P. 122 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 12101-12200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prod3fmul 12101*	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 12102	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 12103	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 12104	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 12105*	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 12106*	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 12107*	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 12108*	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 12109*	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 12110	Extend class notation to include complex products.
class prod_ k e. A B
Definition	df-proddc 12111*	Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 11914 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 12112	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 12113*	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 12114*	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 12115*	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 12116*	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 12117*	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 12118*	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 12119*	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 12120*	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 12121*	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 12122*	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 12123*	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 12124*	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 12125*	Equality deduction for product. Note that unlike prodeq2dv 12126, 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 12126*	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 12127*	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 12128*	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 12129*	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 12130*	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 12131*	Lemma for prodrbdc 12134. (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 12132*	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 12133*	Lemma for prodrbdc 12134. (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 12134*	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 12135*	Lemma for prodmodc 12138. (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 12136*	Lemma for prodmodc 12138. (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 12137*	Lemma for prodmodc 12138. (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 12138*	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 12139*	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 12140*	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 12141*	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 12142*	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 12143*	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 12144*	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 12145	A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- prod_ k e. (/) A = 1
Theorem	prod1dc 12146*	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 12147*	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 12148*	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 12149*	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 12150*	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 12151*	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 12152*	A product of a singleton is the term. A version of prodsn 12153 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 12153*	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 12154*	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 12155	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 12156*	Split a finite product into two parts. New proofs should use fprodsplit 12157 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 12157*	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 12158*	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 12159*	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 12160*	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 12161*	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 12162*	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 12163*	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 12164*	Multiply in an additional term in a finite product. See also fprodsplitsn 12193 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 12165*	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 12166*	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 12167*	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 12168*	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 12169*	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 12170*	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 12171*	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 12172*	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 12173*	Finite product closure lemma. A version of fprodcllem 12166 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 12174*	Closure of a finite product of real numbers. A version of fprodrecl 12168 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 )
Theorem	fprodfac 12175*	Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )
Theorem	fprodabs 12176*	The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   =>    |- ( ph -> ( abs ` prod_ k e. ( M ... N ) A ) = prod_ k e. ( M ... N ) ( abs ` A ) )
Theorem	fprodeq0 12177*	Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ( ph /\ k = N ) -> A = 0 )   =>    |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )
Theorem	fprodshft 12178*	Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( k - K ) -> A = B )   =>    |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( M + K ) ... ( N + K ) ) B )
Theorem	fprodrev 12179*	Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )   &    |- ( j = ( K - k ) -> A = B )   =>    |- ( ph -> prod_ j e. ( M ... N ) A = prod_ k e. ( ( K - N ) ... ( K - M ) ) B )
Theorem	fprodconst 12180*	The product of constant terms ( k is not free in B). (Contributed by Scott Fenton, 12-Jan-2018.)
|- ( ( A e. Fin /\ B e. CC ) -> prod_ k e. A B = ( B ^ (♯ ` A ) ) )
Theorem	fprodap0 12181*	A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A B # 0 )
Theorem	fprod2dlemstep 12182*	Lemma for fprod2d 12183- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   &    |- ( ph -> -. y e. x )   &    |- ( ph -> ( x u. { y } ) C_ A )   &    |- ( ph -> x e. Fin )   &    |- ( ps <-> prod_ j e. x prod_ k e. B C = prod_ z e. U_ j e. x ( { j } X. B ) D )   =>    |- ( ( ph /\ ps ) -> prod_ j e. ( x u. { y } ) prod_ k e. B C = prod_ z e. U_ j e. ( x u. { y } ) ( { j } X. B ) D )
Theorem	fprod2d 12183*	Write a double product as a product over a two-dimensional region. Compare fsum2d 11995. (Contributed by Scott Fenton, 30-Jan-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ z e. U_ j e. A ( { j } X. B ) D )
Theorem	fprodxp 12184*	Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
|- ( z = <. j , k >. -> D = C )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ z e. ( A X. B ) D )
Theorem	fprodcnv 12185*	Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
|- ( x = <. j , k >. -> B = D )   &    |- ( y = <. k , j >. -> C = D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> Rel A )   &    |- ( ( ph /\ x e. A ) -> B e. CC )   =>    |- ( ph -> prod_ x e. A B = prod_ y e. `' A C )
Theorem	fprodcom2fi 12186*	Interchange order of multiplication. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ j e. A ) -> B e. Fin )   &    |- ( ( ph /\ k e. C ) -> D e. Fin )   &    |- ( ph -> ( ( j e. A /\ k e. B ) <-> ( k e. C /\ j e. D ) ) )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> E e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B E = prod_ k e. C prod_ j e. D E )
Theorem	fprodcom 12187*	Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. CC )   =>    |- ( ph -> prod_ j e. A prod_ k e. B C = prod_ k e. B prod_ j e. A C )
Theorem	fprod0diagfz 12188*	Two ways to express "the product of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N. Compare fisum0diag 12001. (Contributed by Scott Fenton, 2-Feb-2018.)
|- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> prod_ j e. ( 0 ... N ) prod_ k e. ( 0 ... ( N - j ) ) A = prod_ k e. ( 0 ... N ) prod_ j e. ( 0 ... ( N - k ) ) A )
Theorem	fprodrec 12189*	The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A ( 1 / B ) = ( 1 / prod_ k e. A B ) )
Theorem	fproddivap 12190*	The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> C # 0 )   =>    |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )
Theorem	fproddivapf 12191*	The quotient of two finite products. A version of fproddivap 12190 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. CC )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ( ph /\ k e. A ) -> C # 0 )   =>    |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )
Theorem	fprodsplitf 12192*	Split a finite product into two parts. A version of fprodsplit 12157 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( 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	fprodsplitsn 12193*	Separate out a term in a finite product. See also fprodunsn 12164 which is the same but with a distinct variable condition in place of F/ k ph. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( k = B -> C = D )   &    |- ( ph -> D e. CC )   =>    |- ( ph -> prod_ k e. ( A u. { B } ) C = ( prod_ k e. A C x. D ) )
Theorem	fprodsplit1f 12194*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> F/_ k D )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ k = C ) -> B = D )   =>    |- ( ph -> prod_ k e. A B = ( D x. prod_ k e. ( A \ { C } ) B ) )
Theorem	fprodclf 12195*	Closure of a finite product of complex numbers. A version of fprodcl 12167 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. CC )   =>    |- ( ph -> prod_ k e. A B e. CC )
Theorem	fprodap0f 12196*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12181 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A B # 0 )
Theorem	fprodge0 12197*	If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   =>    |- ( ph -> 0 <_ prod_ k e. A B )
Theorem	fprodeq0g 12198*	Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ k = C ) -> B = 0 )   =>    |- ( ph -> prod_ k e. A B = 0 )
Theorem	fprodge1 12199*	If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 1 <_ B )   =>    |- ( ph -> 1 <_ prod_ k e. A B )
Theorem	fprodle 12200*	If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. RR )   &    |- ( ( ph /\ k e. A ) -> 0 <_ B )   &    |- ( ( ph /\ k e. A ) -> C e. RR )   &    |- ( ( ph /\ k e. A ) -> B <_ C )   =>    |- ( ph -> prod_ k e. A B <_ prod_ k e. A C )