P. 386 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	amgm2d 38501	Arithmetic-geometric mean inequality for n = 2, derived from amgmlem 24716. (Contributed by Stanislas Polu, 8-Sep-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   =>    |- ( ph -> ( ( A x. B ) ^c ( 1 / 2 ) ) <_ ( ( A + B ) / 2 ) )
Theorem	amgm3d 38502	Arithmetic-geometric mean inequality for n = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> C e. RR+ )   =>    |- ( ph -> ( ( A x. ( B x. C ) ) ^c ( 1 / 3 ) ) <_ ( ( A + ( B + C ) ) / 3 ) )
Theorem	amgm4d 38503	Arithmetic-geometric mean inequality for n = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> C e. RR+ )   &    |- ( ph -> D e. RR+ )   =>    |- ( ph -> ( ( A x. ( B x. ( C x. D ) ) ) ^c ( 1 / 4 ) ) <_ ( ( A + ( B + ( C + D ) ) ) / 4 ) )
20.29  Mathbox for Steve Rodriguez
20.29.1  Miscellanea
Theorem	nanorxor 38504	'nand' is equivalent to the equivalence of inclusive and exclusive or. (Contributed by Steve Rodriguez, 28-Feb-2020.)
|- ( ( ph -/\ ps ) <-> ( ( ph \/ ps ) <-> ( ph \/_ ps ) ) )
Theorem	undisjrab 38505	Union of two disjoint restricted class abstractions; compare unrab 3898. (Contributed by Steve Rodriguez, 28-Feb-2020.)
|- ( ( { x e. A | ph } i^i { x e. A | ps } ) = (/) <-> ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/_ ps ) } )
Theorem	iso0 38506	The empty set is an R , S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
|- (/) Isom R , S ( (/) , (/) )
Theorem	ssrecnpr 38507	RR is a subset of both RR and CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
|- ( S e. { RR , CC } -> RR C_ S )
Theorem	seff 38508	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.)
|- ( ph -> S e. { RR , CC } )   =>    |- ( ph -> ( exp |` S ) : S --> S )
Theorem	sblpnf 38509	The infinity ball in the absolute value metric is just the whole space. S analogue of blpnf 22202. (Contributed by Steve Rodriguez, 8-Nov-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- D = ( ( abs o. - ) |` ( S X. S ) )   =>    |- ( ( ph /\ P e. S ) -> ( P ( ball ` D ) +oo ) = S )
Theorem	prmunb2 38510*	The primes are unbounded. This generalizes prmunb 15618 to real A with arch 11289 and lttrd 10198: every real is less than some positive integer, itself less than some prime. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( A e. RR -> E. p e. Prime A < p )
20.29.2  Ratio test for infinite series convergence and divergence
Theorem	dvgrat 38511*	Ratio test for divergence of a complex infinite series. See e.g. remark "if ( abs ` ( ( a ` ( n + 1 ) ) / ( a ` n ) ) ) >_ 1 for all large n..." in https://en.wikipedia.org/wiki/Ratio_test#The_test. (Contributed by Steve Rodriguez, 28-Feb-2020.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> N e. Z )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. W ) -> ( F ` k ) =/= 0 )   &    |- ( ( ph /\ k e. W ) -> ( abs ` ( F ` k ) ) <_ ( abs ` ( F ` ( k + 1 ) ) ) )   =>    |- ( ph -> seq M ( + , F ) e/ dom ~~> )
Theorem	cvgdvgrat 38512*	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 14615 and dvgrat 38511 directly uses the limit of the ratio. (It also demonstrates how to use climi2 14242 and absltd 14168 to transform a limit to an inequality cf. https://math.stackexchange.com/q/2215191, and how to use r19.29a 3078 in a similar fashion to Mario Carneiro's proof sketch with rexlimdva 3031 at https://groups.google.com/forum/#!topic/metamath/2RPikOiXLMo.) (Contributed by Steve Rodriguez, 28-Feb-2020.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> N e. Z )   &    |- ( ph -> F e. V )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. W ) -> ( F ` k ) =/= 0 )   &    |- R = ( k e. W |-> ( abs ` ( ( F ` ( k + 1 ) ) / ( F ` k ) ) ) )   &    |- ( ph -> R ~~> L )   &    |- ( ph -> L =/= 1 )   =>    |- ( ph -> ( L < 1 <-> seq M ( + , F ) e. dom ~~> ) )
Theorem	radcnvrat 38513*	Let L be the limit, if one exists, of the ratio ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) (as in the ratio test cvgdvgrat 38512) as k increases. Then the radius of convergence of power series sum_ n e. NN0 ( ( A ` n ) x. ( x ^ 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 —a few lines that evidently hide quite an involved process to confirm. (Contributed by Steve Rodriguez, 8-Mar-2020.)
|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )   &    |- ( ph -> A : NN0 --> CC )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )   &    |- D = ( k e. NN0 |-> ( abs ` ( ( A ` ( k + 1 ) ) / ( A ` k ) ) ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. NN0 )   &    |- ( ( ph /\ k e. Z ) -> ( A ` k ) =/= 0 )   &    |- ( ph -> D ~~> L )   &    |- ( ph -> L =/= 0 )   =>    |- ( ph -> R = ( 1 / L ) )
20.29.3  Multiples
Theorem	reldvds 38514	The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- Rel ||
Theorem	nznngen 38515	All positive integers in the set of multiples of n, nℤ, are the absolute value of n or greater. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> N e. ZZ )   =>    |- ( ph -> ( ( || " { N } ) i^i NN ) C_ ( ZZ>= ` ( abs ` N ) ) )
Theorem	nzss 38516	The set of multiples of m, mℤ, is a subset of those of n, nℤ, iff n divides m. Lemma 2.1(a) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. V )   =>    |- ( ph -> ( ( || " { M } ) C_ ( || " { N } ) <-> N || M ) )
Theorem	nzin 38517	The intersection of the set of multiples of m, mℤ, and those of n, nℤ, 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 mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( ( || " { M } ) i^i ( || " { N } ) ) = ( || " { ( M lcm N ) } ) )
Theorem	nzprmdif 38518	Subtract one prime's multiples from an unequal prime's. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> M e. Prime )   &    |- ( ph -> N e. Prime )   &    |- ( ph -> M =/= N )   =>    |- ( ph -> ( ( || " { M } ) \ ( || " { N } ) ) = ( ( || " { M } ) \ ( || " { ( M x. N ) } ) ) )
Theorem	hashnzfz 38519	Special case of hashdvds 15480: the count of multiples in nℤ restricted to an interval. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> N e. NN )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ( ZZ>= ` ( J - 1 ) ) )   =>    |- ( ph -> ( # ` ( ( || " { N } ) i^i ( J ... K ) ) ) = ( ( |_ ` ( K / N ) ) - ( |_ ` ( ( J - 1 ) / N ) ) ) )
Theorem	hashnzfz2 38520	Special case of hashnzfz 38519: the count of multiples in nℤ, n greater than one, restricted to an interval starting at two. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> N e. ( ZZ>= ` 2 ) )   &    |- ( ph -> K e. NN )   =>    |- ( ph -> ( # ` ( ( || " { N } ) i^i ( 2 ... K ) ) ) = ( |_ ` ( K / N ) ) )
Theorem	hashnzfzclim 38521*	As the upper bound K of the constraint interval ( J ... K ) in hashnzfz 38519 increases, the resulting count of multiples tends to ( K / M ) —that is, there are approximately ( K / M ) multiples of M in a finite interval of integers. (Contributed by Steve Rodriguez, 20-Jan-2020.)
|- ( ph -> M e. NN )   &    |- ( ph -> J e. ZZ )   =>    |- ( ph -> ( k e. ( ZZ>= ` ( J - 1 ) ) |-> ( ( # ` ( ( || " { M } ) i^i ( J ... k ) ) ) / k ) ) ~~> ( 1 / M ) )
20.29.4  Function operations
Theorem	caofcan 38522*	Transfer a cancellation law like mulcan 10664 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> T )   &    |- ( ph -> G : A --> S )   &    |- ( ph -> H : A --> S )   &    |- ( ( ph /\ ( x e. T /\ y e. S /\ z e. S ) ) -> ( ( x R y ) = ( x R z ) <-> y = z ) )   =>    |- ( ph -> ( ( F oF R G ) = ( F oF R H ) <-> G = H ) )
Theorem	ofsubid 38523	Function analogue of subid 10300. (Contributed by Steve Rodriguez, 5-Nov-2015.)
|- ( ( A e. V /\ F : A --> CC ) -> ( F oF - F ) = ( A X. { 0 } ) )
Theorem	ofmul12 38524	Function analogue of mul12 10202. (Contributed by Steve Rodriguez, 13-Nov-2015.)
|- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> CC /\ H : A --> CC ) ) -> ( F oF x. ( G oF x. H ) ) = ( G oF x. ( F oF x. H ) ) )
Theorem	ofdivrec 38525	Function analogue of divrec 10701, a division analogue of ofnegsub 11018. (Contributed by Steve Rodriguez, 3-Nov-2015.)
|- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> ( F oF x. ( ( A X. { 1 } ) oF / G ) ) = ( F oF / G ) )
Theorem	ofdivcan4 38526	Function analogue of divcan4 10712. (Contributed by Steve Rodriguez, 4-Nov-2015.)
|- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> ( ( F oF x. G ) oF / G ) = F )
Theorem	ofdivdiv2 38527	Function analogue of divdiv2 10737. (Contributed by Steve Rodriguez, 23-Nov-2015.)
|- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> ( F oF / ( G oF / H ) ) = ( ( F oF x. H ) oF / G ) )
20.29.5  Calculus
Theorem	lhe4.4ex1a 38528	Example of the Fundamental Theorem of Calculus, part two (ftc2 23807): S. ( 1 (,) 2 ) ( ( x ^ 2 ) - 3 ) _d x = -u ( 2 / 3 ). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 23807 as simply the "Fundamental Theorem of Calculus", then ftc1 23805 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
|- S. ( 1 (,) 2 ) ( ( x ^ 2 ) - 3 ) _d x = -u ( 2 / 3 )
Theorem	dvsconst 38529	Derivative of a constant function on the real or complex numbers. The function may return a complex A even if S is RR. (Contributed by Steve Rodriguez, 11-Nov-2015.)
|- ( ( S e. { RR , CC } /\ A e. CC ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
Theorem	dvsid 38530	Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.)
|- ( S e. { RR , CC } -> ( S _D ( _I |` S ) ) = ( S X. { 1 } ) )
Theorem	dvsef 38531	Derivative of the exponential function on the real or complex numbers. (Contributed by Steve Rodriguez, 12-Nov-2015.)
|- ( S e. { RR , CC } -> ( S _D ( exp |` S ) ) = ( exp |` S ) )
Theorem	expgrowthi 38532*	Exponential growth and decay model. See expgrowth 38534 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> K e. CC )   &    |- ( ph -> C e. CC )   &    |- Y = ( t e. S |-> ( C x. ( exp ` ( K x. t ) ) ) )   =>    |- ( ph -> ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) )
Theorem	dvconstbi 38533*	The derivative of a function on S is zero iff it is a constant function. Roughly a biconditional S analogue of dvconst 23680 and dveq0 23763. Corresponds to integration formula " S. 0 _d x = C " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> Y : S --> CC )   &    |- ( ph -> dom ( S _D Y ) = S )   =>    |- ( ph -> ( ( S _D Y ) = ( S X. { 0 } ) <-> E. c e. CC Y = ( S X. { c } ) ) )
Theorem	expgrowth 38534*	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 38532 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); 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 λ. Here y' is given as ( S _D Y ), C as c, and ky as ( ( S X. { K } ) oF x. Y ). ( S X. { K } ) is the constant function that maps any real or complex input to k and oF x. 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 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. Statements for this and expgrowthi 38532 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)
|- ( ph -> S e. { RR , CC } )   &    |- ( ph -> K e. CC )   &    |- ( ph -> Y : S --> CC )   &    |- ( ph -> dom ( S _D Y ) = S )   =>    |- ( ph -> ( ( S _D Y ) = ( ( S X. { K } ) oF x. Y ) <-> E. c e. CC Y = ( t e. S |-> ( c x. ( exp ` ( K x. t ) ) ) ) ) )
20.29.6  The generalized binomial coefficient operation
Syntax	cbcc 38535	Extend class notation to include the generalized binomial coefficient operation.
class C𝑐
Definition	df-bcc 38536*	Define a generalized binomial coefficient operation, which unlike df-bc 13090 allows complex numbers for the first argument. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- C𝑐 = ( c e. CC , k e. NN0 |-> ( ( c FallFac k ) / ( ! ` k ) ) )
Theorem	bccval 38537	Value of the generalized binomial coefficient, C choose K. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   &    |- ( ph -> K e. NN0 )   =>    |- ( ph -> ( CC𝑐 K ) = ( ( C FallFac K ) / ( ! ` K ) ) )
Theorem	bcccl 38538	Closure of the generalized binomial coefficient. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   &    |- ( ph -> K e. NN0 )   =>    |- ( ph -> ( CC𝑐 K ) e. CC )
Theorem	bcc0 38539	The generalized binomial coefficient C choose K is zero iff C is an integer between zero and ( K - 1 ) inclusive. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   &    |- ( ph -> K e. NN0 )   =>    |- ( ph -> ( ( CC𝑐 K ) = 0 <-> C e. ( 0 ... ( K - 1 ) ) ) )
Theorem	bccp1k 38540	Generalized binomial coefficient: C choose ( K + 1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   &    |- ( ph -> K e. NN0 )   =>    |- ( ph -> ( CC𝑐 ( K + 1 ) ) = ( ( CC𝑐 K ) x. ( ( C - K ) / ( K + 1 ) ) ) )
Theorem	bccm1k 38541	Generalized binomial coefficient: C choose ( K - 1 ), when C is not ( K - 1 ). (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. ( CC \ { ( K - 1 ) } ) )   &    |- ( ph -> K e. NN )   =>    |- ( ph -> ( CC𝑐 ( K - 1 ) ) = ( ( CC𝑐 K ) / ( ( C - ( K - 1 ) ) / K ) ) )
Theorem	bccn0 38542	Generalized binomial coefficient: C choose 0. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   =>    |- ( ph -> ( CC𝑐 0 ) = 1 )
Theorem	bccn1 38543	Generalized binomial coefficient: C choose 1. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   =>    |- ( ph -> ( CC𝑐 1 ) = C )
Theorem	bccbc 38544	The binomial coefficient and generalized binomial coefficient are equal when their arguments are nonnegative integers. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> N e. NN0 )   &    |- ( ph -> K e. NN0 )   =>    |- ( ph -> ( NC𝑐 K ) = ( N _C K ) )
20.29.7  Binomial series
Theorem	uzmptshftfval 38545*	When F is a maps-to function on some set of upper integers Z that returns a set B, ( F shift N ) is another maps-to function on the shifted set of upper integers W. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- F = ( x e. Z |-> B )   &    |- B e. _V   &    |- ( x = ( y - N ) -> B = C )   &    |- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` ( M + N ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( F shift N ) = ( y e. W |-> C ) )
Theorem	dvradcnv2 38546*	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 24175 uses a shifted version of H to match the sum form of ( CC _D F ) in pserdv2 24184 (and shows how to use uzmptshftfval 38545 to shift a maps-to function on a set of upper integers). (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < )   &    |- H = ( n e. NN |-> ( ( n x. ( A ` n ) ) x. ( X ^ ( n - 1 ) ) ) )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> X e. CC )   &    |- ( ph -> ( abs ` X ) < R )   =>    |- ( ph -> seq 1 ( + , H ) e. dom ~~> )
Theorem	binomcxplemwb 38547	Lemma for binomcxp 38556. The lemma in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> C e. CC )   &    |- ( ph -> K e. NN )   =>    |- ( ph -> ( ( ( C - K ) x. ( CC𝑐 K ) ) + ( ( C - ( K - 1 ) ) x. ( CC𝑐 ( K - 1 ) ) ) ) = ( C x. ( CC𝑐 K ) ) )
Theorem	binomcxplemnn0 38548*	Lemma for binomcxp 38556. When C is a nonnegative integer, the binomial's finite sum value by the standard binomial theorem binom 14562 equals this generalized infinite sum: the generalized binomial coefficient and exponentiation operators give exactly the same values in the standard index set ( 0 ... C ), and when the index set is widened beyond C the additional values are just zeroes. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   =>    |- ( ( ph /\ C e. NN0 ) -> ( ( A + B ) ^c C ) = sum_ k e. NN0 ( ( CC𝑐 k ) x. ( ( A ^c ( C - k ) ) x. ( B ^ k ) ) ) )
Theorem	binomcxplemrat 38549*	Lemma for binomcxp 38556. As k 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.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( k e. NN0 |-> ( abs ` ( ( C - k ) / ( k + 1 ) ) ) ) ~~> 1 )
Theorem	binomcxplemfrat 38550*	Lemma for binomcxp 38556. binomcxplemrat 38549 implies that when C is not a nonnegative integer, the absolute value of the ratio ( ( F ` ( k + 1 ) ) / ( F ` k ) ) converges to one. The rest of equation "Since continuity of the absolute value..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   =>    |- ( ( ph /\ -. C e. NN0 ) -> ( k e. NN0 |-> ( abs ` ( ( F ` ( k + 1 ) ) / ( F ` k ) ) ) ) ~~> 1 )
Theorem	binomcxplemradcnv 38551*	Lemma for binomcxp 38556. By binomcxplemfrat 38550 and radcnvrat 38513 the radius of convergence of power series sum_ k e. NN0 ( ( F ` k ) x. ( b ^ k ) ) is one. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   &    |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )   =>    |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )
Theorem	binomcxplemdvbinom 38552*	Lemma for binomcxp 38556. By the power and chain rules, calculate the derivative of ( ( 1 + b ) ^c -u C ), with respect to b in the disk of convergence D. We later multiply the derivative in the later binomcxplemdvsum 38554 by this derivative to show that ( ( 1 + b ) ^c C ) (with a non-negated C) and the later sum, since both at b = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   &    |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )   &    |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )   &    |- D = ( `' abs " ( 0 [,) R ) )   =>    |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
Theorem	binomcxplemcvg 38553*	Lemma for binomcxp 38556. The sum in binomcxplemnn0 38548 and its derivative (see the next theorem, binomcxplemdvsum 38554) 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.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   &    |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )   &    |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )   &    |- D = ( `' abs " ( 0 [,) R ) )   =>    |- ( ( ph /\ J e. D ) -> ( seq 0 ( + , ( S ` J ) ) e. dom ~~> /\ seq 1 ( + , ( E ` J ) ) e. dom ~~> ) )
Theorem	binomcxplemdvsum 38554*	Lemma for binomcxp 38556. The derivative of the generalized sum in binomcxplemnn0 38548. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   &    |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )   &    |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )   &    |- D = ( `' abs " ( 0 [,) R ) )   &    |- P = ( b e. D |-> sum_ k e. NN0 ( ( S ` b ) ` k ) )   =>    |- ( ph -> ( CC _D P ) = ( b e. D |-> sum_ k e. NN ( ( E ` b ) ` k ) ) )
Theorem	binomcxplemnotnn0 38555*	Lemma for binomcxp 38556. When C is not a nonnegative integer, the generalized sum in binomcxplemnn0 38548 —which we will call P —is a convergent power series: its base b is always of smaller absolute value than the radius of convergence. pserdv2 24184 gives the derivative of P, which by dvradcnv 24175 also converges in that radius. When A is fixed at one, ( A + b ) times that derivative equals ( C x. P ) and fraction ( P / ( ( A + b ) ^c C ) ) is always defined with derivative zero, so the fraction is a constant—specifically one, because ( ( 1 + 0 ) ^c C ) = 1. Thus ( ( 1 + b ) ^c C ) = ( P ` b ). Finally, let b be ( B / A ), and multiply both the binomial ( ( 1 + ( B / A ) ) ^c C ) and the sum ( P ` ( B / A ) ) by ( A ^c C ) to get the result. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   &    |- F = ( j e. NN0 |-> ( CC𝑐 j ) )   &    |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )   &    |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )   &    |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )   &    |- D = ( `' abs " ( 0 [,) R ) )   &    |- P = ( b e. D |-> sum_ k e. NN0 ( ( S ` b ) ` k ) )   =>    |- ( ( ph /\ -. C e. NN0 ) -> ( ( A + B ) ^c C ) = sum_ k e. NN0 ( ( CC𝑐 k ) x. ( ( A ^c ( C - k ) ) x. ( B ^ k ) ) ) )
Theorem	binomcxp 38556*	Generalize the binomial theorem binom 14562 to positive real summand A, real summand B, and complex exponent C. Proof in https://en.wikibooks.org/wiki/Advanced_Calculus; see also https://en.wikipedia.org/wiki/Binomial_series, https://en.wikipedia.org/wiki/Binomial_theorem (sections "Newton's generalized binomial theorem" and "Future generalizations"), and proof "General Binomial Theorem" in https://proofwiki.org/wiki/Binomial_Theorem. (Contributed by Steve Rodriguez, 22-Apr-2020.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` B ) < ( abs ` A ) )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( ( A + B ) ^c C ) = sum_ k e. NN0 ( ( CC𝑐 k ) x. ( ( A ^c ( C - k ) ) x. ( B ^ k ) ) ) )
20.30  Mathbox for Andrew Salmon
20.30.1  Principia Mathematica * 10
Theorem	pm10.12 38557*	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.)
|- ( A. x ( ph \/ ps ) -> ( ph \/ A. x ps ) )
Theorem	pm10.14 38558	Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ( ( A. x ph /\ A. x ps ) -> ( [ y / x ] ph /\ [ y / x ] ps ) )
Theorem	pm10.251 38559	Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ( A. x -. ph -> -. A. x ph )
Theorem	pm10.252 38560	Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.)
|- ( -. E. x ph <-> A. x -. ph )
Theorem	pm10.253 38561	Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ( -. A. x ph <-> E. x -. ph )
Theorem	albitr 38562	Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( A. x ( ph <-> ps ) /\ A. x ( ps <-> ch ) ) -> A. x ( ph <-> ch ) )
Theorem	pm10.42 38563	Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ( ( E. x ph \/ E. x ps ) <-> E. x ( ph \/ ps ) )
Theorem	pm10.52 38564*	Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x ph -> ( A. x ( ph -> ps ) <-> ps ) )
Theorem	pm10.53 38565	Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( -. E. x ph -> A. x ( ph -> ps ) )
Theorem	pm10.541 38566*	Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x ( ph -> ( ch \/ ps ) ) <-> ( ch \/ A. x ( ph -> ps ) ) )
Theorem	pm10.542 38567*	Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x ( ph -> ( ch -> ps ) ) <-> ( ch -> A. x ( ph -> ps ) ) )
Theorem	pm10.55 38568	Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( E. x ( ph /\ ps ) /\ A. x ( ph -> ps ) ) <-> ( E. x ph /\ A. x ( ph -> ps ) ) )
Theorem	pm10.56 38569	Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( A. x ( ph -> ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ps /\ ch ) )
Theorem	pm10.57 38570	Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x ( ph -> ( ps \/ ch ) ) -> ( A. x ( ph -> ps ) \/ E. x ( ph /\ ch ) ) )
20.30.2  Principia Mathematica * 11
Theorem	2alanimi 38571	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x A. y ph /\ A. x A. y ps ) -> A. x A. y ch )
Theorem	2al2imi 38572	Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x A. y ph -> ( A. x A. y ps -> A. x A. y ch ) )
Theorem	pm11.11 38573	Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ph   =>    |- A. z A. w [ z / x ] [ w / y ] ph
Theorem	pm11.12 38574*	Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
|- ( A. x A. y ( ph \/ ps ) -> ( ph \/ A. x A. y ps ) )
Theorem	19.21vv 38575*	Compare Theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with two quantifiers. See 19.21v 1868. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ps -> ph ) <-> ( ps -> A. x A. y ph ) )
Theorem	2alim 38576	Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph -> ps ) -> ( A. x A. y ph -> A. x A. y ps ) )
Theorem	2albi 38577	Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph <-> ps ) -> ( A. x A. y ph <-> A. x A. y ps ) )
Theorem	2exim 38578	Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph -> ps ) -> ( E. x E. y ph -> E. x E. y ps ) )
Theorem	2exbi 38579	Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph <-> ps ) -> ( E. x E. y ph <-> E. x E. y ps ) )
Theorem	spsbce-2 38580	Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( [ z / x ] [ w / y ] ph -> E. x E. y ph )
Theorem	19.33-2 38581	Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( A. x A. y ph \/ A. x A. y ps ) -> A. x A. y ( ph \/ ps ) )
Theorem	19.36vv 38582*	Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
|- ( E. x E. y ( ph -> ps ) <-> ( A. x A. y ph -> ps ) )
Theorem	19.31vv 38583*	Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ph \/ ps ) <-> ( A. x A. y ph \/ ps ) )
Theorem	19.37vv 38584*	Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ( ps -> ph ) <-> ( ps -> E. x E. y ph ) )
Theorem	19.28vv 38585*	Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ps /\ ph ) <-> ( ps /\ A. x A. y ph ) )
Theorem	pm11.52 38586	Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ( ph /\ ps ) <-> -. A. x A. y ( ph -> -. ps ) )
Theorem	2exanali 38587	Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( -. E. x E. y ( ph /\ -. ps ) <-> A. x A. y ( ph -> ps ) )
Theorem	aaanv 38588*	Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2170. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( A. x ph /\ A. y ps ) <-> A. x A. y ( ph /\ ps ) )
Theorem	pm11.57 38589*	Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x ph <-> A. x A. y ( ph /\ [ y / x ] ph ) )
Theorem	pm11.58 38590*	Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x ph <-> E. x E. y ( ph /\ [ y / x ] ph ) )
Theorem	pm11.59 38591*	Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
|- ( A. x ( ph -> ps ) -> A. y A. x ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
Theorem	pm11.6 38592*	Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
|- ( E. x ( E. y ( ph /\ ps ) /\ ch ) <-> E. y ( E. x ( ph /\ ch ) /\ ps ) )
Theorem	pm11.61 38593*	Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. y A. x ( ph -> ps ) -> A. x ( ph -> E. y ps ) )
Theorem	pm11.62 38594*	Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y ( ( ph /\ ps ) -> ch ) <-> A. x ( ph -> A. y ( ps -> ch ) ) )
Theorem	pm11.63 38595	Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( -. E. x E. y ph -> A. x A. y ( ph -> ps ) )
Theorem	pm11.7 38596	Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ( ph \/ ph ) <-> E. x E. y ph )
Theorem	pm11.71 38597*	Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( ( E. x ph /\ E. y ch ) -> ( ( A. x ( ph -> ps ) /\ A. y ( ch -> th ) ) <-> A. x A. y ( ( ph /\ ch ) -> ( ps /\ th ) ) ) )
20.30.3  Predicate Calculus
Theorem	sbeqal1 38598*	If x = y always implies x = z, then y = z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
|- ( A. x ( x = y -> x = z ) -> y = z )
Theorem	sbeqal1i 38599*	Suppose you know x = y implies x = z, assuming x and z are distinct. Then, y = z. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( x = y -> x = z )   =>    |- y = z
Theorem	sbeqal2i 38600*	If x = y implies x = z, then we can infer z = y. (Contributed by Andrew Salmon, 3-Jun-2011.)
|- ( x = y -> x = z )   =>    |- z = y