P. 124 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12301-12400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	geo2sum2 12301*	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 12302*	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 12303*	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 12304*	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 12305*	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 12306*	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... 12307	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 12308	Prove that the infinite geometric series of 1/2, 1/2 + 1/4 + 1/8 + ... = 1. Uses geoisum1 12305. This is a representation of .111... in binary with an infinite number of 1's. Theorem 0.999... 12307 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 12309	Lemma for cvgratnn 12317. 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 12310*	Lemma for cvgratnn 12317. (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 12311*	Lemma for cvgratnn 12317. (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 12312*	Lemma for cvgratnn 12317. (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 12313*	Lemma for cvgratnn 12317. (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 12314*	Lemma for cvgratnn 12317. (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 12315*	Lemma for cvgratnn 12317. (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 12316*	Lemma for cvgratnn 12317. (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 12317*	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 12318 and cvgratgt0 12319, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 12135 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 12318*	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 12319*	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 12320*	Lemma for mertensabs 12323. 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 12321*	Lemma for mertensabs 12323. (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 12322*	Lemma for mertensabs 12323. (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 12323*	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 12324*	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 12325*	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 12326*	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 12327*	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 12328	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 12329	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 12330	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 12331*	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 12332*	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 12333*	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 12334*	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 12335*	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 12336	Extend class notation to include complex products.
class prod_ k e. A B
Definition	df-proddc 12337*	Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 12139 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 12338	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 12339*	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 12340*	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 12341*	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 12342*	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 12343*	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 12344*	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 12345*	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 12346*	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 12347*	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 12348*	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 12349*	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 12350*	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 12351*	Equality deduction for product. Note that unlike prodeq2dv 12352, 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 12352*	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 12353*	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 12354*	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 12355*	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 12356*	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 12357*	Lemma for prodrbdc 12360. (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 12358*	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 12359*	Lemma for prodrbdc 12360. (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 12360*	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 12361*	Lemma for prodmodc 12364. (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 12362*	Lemma for prodmodc 12364. (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 12363*	Lemma for prodmodc 12364. (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 12364*	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 12365*	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 12366*	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 12367*	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 12368*	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 12369*	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 12370*	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 12371	A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- prod_ k e. (/) A = 1
Theorem	prod1dc 12372*	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 12373*	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 12374*	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 12375*	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 12376*	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 12377*	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 12378*	A product of a singleton is the term. A version of prodsn 12379 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 12379*	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 12380*	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 )
Theorem	climprod1 12381	The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   =>    |- ( ph -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )
Theorem	fprodsplitdc 12382*	Split a finite product into two parts. New proofs should use fprodsplit 12383 which is the same but with one fewer hypothesis. (Contributed by Scott Fenton, 16-Dec-2017.) (New usage is discouraged.)
|- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ph -> A. j e. U DECID j e. A )   &    |- ( ( ph /\ k e. U ) -> C e. CC )   =>    |- ( ph -> prod_ k e. U C = ( prod_ k e. A C x. prod_ k e. B C ) )
Theorem	fprodsplit 12383*	Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> U = ( A u. B ) )   &    |- ( ph -> U e. Fin )   &    |- ( ( ph /\ k e. U ) -> C e. CC )   =>    |- ( ph -> prod_ k e. U C = ( prod_ k e. A C x. prod_ k e. B C ) )
Theorem	fprodm1 12384*	Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = N -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )
Theorem	fprod1p 12385*	Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   &    |- ( k = M -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( B x. prod_ k e. ( ( M + 1 ) ... N ) A ) )
Theorem	fprodp1 12386*	Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> prod_ k e. ( M ... ( N + 1 ) ) A = ( prod_ k e. ( M ... N ) A x. B ) )
Theorem	fprodm1s 12387*	Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )
Theorem	fprodp1s 12388*	Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. CC )   =>    |- ( ph -> prod_ k e. ( M ... ( N + 1 ) ) A = ( prod_ k e. ( M ... N ) A x. [_ ( N + 1 ) / k ]_ A ) )
Theorem	prodsns 12389*	A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)
|- ( ( M e. V /\ [_ M / k ]_ A e. CC ) -> prod_ k e. { M } A = [_ M / k ]_ A )
Theorem	fprodunsn 12390*	Multiply in an additional term in a finite product. See also fprodsplitsn 12419 which is the same but with a F/ k ph hypothesis in place of the distinct variable condition between ph and k. (Contributed by Jim Kingdon, 16-Aug-2024.)
|- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( k = B -> C = D )   =>    |- ( ph -> prod_ k e. ( A u. { B } ) C = ( prod_ k e. A C x. D ) )
Theorem	fprodcl2lem 12391*	Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodcllem 12392*	Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> 1 e. S )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodcl 12393*	Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> prod_ k e. A B e. CC )
Theorem	fprodrecl 12394*	Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   =>    |- ( ph -> prod_ k e. A B e. RR )
Theorem	fprodzcl 12395*	Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   =>    |- ( ph -> prod_ k e. A B e. ZZ )
Theorem	fprodnncl 12396*	Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. NN )   =>    |- ( ph -> prod_ k e. A B e. NN )
Theorem	fprodrpcl 12397*	Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR+ )   =>    |- ( ph -> prod_ k e. A B e. RR+ )
Theorem	fprodnn0cl 12398*	Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. NN0 )   =>    |- ( ph -> prod_ k e. A B e. NN0 )
Theorem	fprodcllemf 12399*	Finite product closure lemma. A version of fprodcllem 12392 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> 1 e. S )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodreclf 12400*	Closure of a finite product of real numbers. A version of fprodrecl 12394 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   =>    |- ( ph -> prod_ k e. A B e. RR )