P. 121 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12001-12100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fsum2d 12001*	Write a double sum as a sum over a two-dimensional region. Note that B ( j ) is a function of j. (Contributed by Mario Carneiro, 27-Apr-2014.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ z e. U_ j e. A ( { j } X. B ) D )
Theorem	fsumxp 12002*	Combine two sums into a single sum over the cartesian product. (Contributed by Mario Carneiro, 23-Apr-2014.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ z e. ( A X. B ) D )
Theorem	fsumcnv 12003*	Transform a region of summation by using the converse operation. (Contributed by Mario Carneiro, 23-Apr-2014.)
|- ( x = <. j , k >. -> B = D )   &    |- ( y = <. k , j >. -> C = D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> sum_ x e. A B = sum_ y e. `' A C )
Theorem	fisumcom2 12004*	Interchange order of summation. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.) (Proof shortened by JJ, 2-Aug-2021.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ k e. C ) -> D e. Fin )   &    |- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B E = sum_ k e. C sum_ j e. D E )
Theorem	fsumcom 12005*	Interchange order of summation. (Contributed by NM, 15-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B C = sum_ k e. B sum_ j e. A C )
Theorem	fsum0diaglem 12006*	Lemma for fisum0diag 12007. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
|- ( ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) -> ( k e. ( 0 ... N ) /\ j e. ( 0 ... ( N - k ) ) ) )
Theorem	fisum0diag 12007*	Two ways to express "the sum of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N". (Contributed by NM, 31-Dec-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
|- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> sum_ j e. ( 0 ... N ) sum_ k e. ( 0 ... ( N - j ) ) A = sum_ k e. ( 0 ... N ) sum_ j e. ( 0 ... ( N - k ) ) A )
Theorem	mptfzshft 12008*	1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( j e. ( ( M + K ) ... ( N + K ) ) |-> ( j - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )
Theorem	fsumrev 12009*	Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( K - k ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( K - N ) ... ( K - M ) ) B )
Theorem	fsumshft 12010*	Index shift of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) (Proof shortened by AV, 8-Sep-2019.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k - K ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M + K ) ... ( N + K ) ) B )
Theorem	fsumshftm 12011*	Negative index shift of a finite sum. (Contributed by NM, 28-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k + K ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( ( M - K ) ... ( N - K ) ) B )
Theorem	fisumrev2 12012*	Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( ( M + N ) - k ) -> A = B )   =>    |- ( ph -> sum_ j e. ( M ... N ) A = sum_ k e. ( M ... N ) B )
Theorem	fisum0diag2 12013*	Two ways to express "the sum of A ( j , k ) over the triangular region 0 <_ j, 0 <_ k, j + k <_ N". (Contributed by Mario Carneiro, 21-Jul-2014.)
|- ( x = k -> B = A )   &    |- ( x = ( k - j ) -> B = C )   &    |- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> sum_ j e. ( 0 ... N ) sum_ k e. ( 0 ... ( N - j ) ) A = sum_ k e. ( 0 ... N ) sum_ j e. ( 0 ... k ) C )
Theorem	fsummulc2 12014*	A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( C x. sum_ k e. A B ) = sum_ k e. A ( C x. B ) )
Theorem	fsummulc1 12015*	A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )
Theorem	fsumdivapc 12016*	A finite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C # 0 )   =>    |- ( ph -> ( sum_ k e. A B / C ) = sum_ k e. A ( B / C ) )
Theorem	fsumneg 12017*	Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A -u B = -u sum_ k e. A B )
Theorem	fsumsub 12018*	Split a finite sum over a subtraction. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   =>    |- ( ph -> sum_ k e. A ( B - C ) = ( sum_ k e. A B - sum_ k e. A C ) )
Theorem	fsum2mul 12019*	Separate the nested sum of the product C ( j ) x. D ( k ). (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ j e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. B ) -> D e. CC )   =>    |- ( ph -> sum_ j e. A sum_ k e. B ( C x. D ) = ( sum_ j e. A C x. sum_ k e. B D ) )
Theorem	fsumconst 12020*	The sum of constant terms ( k is not free in B). (Contributed by NM, 24-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ( A e. Fin /\ B e. CC ) -> sum_ k e. A B = ( (♯ ` A ) x. B ) )
Theorem	fsumdifsnconst 12021*	The sum of constant terms ( k is not free in C) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.)
|- ( ( A e. Fin /\ B e. A /\ C e. CC ) -> sum_ k e. ( A \ { B } ) C = ( ( (♯ ` A ) - 1 ) x. C ) )
Theorem	modfsummodlem1 12022*	Lemma for modfsummod 12024. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
|- ( A. k e. ( A u. { z } ) B e. ZZ -> [_ z / k ]_ B e. ZZ )
Theorem	modfsummodlemstep 12023*	Induction step for modfsummod 12024. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by Jim Kingdon, 12-Oct-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> N e. NN )   &    |- ( ph -> A. k e. ( A u. { z } ) B e. ZZ )   &    |- ( ph -> -. z e. A )   &    |- ( ph -> ( sum_ k e. A B mod N ) = ( sum_ k e. A ( B mod N ) mod N ) )   =>    |- ( ph -> ( sum_ k e. ( A u. { z } ) B mod N ) = ( sum_ k e. ( A u. { z } ) ( B mod N ) mod N ) )
Theorem	modfsummod 12024*	A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
|- ( ph -> N e. NN )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A. k e. A B e. ZZ )   =>    |- ( ph -> ( sum_ k e. A B mod N ) = ( sum_ k e. A ( B mod N ) mod N ) )
Theorem	fsumge0 12025*	If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> 0 <_ sum_ k e. A B )
Theorem	fsumlessfi 12026*	A shorter sum of nonnegative terms is no greater than a longer one. (Contributed by NM, 26-Dec-2005.) (Revised by Jim Kingdon, 12-Oct-2022.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( ph -> C C_ A )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> sum_ k e. C B <_ sum_ k e. A B )
Theorem	fsumge1 12027*	A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( k = M -> B = C )   &    |- ( ph -> M e. A )   =>    |- ( ph -> C <_ sum_ k e. A B )
Theorem	fsum00 12028*	A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> ( sum_ k e. A B = 0 <-> A. k e. A B = 0 ) )
Theorem	fsumle 12029*	If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B <_ C )   =>    |- ( ph -> sum_ k e. A B <_ sum_ k e. A C )
Theorem	fsumlt 12030*	If every term in one finite sum is less than the corresponding term in another, then the first sum is less than the second. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B < C )   =>    |- ( ph -> sum_ k e. A B < sum_ k e. A C )
Theorem	fsumabs 12031*	Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by NM, 9-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( abs ` sum_ k e. A B ) <_ sum_ k e. A ( abs ` B ) )
Theorem	telfsumo 12032*	Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = N -> A = E )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( B - C ) = ( D - E ) )
Theorem	telfsumo2 12033*	Sum of a telescoping series. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = N -> A = E )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( C - B ) = ( E - D ) )
Theorem	telfsum 12034*	Sum of a telescoping series. (Contributed by Scott Fenton, 24-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = ( N + 1 ) -> A = E )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M ... N ) ( B - C ) = ( D - E ) )
Theorem	telfsum2 12035*	Sum of a telescoping series. (Contributed by Mario Carneiro, 15-Jun-2014.) (Revised by Mario Carneiro, 2-May-2016.)
|- ( k = j -> A = B )   &    |- ( k = ( j + 1 ) -> A = C )   &    |- ( k = M -> A = D )   &    |- ( k = ( N + 1 ) -> A = E )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> sum_ j e. ( M ... N ) ( C - B ) = ( E - D ) )
Theorem	fsumparts 12036*	Summation by parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
|- ( k = j -> ( A = B /\ V = W ) )   &    |- ( k = ( j + 1 ) -> ( A = C /\ V = X ) )   &    |- ( k = M -> ( A = D /\ V = Y ) )   &    |- ( k = N -> ( A = E /\ V = Z ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> V e. CC )   =>    |- ( ph -> sum_ j e. ( M..^ N ) ( B x. ( X - W ) ) = ( ( ( E x. Z ) - ( D x. Y ) ) - sum_ j e. ( M..^ N ) ( ( C - B ) x. X ) ) )
Theorem	fsumrelem 12037*	Lemma for fsumre 12038, fsumim 12039, and fsumcj 12040. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- F : CC --> CC   &    |- ( ( x e. CC /\ y e. CC ) -> ( F ` ( x + y ) ) = ( ( F ` x ) + ( F ` y ) ) )   =>    |- ( ph -> ( F ` sum_ k e. A B ) = sum_ k e. A ( F ` B ) )
Theorem	fsumre 12038*	The real part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( Re ` sum_ k e. A B ) = sum_ k e. A ( Re ` B ) )
Theorem	fsumim 12039*	The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( Im ` sum_ k e. A B ) = sum_ k e. A ( Im ` B ) )
Theorem	fsumcj 12040*	The complex conjugate of a sum. (Contributed by Paul Chapman, 9-Nov-2007.) (Revised by Mario Carneiro, 25-Jul-2014.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> ( * ` sum_ k e. A B ) = sum_ k e. A ( * ` B ) )
Theorem	iserabs 12041*	Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ph -> seq M ( + , G ) ~~> B )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )   =>    |- ( ph -> ( abs ` A ) <_ B )
Theorem	cvgcmpub 12042*	An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ph -> seq M ( + , G ) ~~> B )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ ( F ` k ) )   =>    |- ( ph -> B <_ A )
Theorem	fsumiun 12043*	Sum over a disjoint indexed union. (Contributed by Mario Carneiro, 1-Jul-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> B e. Fin )   &    |- ( ph -> Disj x e. A B )   &    |- ( ( ph /\ ( x e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> sum_ k e. U_ x e. A B C = sum_ x e. A sum_ k e. B C )
Theorem	hashiun 12044*	The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> B e. Fin )   &    |- ( ph -> Disj x e. A B )   =>    |- ( ph -> (♯ ` U_ x e. A B ) = sum_ x e. A (♯ ` B ) )
Theorem	hash2iun 12045*	The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ x e. A ) -> B e. Fin )   &    |- ( ( ph /\ x e. A /\ y e. B ) -> C e. Fin )   &    |- ( ph -> Disj x e. A U_ y e. B C )   &    |- ( ( ph /\ x e. A ) -> Disj y e. B C )   =>    |- ( ph -> (♯ ` U_ x e. A U_ y e. B C ) = sum_ x e. A sum_ y e. B (♯ ` C ) )
Theorem	hash2iun1dif1 12046*	The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.)
|- ( ph -> A e. Fin )   &    |- B = ( A \ { x } )   &    |- ( ( ph /\ x e. A /\ y e. B ) -> C e. Fin )   &    |- ( ph -> Disj x e. A U_ y e. B C )   &    |- ( ( ph /\ x e. A ) -> Disj y e. B C )   &    |- ( ( ph /\ x e. A /\ y e. B ) -> (♯ ` C ) = 1 )   =>    |- ( ph -> (♯ ` U_ x e. A U_ y e. B C ) = ( (♯ ` A ) x. ( (♯ ` A ) - 1 ) ) )
Theorem	hashrabrex 12047*	The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
|- ( ph -> Y e. Fin )   &    |- ( ( ph /\ y e. Y ) -> { x e. X | ps } e. Fin )   &    |- ( ph -> Disj y e. Y { x e. X | ps } )   =>    |- ( ph -> (♯ ` { x e. X | E. y e. Y ps } ) = sum_ y e. Y (♯ ` { x e. X | ps } ) )
Theorem	hashuni 12048*	The cardinality of a disjoint union. (Contributed by Mario Carneiro, 24-Jan-2015.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A C_ Fin )   &    |- ( ph -> Disj x e. A x )   =>    |- ( ph -> (♯ ` U. A ) = sum_ x e. A (♯ ` x ) )
4.9.3  The binomial theorem
Theorem	binomlem 12049*	Lemma for binom 12050 (binomial theorem). Inductive step. (Contributed by NM, 6-Dec-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ps -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )   =>    |- ( ( ph /\ ps ) -> ( ( A + B ) ^ ( N + 1 ) ) = sum_ k e. ( 0 ... ( N + 1 ) ) ( ( ( N + 1 ) _C k ) x. ( ( A ^ ( ( N + 1 ) - k ) ) x. ( B ^ k ) ) ) )
Theorem	binom 12050*	The binomial theorem: ( A + B ) ^ N is the sum from k = 0 to N of ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ). Theorem 15-2.8 of [Gleason] p. 296. This part of the proof sets up the induction and does the base case, with the bulk of the work (the induction step) in binomlem 12049. This is Metamath 100 proof #44. (Contributed by NM, 7-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
|- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A + B ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( A ^ ( N - k ) ) x. ( B ^ k ) ) ) )
Theorem	binom1p 12051*	Special case of the binomial theorem for ( 1 + A ) ^ N. (Contributed by Paul Chapman, 10-May-2007.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( ( 1 + A ) ^ N ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( A ^ k ) ) )
Theorem	binom11 12052*	Special case of the binomial theorem for 2 ^ N. (Contributed by Mario Carneiro, 13-Mar-2014.)
|- ( N e. NN0 -> ( 2 ^ N ) = sum_ k e. ( 0 ... N ) ( N _C k ) )
Theorem	binom1dif 12053*	A summation for the difference between ( ( A + 1 ) ^ N ) and ( A ^ N ). (Contributed by Scott Fenton, 9-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
|- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( A + 1 ) ^ N ) - ( A ^ N ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( N _C k ) x. ( A ^ k ) ) )
Theorem	bcxmaslem1 12054	Lemma for bcxmas 12055. (Contributed by Paul Chapman, 18-May-2007.)
|- ( A = B -> ( ( N + A ) _C A ) = ( ( N + B ) _C B ) )
Theorem	bcxmas 12055*	Parallel summation (Christmas Stocking) theorem for Pascal's Triangle. (Contributed by Paul Chapman, 18-May-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( ( N e. NN0 /\ M e. NN0 ) -> ( ( ( N + 1 ) + M ) _C M ) = sum_ j e. ( 0 ... M ) ( ( N + j ) _C j ) )
4.9.4  Infinite sums (cont.)
Theorem	isumshft 12056*	Index shift of an infinite sum. (Contributed by Paul Chapman, 31-Oct-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` ( M + K ) )   &    |- ( j = ( K + k ) -> A = B )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ j e. W ) -> A e. CC )   =>    |- ( ph -> sum_ j e. W A = sum_ k e. Z B )
Theorem	isumsplit 12057*	Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
Theorem	isum1p 12058*	The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A = ( ( F ` M ) + sum_ k e. ( ZZ>= ` ( M + 1 ) ) A ) )
Theorem	isumnn0nn 12059*	Sum from 0 to infinity in terms of sum from 1 to infinity. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- ( k = 0 -> A = B )   &    |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. NN0 ) -> A e. CC )   &    |- ( ph -> seq 0 ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. NN0 A = ( B + sum_ k e. NN A ) )
Theorem	isumrpcl 12060*	The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. RR+ )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. W A e. RR+ )
Theorem	isumle 12061*	Comparison of two infinite sums. (Contributed by Paul Chapman, 13-Nov-2007.) (Revised by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. RR )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. RR )   &    |- ( ( ph /\ k e. Z ) -> A <_ B )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   &    |- ( ph -> seq M ( + , G ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A <_ sum_ k e. Z B )
Theorem	isumlessdc 12062*	A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ph -> A. k e. Z DECID k e. A )   &    |- ( ( ph /\ k e. Z ) -> B e. RR )   &    |- ( ( ph /\ k e. Z ) -> 0 <_ B )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. A B <_ sum_ k e. Z B )
4.9.5  Miscellaneous converging and diverging sequences
Theorem	divcnv 12063*	The sequence of reciprocals of positive integers, multiplied by the factor A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
|- ( A e. CC -> ( n e. NN |-> ( A / n ) ) ~~> 0 )
4.9.6  Arithmetic series
Theorem	arisum 12064*	Arithmetic series sum of the first N positive integers. This is Metamath 100 proof #68. (Contributed by FL, 16-Nov-2006.) (Proof shortened by Mario Carneiro, 22-May-2014.)
|- ( N e. NN0 -> sum_ k e. ( 1 ... N ) k = ( ( ( N ^ 2 ) + N ) / 2 ) )
Theorem	arisum2 12065*	Arithmetic series sum of the first N nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.)
|- ( N e. NN0 -> sum_ k e. ( 0 ... ( N - 1 ) ) k = ( ( ( N ^ 2 ) - N ) / 2 ) )
Theorem	trireciplem 12066	Lemma for trirecip 12067. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
|- F = ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) )   =>    |- seq 1 ( + , F ) ~~> 1
Theorem	trirecip 12067	The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
|- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2
4.9.7  Geometric series
Theorem	expcnvap0 12068*	A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
Theorem	expcnvre 12069*	A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
Theorem	expcnv 12070*	A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
Theorem	explecnv 12071*	A sequence of terms converges to zero when it is less than powers of a number A whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> F e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( abs ` A ) < 1 )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) <_ ( A ^ k ) )   =>    |- ( ph -> F ~~> 0 )
Theorem	geosergap 12072*	The value of the finite geometric series A ^ M + A ^ ( M + 1 ) +... + A ^ ( N - 1 ). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 1 )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )
Theorem	geoserap 12073*	The value of the finite geometric series 1 + A ^ 1 + A ^ 2 +... + A ^ ( N - 1 ). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 1 )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
Theorem	pwm1geoserap1 12074*	The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + A ^ 1 + A ^ 2 +... + A ^ ( N - 1 ). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )
Theorem	absltap 12075	Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` A ) < B )   =>    |- ( ph -> A # B )
Theorem	absgtap 12076	Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> B < ( abs ` A ) )   =>    |- ( ph -> A # B )
Theorem	geolim 12077*	The partial sums in the infinite series 1 + A ^ 1 + A ^ 2... converge to ( 1 / ( 1 - A ) ). (Contributed by NM, 15-May-2006.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   &    |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( A ^ k ) )   =>    |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - A ) ) )
Theorem	geolim2 12078*	The partial sums in the geometric series A ^ M + A ^ ( M + 1 )... converge to ( ( A ^ M ) / ( 1 - A ) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   &    |- ( ph -> M e. NN0 )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( A ^ k ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( ( A ^ M ) / ( 1 - A ) ) )
Theorem	georeclim 12079*	The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ph -> A e. CC )   &    |- ( ph -> 1 < ( abs ` A ) )   &    |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( 1 / A ) ^ k ) )   =>    |- ( ph -> seq 0 ( + , F ) ~~> ( A / ( A - 1 ) ) )
Theorem	geo2sum 12080*	The value of the finite geometric series 2 ^ -u 1 + 2 ^ -u 2 +... + 2 ^ -u N, multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ( N e. NN /\ A e. CC ) -> sum_ k e. ( 1 ... N ) ( A / ( 2 ^ k ) ) = ( A - ( A / ( 2 ^ N ) ) ) )
Theorem	geo2sum2 12081*	The value of the finite geometric series 1 + 2 + 4 + 8 +... + 2 ^ ( N - 1 ). (Contributed by Mario Carneiro, 7-Sep-2016.)
|- ( N e. NN0 -> sum_ k e. ( 0..^ N ) ( 2 ^ k ) = ( ( 2 ^ N ) - 1 ) )
Theorem	geo2lim 12082*	The value of the infinite geometric series 2 ^ -u 1 + 2 ^ -u 2 +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
|- F = ( k e. NN |-> ( A / ( 2 ^ k ) ) )   =>    |- ( A e. CC -> seq 1 ( + , F ) ~~> A )
Theorem	geoisum 12083*	The infinite sum of 1 + A ^ 1 + A ^ 2... is ( 1 / ( 1 - A ) ). (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
Theorem	geoisumr 12084*	The infinite sum of reciprocals 1 + ( 1 / A ) ^ 1 + ( 1 / A ) ^ 2... is A / ( A - 1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> sum_ k e. NN0 ( ( 1 / A ) ^ k ) = ( A / ( A - 1 ) ) )
Theorem	geoisum1 12085*	The infinite sum of A ^ 1 + A ^ 2... is ( A / ( 1 - A ) ). (Contributed by NM, 1-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ( A e. CC /\ ( abs ` A ) < 1 ) -> sum_ k e. NN ( A ^ k ) = ( A / ( 1 - A ) ) )
Theorem	geoisum1c 12086*	The infinite sum of A x. ( R ^ 1 ) + A x. ( R ^ 2 )... is ( A x. R ) / ( 1 - R ). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
|- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( A x. ( R ^ k ) ) = ( ( A x. R ) / ( 1 - R ) ) )
Theorem	0.999... 12087	The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 1 0 ^ 1 + 9 / 1 0 ^ 2 + 9 / 1 0 ^ 3 + ..., is exactly equal to 1. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.)
|- sum_ k e. NN ( 9 / (; 1 0 ^ k ) ) = 1
Theorem	geoihalfsum 12088	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12085. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12087 proves a similar claim for .999... in base 10. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by AV, 9-Jul-2022.)
|- sum_ k e. NN ( 1 / ( 2 ^ k ) ) = 1
4.9.8  Ratio test for infinite series convergence
Theorem	cvgratnnlembern 12089	Lemma for cvgratnn 12097. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )
Theorem	cvgratnnlemnexp 12090*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 15-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( abs ` ( F ` N ) ) <_ ( ( abs ` ( F ` 1 ) ) x. ( A ^ ( N - 1 ) ) ) )
Theorem	cvgratnnlemmn 12091*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 15-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( F ` N ) ) <_ ( ( abs ` ( F ` M ) ) x. ( A ^ ( N - M ) ) ) )
Theorem	cvgratnnlemseq 12092*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
Theorem	cvgratnnlemabsle 12093*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )
Theorem	cvgratnnlemsumlt 12094*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 23-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
Theorem	cvgratnnlemfm 12095*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 23-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
Theorem	cvgratnnlemrate 12096*	Lemma for cvgratnn 12097. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) )
Theorem	cvgratnn 12097*	Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms, then the infinite sum of the terms of F converges to a complex number. Although this theorem is similar to cvgratz 12098 and cvgratgt0 12099, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11915 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq 1 ( + , F ) e. dom ~~> )
Theorem	cvgratz 12098*	Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms, then the infinite sum of the terms of F converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
Theorem	cvgratgt0 12099*	Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms beyond some index B, then the infinite sum of the terms of F converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. W ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
4.9.9  Mertens' theorem
Theorem	mertenslemub 12100*	Lemma for mertensabs 12103. An upper bound for T. (Contributed by Jim Kingdon, 3-Dec-2022.)
|- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }   &    |- ( ph -> X e. T )   &    |- ( ph -> S e. NN )   =>    |- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )