P. 115 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11401-11500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fisumcom2 11401*	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 11402*	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 11403*	Lemma for fisum0diag 11404. (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 11404*	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 11405*	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 11406*	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 11407*	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 11408*	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 11409*	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 11410*	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 11411*	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 11412*	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 11413*	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 11414*	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 11415*	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 11416*	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 11417*	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 11418*	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 11419*	Lemma for modfsummod 11421. (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 11420*	Induction step for modfsummod 11421. (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 11421*	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 11422*	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 11423*	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 11424*	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 11425*	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 11426*	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 11427*	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 11428*	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 11429*	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 11430*	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 11431*	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 11432*	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 11433*	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 11434*	Lemma for fsumre 11435, fsumim 11436, and fsumcj 11437. (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 11435*	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 11436*	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 11437*	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 11438*	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 11439*	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 11440*	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 11441*	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 11442*	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 11443*	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 11444*	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 11445*	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 ) )
4.8.3  The binomial theorem
Theorem	binomlem 11446*	Lemma for binom 11447 (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 11447*	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 11446. 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 11448*	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 11449*	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 11450*	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 11451	Lemma for bcxmas 11452. (Contributed by Paul Chapman, 18-May-2007.)
|- ( A = B -> ( ( N + A ) _C A ) = ( ( N + B ) _C B ) )
Theorem	bcxmas 11452*	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 ) )
4.8.4  Infinite sums (cont.)
Theorem	isumshft 11453*	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 11454*	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 11455*	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 11456*	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 11457*	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 11458*	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 11459*	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 )
4.8.5  Miscellaneous converging and diverging sequences
Theorem	divcnv 11460*	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 )
4.8.6  Arithmetic series
Theorem	arisum 11461*	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 11462*	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 11463	Lemma for trirecip 11464. 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 11464	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
4.8.7  Geometric series
Theorem	expcnvap0 11465*	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 11466*	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 11467*	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 11468*	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 11469*	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 11470*	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 11471*	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 11472	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 11473	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 11474*	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 11475*	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 11476*	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 11477*	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 11478*	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 11479*	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 11480*	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 11481*	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 11482*	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 11483*	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... 11484	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 11485	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 11482. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 11484 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
4.8.8  Ratio test for infinite series convergence
Theorem	cvgratnnlembern 11486	Lemma for cvgratnn 11494. 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 11487*	Lemma for cvgratnn 11494. (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 11488*	Lemma for cvgratnn 11494. (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 11489*	Lemma for cvgratnn 11494. (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 11490*	Lemma for cvgratnn 11494. (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 11491*	Lemma for cvgratnn 11494. (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 11492*	Lemma for cvgratnn 11494. (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 11493*	Lemma for cvgratnn 11494. (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 11494*	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 11495 and cvgratgt0 11496, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11313 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 11495*	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 11496*	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 ~~> )
4.8.9  Mertens' theorem
Theorem	mertenslemub 11497*	Lemma for mertensabs 11500. 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 ) ) )
Theorem	mertenslemi1 11498*	Lemma for mertensabs 11500. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- ( ( ph /\ j e. NN0 ) -> ( F ` j ) = A )   &    |- ( ( ph /\ j e. NN0 ) -> ( K ` j ) = ( abs ` A ) )   &    |- ( ( ph /\ j e. NN0 ) -> A e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( H ` k ) = sum_ j e. ( 0 ... k ) ( A x. ( G ` ( k - j ) ) ) )   &    |- ( ph -> seq 0 ( + , K ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- ( ph -> E e. RR+ )   &    |- T = { z | E. n e. ( 0 ... ( s - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }   &    |- ( ps <-> ( s e. NN /\ A. n e. ( ZZ>= ` s ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) < ( ( E / 2 ) / ( sum_ j e. NN0 ( K ` j ) + 1 ) ) ) )   &    |- ( ph -> P e. RR )   &    |- ( ph -> ( ps /\ ( t e. NN0 /\ A. m e. ( ZZ>= ` t ) ( K ` m ) < ( ( ( E / 2 ) / s ) / ( P + 1 ) ) ) ) )   &    |- ( ph -> 0 <_ P )   &    |- ( ph -> A. w e. T w <_ P )   =>    |- ( ph -> E. y e. NN0 A. m e. ( ZZ>= ` y ) ( abs ` sum_ j e. ( 0 ... m ) ( A x. sum_ k e. ( ZZ>= ` ( ( m - j ) + 1 ) ) B ) ) < E )
Theorem	mertenslem2 11499*	Lemma for mertensabs 11500. (Contributed by Mario Carneiro, 28-Apr-2014.)
|- ( ( ph /\ j e. NN0 ) -> ( F ` j ) = A )   &    |- ( ( ph /\ j e. NN0 ) -> ( K ` j ) = ( abs ` A ) )   &    |- ( ( ph /\ j e. NN0 ) -> A e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( H ` k ) = sum_ j e. ( 0 ... k ) ( A x. ( G ` ( k - j ) ) ) )   &    |- ( ph -> seq 0 ( + , K ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- ( ph -> E e. RR+ )   &    |- T = { z | E. n e. ( 0 ... ( s - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }   &    |- ( ps <-> ( s e. NN /\ A. n e. ( ZZ>= ` s ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) < ( ( E / 2 ) / ( sum_ j e. NN0 ( K ` j ) + 1 ) ) ) )   =>    |- ( ph -> E. y e. NN0 A. m e. ( ZZ>= ` y ) ( abs ` sum_ j e. ( 0 ... m ) ( A x. sum_ k e. ( ZZ>= ` ( ( m - j ) + 1 ) ) B ) ) < E )
Theorem	mertensabs 11500*	Mertens' theorem. If A ( j ) is an absolutely convergent series and B ( k ) is convergent, then ( sum_ j e. NN0 A ( j ) x. sum_ k e. NN0 B ( k ) ) = sum_ k e. NN0 sum_ j e. ( 0 ... k ) ( A ( j ) x. B ( k - j ) ) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- ( ( ph /\ j e. NN0 ) -> ( F ` j ) = A )   &    |- ( ( ph /\ j e. NN0 ) -> ( K ` j ) = ( abs ` A ) )   &    |- ( ( ph /\ j e. NN0 ) -> A e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( H ` k ) = sum_ j e. ( 0 ... k ) ( A x. ( G ` ( k - j ) ) ) )   &    |- ( ph -> seq 0 ( + , K ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , F ) e. dom ~~> )   =>    |- ( ph -> seq 0 ( + , H ) ~~> ( sum_ j e. NN0 A x. sum_ k e. NN0 B ) )