ||
Theorem | binomcxplemdvsum 43853* | Lemma for binomcxp 43855. The derivative of the generalized sum in binomcxplemnn0 43847. Part of remark "This convergence allows to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) & β’ π = (π β π· β¦ Ξ£π β β0 ((πβπ)βπ)) β β’ (π β (β D π) = (π β π· β¦ Ξ£π β β ((πΈβπ)βπ))) | ||
Theorem | binomcxplemnotnn0 43854* |
Lemma for binomcxp 43855. When πΆ is not a nonnegative integer, the
generalized sum in binomcxplemnn0 43847 βwhich we will call π
βis a convergent power series: its base π is always of
smaller absolute value than the radius of convergence.
pserdv2 26380 gives the derivative of π, which by dvradcnv 26370 also converges in that radius. When π΄ is fixed at one, (π΄ + π) times that derivative equals (πΆ Β· π) and fraction (π / ((π΄ + π)βππΆ)) is always defined with derivative zero, so the fraction is a constantβspecifically one, because ((1 + 0)βππΆ) = 1. Thus ((1 + π)βππΆ) = (πβπ). Finally, let π be (π΅ / π΄), and multiply both the binomial ((1 + (π΅ / π΄))βππΆ) and the sum (πβ(π΅ / π΄)) by (π΄βππΆ) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) & β’ πΉ = (π β β0 β¦ (πΆCππ)) & β’ π = (π β β β¦ (π β β0 β¦ ((πΉβπ) Β· (πβπ)))) & β’ π = sup({π β β β£ seq0( + , (πβπ)) β dom β }, β*, < ) & β’ πΈ = (π β β β¦ (π β β β¦ ((π Β· (πΉβπ)) Β· (πβ(π β 1))))) & β’ π· = (β‘abs β (0[,)π )) & β’ π = (π β π· β¦ Ξ£π β β0 ((πβπ)βπ)) β β’ ((π β§ Β¬ πΆ β β0) β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | binomcxp 43855* | Generalize the binomial theorem binom 15803 to positive real summand π΄, real summand π΅, and complex exponent πΆ. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus 15803; see also https://en.wikipedia.org/wiki/Binomial_series 15803, https://en.wikipedia.org/wiki/Binomial_theorem 15803 (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem 15803. (Contributed by Steve Rodriguez, 22-Apr-2020.) |
β’ (π β π΄ β β+) & β’ (π β π΅ β β) & β’ (π β (absβπ΅) < (absβπ΄)) & β’ (π β πΆ β β) β β’ (π β ((π΄ + π΅)βππΆ) = Ξ£π β β0 ((πΆCππ) Β· ((π΄βπ(πΆ β π)) Β· (π΅βπ)))) | ||
Theorem | pm10.12 43856* | 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.) |
β’ (βπ₯(π β¨ π) β (π β¨ βπ₯π)) | ||
Theorem | pm10.14 43857 | Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ ((βπ₯π β§ βπ₯π) β ([π¦ / π₯]π β§ [π¦ / π₯]π)) | ||
Theorem | pm10.251 43858 | Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (βπ₯ Β¬ π β Β¬ βπ₯π) | ||
Theorem | pm10.252 43859 | Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) |
β’ (Β¬ βπ₯π β βπ₯ Β¬ π) | ||
Theorem | pm10.253 43860 | Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (Β¬ βπ₯π β βπ₯ Β¬ π) | ||
Theorem | albitr 43861 | Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β π) β§ βπ₯(π β π)) β βπ₯(π β π)) | ||
Theorem | pm10.42 43862 | Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ ((βπ₯π β¨ βπ₯π) β βπ₯(π β¨ π)) | ||
Theorem | pm10.52 43863* | Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β (βπ₯(π β π) β π)) | ||
Theorem | pm10.53 43864 | Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (Β¬ βπ₯π β βπ₯(π β π)) | ||
Theorem | pm10.541 43865* | Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β¨ π)) β (π β¨ βπ₯(π β π))) | ||
Theorem | pm10.542 43866* | Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β π)) β (π β βπ₯(π β π))) | ||
Theorem | pm10.55 43867 | Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β§ π) β§ βπ₯(π β π)) β (βπ₯π β§ βπ₯(π β π))) | ||
Theorem | pm10.56 43868 | Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯(π β π) β§ βπ₯(π β§ π)) β βπ₯(π β§ π)) | ||
Theorem | pm10.57 43869 | Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯(π β (π β¨ π)) β (βπ₯(π β π) β¨ βπ₯(π β§ π))) | ||
Theorem | 2alanimi 43870 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((π β§ π) β π) β β’ ((βπ₯βπ¦π β§ βπ₯βπ¦π) β βπ₯βπ¦π) | ||
Theorem | 2al2imi 43871 | Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (π β (π β π)) β β’ (βπ₯βπ¦π β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | pm11.11 43872 | Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ π β β’ βπ§βπ€[π§ / π₯][π€ / π¦]π | ||
Theorem | pm11.12 43873* | Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β (π β¨ βπ₯βπ¦π)) | ||
Theorem | 19.21vv 43874* | Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1934. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (π β βπ₯βπ¦π)) | ||
Theorem | 2alim 43875 | Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2albi 43876 | Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2exim 43877 | Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | 2exbi 43878 | Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | spsbce-2 43879 | Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ([π§ / π₯][π€ / π¦]π β βπ₯βπ¦π) | ||
Theorem | 19.33-2 43880 | Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯βπ¦π β¨ βπ₯βπ¦π) β βπ₯βπ¦(π β¨ π)) | ||
Theorem | 19.36vv 43881* | Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β π)) | ||
Theorem | 19.31vv 43882* | Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β (βπ₯βπ¦π β¨ π)) | ||
Theorem | 19.37vv 43883* | Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β π) β (π β βπ₯βπ¦π)) | ||
Theorem | 19.28vv 43884* | Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β§ π) β (π β§ βπ₯βπ¦π)) | ||
Theorem | pm11.52 43885 | Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β§ π) β Β¬ βπ₯βπ¦(π β Β¬ π)) | ||
Theorem | aaanv 43886* | Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2321. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯π β§ βπ¦π) β βπ₯βπ¦(π β§ π)) | ||
Theorem | pm11.57 43887* | Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β βπ₯βπ¦(π β§ [π¦ / π₯]π)) | ||
Theorem | pm11.58 43888* | Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯π β βπ₯βπ¦(π β§ [π¦ / π₯]π)) | ||
Theorem | pm11.59 43889* | Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯(π β π) β βπ¦βπ₯((π β§ [π¦ / π₯]π) β (π β§ [π¦ / π₯]π))) | ||
Theorem | pm11.6 43890* | Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.) |
β’ (βπ₯(βπ¦(π β§ π) β§ π) β βπ¦(βπ₯(π β§ π) β§ π)) | ||
Theorem | pm11.61 43891* | Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ¦βπ₯(π β π) β βπ₯(π β βπ¦π)) | ||
Theorem | pm11.62 43892* | Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦((π β§ π) β π) β βπ₯(π β βπ¦(π β π))) | ||
Theorem | pm11.63 43893 | Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (Β¬ βπ₯βπ¦π β βπ₯βπ¦(π β π)) | ||
Theorem | pm11.7 43894 | Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ (βπ₯βπ¦(π β¨ π) β βπ₯βπ¦π) | ||
Theorem | pm11.71 43895* | Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
β’ ((βπ₯π β§ βπ¦π) β ((βπ₯(π β π) β§ βπ¦(π β π)) β βπ₯βπ¦((π β§ π) β (π β§ π)))) | ||
Theorem | sbeqal1 43896* | If π₯ = π¦ always implies π₯ = π§, then π¦ = π§. (Contributed by Andrew Salmon, 2-Jun-2011.) |
β’ (βπ₯(π₯ = π¦ β π₯ = π§) β π¦ = π§) | ||
Theorem | sbeqal1i 43897* | Suppose you know π₯ = π¦ implies π₯ = π§, assuming π₯ and π§ are distinct. Then, π¦ = π§. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (π₯ = π¦ β π₯ = π§) β β’ π¦ = π§ | ||
Theorem | sbeqal2i 43898* | If π₯ = π¦ implies π₯ = π§, then we can infer π§ = π¦. (Contributed by Andrew Salmon, 3-Jun-2011.) |
β’ (π₯ = π¦ β π₯ = π§) β β’ π§ = π¦ | ||
Theorem | axc5c4c711 43899 | Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1789 as the inference rule. This proof extends the idea of axc5c711 38442 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.) |
β’ ((βπ₯βπ¦ Β¬ βπ₯βπ¦(βπ¦π β π) β (π β βπ¦(βπ¦π β π))) β (βπ¦π β βπ¦π)) | ||
Theorem | axc5c4c711toc5 43900 | Rederivation of sp 2171 from axc5c4c711 43899. Note that ax6 2377 is used for the rederivation. (Contributed by Andrew Salmon, 14-Jul-2011.) Revised to use ax6v 1964 instead of ax6 2377, so that this rederivation requires only ax6v 1964 and propositional calculus. (Revised by BJ, 14-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯π β π) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |