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	isumle 12001*	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 12002*	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 12003*	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 12004*	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 12005*	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 12006	Lemma for trirecip 12007. 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 12007	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 12008*	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 12009*	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 12010*	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 12011*	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 12012*	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 12013*	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 12014*	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 12015	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 12016	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 12017*	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 12018*	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 12019*	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 12020*	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 12021*	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 12022*	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 12023*	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 12024*	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 12025*	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 12026*	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... 12027	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 12028	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12025. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12027 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 12029	Lemma for cvgratnn 12037. 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 12030*	Lemma for cvgratnn 12037. (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 12031*	Lemma for cvgratnn 12037. (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 12032*	Lemma for cvgratnn 12037. (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 12033*	Lemma for cvgratnn 12037. (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 12034*	Lemma for cvgratnn 12037. (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 12035*	Lemma for cvgratnn 12037. (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 12036*	Lemma for cvgratnn 12037. (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 12037*	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 12038 and cvgratgt0 12039, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11856 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 12038*	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 12039*	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 12040*	Lemma for mertensabs 12043. 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 12041*	Lemma for mertensabs 12043. (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 12042*	Lemma for mertensabs 12043. (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 12043*	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 ) )
4.9.10  Finite and infinite products
4.9.10.1  Product sequences
Theorem	prodf 12044*	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 12045*	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 12046*	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 12047*	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 12048	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 12049	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 12050	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 12051*	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 12052*	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 12053*	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 ) ) )
4.9.10.2  Non-trivial convergence
Theorem	ntrivcvgap 12054*	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 12055*	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 ) )
4.9.10.3  Complex products
Syntax	cprod 12056	Extend class notation to include complex products.
class prod_ k e. A B
Definition	df-proddc 12057*	Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 11860 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 12058	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 12059*	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 12060*	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 12061*	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 12062*	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 12063*	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 )
Theorem	cbvprod 12064*	Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( j = k -> B = C )   &    |- F/_ k A   &    |- F/_ j A   &    |- F/_ k B   &    |- F/_ j C   =>    |- prod_ j e. A B = prod_ k e. A C
Theorem	cbvprodv 12065*	Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( j = k -> B = C )   =>    |- prod_ j e. A B = prod_ k e. A C
Theorem	cbvprodi 12066*	Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F/_ k B   &    |- F/_ j C   &    |- ( j = k -> B = C )   =>    |- prod_ j e. A B = prod_ k e. A C
Theorem	prodeq1i 12067*	Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- A = B   =>    |- prod_ k e. A C = prod_ k e. B C
Theorem	prodeq2i 12068*	Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( k e. A -> B = C )   =>    |- prod_ k e. A B = prod_ k e. A C
Theorem	prodeq12i 12069*	Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- A = B   &    |- ( k e. A -> C = D )   =>    |- prod_ k e. A C = prod_ k e. B D
Theorem	prodeq1d 12070*	Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	prodeq2d 12071*	Equality deduction for product. Note that unlike prodeq2dv 12072, k may occur in ph. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> A. k e. A B = C )   =>    |- ( ph -> prod_ k e. A B = prod_ k e. A C )
Theorem	prodeq2dv 12072*	Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ( ph /\ k e. A ) -> B = C )   =>    |- ( ph -> prod_ k e. A B = prod_ k e. A C )
Theorem	prodeq2sdv 12073*	Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> B = C )   =>    |- ( ph -> prod_ k e. A B = prod_ k e. A C )
Theorem	2cprodeq2dv 12074*	Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ( ph /\ j e. A /\ k e. B ) -> C = D )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ j e. A prod_ k e. B D )
Theorem	prodeq12dv 12075*	Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> A = B )   &    |- ( ( ph /\ k e. A ) -> C = D )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B D )
Theorem	prodeq12rdv 12076*	Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> A = B )   &    |- ( ( ph /\ k e. B ) -> C = D )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B D )
Theorem	prodrbdclem 12077*	Lemma for prodrbdc 12080. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) )
Theorem	fproddccvg 12078*	The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) )
Theorem	prodrbdclem2 12079*	Lemma for prodrbdc 12080. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( ZZ>= ` N ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )   =>    |- ( ( ph /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ~~> C <-> seq N ( x. , F ) ~~> C ) )
Theorem	prodrbdc 12080*	Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( ZZ>= ` N ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )   =>    |- ( ph -> ( seq M ( x. , F ) ~~> C <-> seq N ( x. , F ) ~~> C ) )
Theorem	prodmodclem3 12081*	Lemma for prodmodc 12084. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( f ` j ) / k ]_ B , 1 ) )   &    |- H = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( K ` j ) / k ]_ B , 1 ) )   &    |- ( ph -> ( M e. NN /\ N e. NN ) )   &    |- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )   =>    |- ( ph -> ( seq 1 ( x. , G ) ` M ) = ( seq 1 ( x. , H ) ` N ) )
Theorem	prodmodclem2a 12082*	Lemma for prodmodc 12084. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( f ` j ) / k ]_ B , 1 ) )   &    |- H = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( K ` j ) / k ]_ B , 1 ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> f : ( 1 ... N ) -1-1-onto-> A )   &    |- ( ph -> K Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   =>    |- ( ph -> seq M ( x. , F ) ~~> ( seq 1 ( x. , G ) ` N ) )
Theorem	prodmodclem2 12083*	Lemma for prodmodc 12084. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( f ` j ) / k ]_ B , 1 ) )   =>    |- ( ( ph /\ 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. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
Theorem	prodmodc 12084*	A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.) (Modified by Jim Kingdon, 14-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( f ` j ) / k ]_ B , 1 ) )   =>    |- ( ph -> E* 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. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , G ) ` m ) ) ) )
Theorem	zproddc 12085*	Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )   &    |- ( ph -> A C_ Z )   &    |- ( ph -> A. j e. Z DECID j e. A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B = ( ~~> ` seq M ( x. , F ) ) )
Theorem	iprodap 12086*	Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> prod_ k e. Z B = ( ~~> ` seq M ( x. , F ) ) )
Theorem	zprodap0 12087*	Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> X # 0 )   &    |- ( ph -> seq M ( x. , F ) ~~> X )   &    |- ( ph -> A. j e. Z DECID j e. A )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B = X )
Theorem	iprodap0 12088*	Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> X # 0 )   &    |- ( ph -> seq M ( x. , F ) ~~> X )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> prod_ k e. Z B = X )
4.9.10.4  Finite products
Theorem	fprodseq 12089*	The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
|- ( k = ( F ` n ) -> B = C )   &    |- ( ph -> M e. NN )   &    |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )   =>    |- ( ph -> prod_ k e. A B = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ M , ( G ` n ) , 1 ) ) ) ` M ) )
Theorem	fprodntrivap 12090*	A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> E. n e. Z E. y ( y # 0 /\ seq n ( x. , ( k e. Z |-> if ( k e. A , B , 1 ) ) ) ~~> y ) )
Theorem	prod0 12091	A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- prod_ k e. (/) A = 1
Theorem	prod1dc 12092*	Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)
|- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> prod_ k e. A 1 = 1 )
Theorem	prodfct 12093*	A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
|- ( A. k e. A B e. CC -> prod_ j e. A ( ( k e. A |-> B ) ` j ) = prod_ k e. A B )
Theorem	fprodf1o 12094*	Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
|- ( k = G -> B = D )   &    |- ( ph -> C e. Fin )   &    |- ( ph -> F : C -1-1-onto-> A )   &    |- ( ( ph /\ n e. C ) -> ( F ` n ) = G )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B = prod_ n e. C D )
Theorem	prodssdc 12095*	Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> E. n e. ( ZZ>= ` M ) E. y ( y # 0 /\ seq n ( x. , ( k e. ( ZZ>= ` M ) |-> if ( k e. B , C , 1 ) ) ) ~~> y ) )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. A )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. B )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	fprodssdc 12096*	Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> A. j e. B DECID j e. A )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	fprodmul 12097*	The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   =>    |- ( ph -> prod_ k e. A ( B x. C ) = ( prod_ k e. A B x. prod_ k e. A C ) )
Theorem	prodsnf 12098*	A product of a singleton is the term. A version of prodsn 12099 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/_ k B   &    |- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = B )
Theorem	prodsn 12099*	A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( k = M -> A = B )   =>    |- ( ( M e. V /\ B e. CC ) -> prod_ k e. { M } A = B )
Theorem	fprod1 12100*	A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( k = M -> A = B )   =>    |- ( ( M e. ZZ /\ B e. CC ) -> prod_ k e. ( M ... M ) A = B )