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	prodeq2w 12301*	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 12302*	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 12303*	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 12304*	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 12305*	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 12306*	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 12307*	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 12308*	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 12309*	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 12310*	Equality deduction for product. Note that unlike prodeq2dv 12311, 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 12311*	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 12312*	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 12313*	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 12314*	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 12315*	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 12316*	Lemma for prodrbdc 12319. (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 12317*	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 12318*	Lemma for prodrbdc 12319. (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 12319*	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 12320*	Lemma for prodmodc 12323. (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 12321*	Lemma for prodmodc 12323. (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 12322*	Lemma for prodmodc 12323. (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 12323*	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 12324*	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 12325*	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 12326*	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 12327*	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 12328*	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 12329*	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 12330	A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- prod_ k e. (/) A = 1
Theorem	prod1dc 12331*	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 12332*	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 12333*	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 12334*	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 12335*	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 12336*	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 12337*	A product of a singleton is the term. A version of prodsn 12338 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 12338*	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 12339*	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 12340	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 12341*	Split a finite product into two parts. New proofs should use fprodsplit 12342 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 12342*	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 12343*	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 12344*	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 12345*	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 12346*	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 12347*	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 12348*	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 12349*	Multiply in an additional term in a finite product. See also fprodsplitsn 12378 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 12350*	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 12351*	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 12352*	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 12353*	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 12354*	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 12355*	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 12356*	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 12357*	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 12358*	Finite product closure lemma. A version of fprodcllem 12351 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 12359*	Closure of a finite product of real numbers. A version of fprodrecl 12353 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 12360*	Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )
Theorem	fprodabs 12361*	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 12362*	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 12363*	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 12364*	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 12365*	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 12366*	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 12367*	Lemma for fprod2d 12368- 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 12368*	Write a double product as a product over a two-dimensional region. Compare fsum2d 12180. (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 12369*	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 12370*	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 12371*	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 12372*	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 12373*	Two ways to express "the product of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N. Compare fisum0diag 12186. (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 12374*	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 12375*	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 12376*	The quotient of two finite products. A version of fproddivap 12375 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 12377*	Split a finite product into two parts. A version of fprodsplit 12342 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 12378*	Separate out a term in a finite product. See also fprodunsn 12349 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 12379*	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 12380*	Closure of a finite product of complex numbers. A version of fprodcl 12352 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 12381*	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12366 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 12382*	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 12383*	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 12384*	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 12385*	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 )
Theorem	fprodmodd 12386*	If all factors of two finite products are equal modulo M, the products are equal modulo M. (Contributed by AV, 7-Jul-2021.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. ZZ )   &    |- ( ( ph /\ k e. A ) -> C e. ZZ )   &    |- ( ph -> M e. NN )   &    |- ( ( ph /\ k e. A ) -> ( B mod M ) = ( C mod M ) )   =>    |- ( ph -> ( prod_ k e. A B mod M ) = ( prod_ k e. A C mod M ) )
4.10  Elementary trigonometry
4.10.1  The exponential, sine, and cosine functions
Syntax	ce 12387	Extend class notation to include the exponential function.
class exp
Syntax	ceu 12388	Extend class notation to include Euler's constant _e = 2.71828....
class _e
Syntax	csin 12389	Extend class notation to include the sine function.
class sin
Syntax	ccos 12390	Extend class notation to include the cosine function.
class cos
Syntax	ctan 12391	Extend class notation to include the tangent function.
class tan
Syntax	cpi 12392	Extend class notation to include the constant pi, pi = 3.14159....
class pi
Definition	df-ef 12393*	Define the exponential function. Its value at the complex number A is ( exp ` A ) and is called the "exponential of A"; see efval 12406. (Contributed by NM, 14-Mar-2005.)
|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
Definition	df-e 12394	Define Euler's constant _e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
|- _e = ( exp ` 1 )
Definition	df-sin 12395	Define the sine function. (Contributed by NM, 14-Mar-2005.)
|- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
Definition	df-cos 12396	Define the cosine function. (Contributed by NM, 14-Mar-2005.)
|- cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )
Definition	df-tan 12397	Define the tangent function. We define it this way for cmpt 4187, which requires the form ( x e. A |-> B ). (Contributed by Mario Carneiro, 14-Mar-2014.)
|- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )
Definition	df-pi 12398	Define the constant pi, pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
|- pi = inf ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , < )
Theorem	eftcl 12399	Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
|- ( ( A e. CC /\ K e. NN0 ) -> ( ( A ^ K ) / ( ! ` K ) ) e. CC )
Theorem	reeftcl 12400	The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
|- ( ( A e. RR /\ K e. NN0 ) -> ( ( A ^ K ) / ( ! ` K ) ) e. RR )