P. 116 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11501-11600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	modfsummod 11501*	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 11502*	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 11503*	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 11504*	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 11505*	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 11506*	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 11507*	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 11508*	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 11509*	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 11510*	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 11511*	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 11512*	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 11513*	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 11514*	Lemma for fsumre 11515, fsumim 11516, and fsumcj 11517. (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 11515*	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 11516*	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 11517*	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 11518*	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 11519*	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 11520*	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 11521*	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 11522*	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 11523*	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 11524*	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 11525*	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 11526*	Lemma for binom 11527 (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 11527*	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 11526. 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 11528*	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 11529*	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 11530*	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 11531	Lemma for bcxmas 11532. (Contributed by Paul Chapman, 18-May-2007.)
|- ( A = B -> ( ( N + A ) _C A ) = ( ( N + B ) _C B ) )
Theorem	bcxmas 11532*	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 11533*	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 11534*	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 11535*	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 11536*	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 11537*	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 11538*	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 11539*	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 11540*	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 11541*	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 11542*	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 11543	Lemma for trirecip 11544. 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 11544	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 11545*	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 11546*	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 11547*	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 11548*	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 11549*	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 11550*	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 11551*	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 11552	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 11553	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 11554*	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 11555*	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 11556*	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 11557*	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 11558*	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 11559*	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 11560*	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 11561*	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 11562*	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 11563*	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... 11564	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 11565	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 11562. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 11564 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 11566	Lemma for cvgratnn 11574. 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 11567*	Lemma for cvgratnn 11574. (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 11568*	Lemma for cvgratnn 11574. (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 11569*	Lemma for cvgratnn 11574. (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 11570*	Lemma for cvgratnn 11574. (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 11571*	Lemma for cvgratnn 11574. (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 11572*	Lemma for cvgratnn 11574. (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 11573*	Lemma for cvgratnn 11574. (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 11574*	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 11575 and cvgratgt0 11576, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11393 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 11575*	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 11576*	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 11577*	Lemma for mertensabs 11580. 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 11578*	Lemma for mertensabs 11580. (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 11579*	Lemma for mertensabs 11580. (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 11580*	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 ) )
4.8.10  Finite and infinite products
4.8.10.1  Product sequences
Theorem	prodf 11581*	An infinite product of complex terms is a function from an upper set of integers to CC. (Contributed by Scott Fenton, 4-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> seq M ( x. , F ) : Z --> CC )
Theorem	clim2prod 11582*	The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> A )   =>    |- ( ph -> seq M ( x. , F ) ~~> ( ( seq M ( x. , F ) ` N ) x. A ) )
Theorem	clim2divap 11583*	The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ph -> seq M ( x. , F ) ~~> A )   &    |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )   =>    |- ( ph -> seq ( N + 1 ) ( x. , F ) ~~> ( A / ( seq M ( x. , F ) ` N ) ) )
Theorem	prod3fmul 11584*	The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , G ) ` N ) ) )
Theorem	prodf1 11585	The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)
|- Z = ( ZZ>= ` M )   =>    |- ( N e. Z -> ( seq M ( x. , ( Z X. { 1 } ) ) ` N ) = 1 )
Theorem	prodf1f 11586	A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> seq M ( x. , ( Z X. { 1 } ) ) = ( Z X. { 1 } ) )
Theorem	prodfclim1 11587	The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- Z = ( ZZ>= ` M )   =>    |- ( M e. ZZ -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )
Theorem	prodfap0 11588*	The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )   =>    |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
Theorem	prodfrecap 11589*	The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   =>    |- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
Theorem	prodfdivap 11590*	The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) # 0 )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) / ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )
4.8.10.2  Non-trivial convergence
Theorem	ntrivcvgap 11591*	A non-trivially converging infinite product converges. (Contributed by Scott Fenton, 18-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   =>    |- ( ph -> seq M ( x. , F ) e. dom ~~> )
Theorem	ntrivcvgap0 11592*	A product that converges to a value apart from zero converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> seq M ( x. , F ) ~~> X )   &    |- ( ph -> X # 0 )   =>    |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )
4.8.10.3  Complex products
Syntax	cprod 11593	Extend class notation to include complex products.
class prod_ k e. A B
Definition	df-proddc 11594*	Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 11397 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.)
|- prod_ k e. A B = ( iota x ( E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( E. n e. ( ZZ>= ` m ) E. y ( y # 0 /\ seq n ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) ) ) )
Theorem	prodeq1f 11595	Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
|- F/_ k A   &    |- F/_ k B   =>    |- ( A = B -> prod_ k e. A C = prod_ k e. B C )
Theorem	prodeq1 11596*	Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
|- ( A = B -> prod_ k e. A C = prod_ k e. B C )
Theorem	nfcprod1 11597*	Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F/_ k A   =>    |- F/_ k prod_ k e. A B
Theorem	nfcprod 11598*	Bound-variable hypothesis builder for product: if x is (effectively) not free in A and B, it is not free in prod_ k e. A B. (Contributed by Scott Fenton, 1-Dec-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x prod_ k e. A B
Theorem	prodeq2w 11599*	Equality theorem for product, when the class expressions B and C are equal everywhere. Proved using only Extensionality. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( A. k B = C -> prod_ k e. A B = prod_ k e. A C )
Theorem	prodeq2 11600*	Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( A. k e. A B = C -> prod_ k e. A B = prod_ k e. A C )