P. 138 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prodeq2w 13701*	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	prodeq2ii 13702*	Equality theorem for product, with the class expressions B and C guarded by _I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( A. k e. A ( _I ` B ) = ( _I ` C ) -> prod_ k e. A B = prod_ k e. A C )
Theorem	prodeq2 13703*	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 13704*	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 13705*	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 13706*	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 13707*	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 13708*	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 13709*	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 13710*	Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> A = B )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	prodeq2d 13711*	Equality deduction for sum. Note that unlike prodeq2dv 13712, 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 13712*	Equality deduction for sum. (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 13713*	Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
|- ( ph -> B = C )   =>    |- ( ph -> prod_ k e. A B = prod_ k e. A C )
Theorem	2cprodeq2dv 13714*	Equality deduction for double sum. (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 13715*	Equality deduction for sum. (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 13716*	Equality deduction for sum. (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	prod2id 13717*	The second class argument to a sum can be chosen so that it is always a set. (Contributed by Scott Fenton, 4-Dec-2017.)
|- prod_ k e. A B = prod_ k e. A ( _I ` B )
Theorem	prodrblem 13718*	Lemma for prodrb 13721. (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 -> N e. ( ZZ>= ` M ) )   =>    |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) )
Theorem	fprodcvg 13719*	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 -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( x. , F ) ~~> ( seq M ( x. , F ) ` N ) )
Theorem	prodrblem2 13720*	Lemma for prodrb 13721. (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 /\ N e. ( ZZ>= ` M ) ) -> ( seq M ( x. , F ) ~~> C <-> seq N ( x. , F ) ~~> C ) )
Theorem	prodrb 13721*	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 -> ( seq M ( x. , F ) ~~> C <-> seq N ( x. , F ) ~~> C ) )
Theorem	prodmolem3 13722*	Lemma for prodmo 13725. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )   &    |- H = ( j e. NN |-> [_ ( K ` j ) / k ]_ B )   &    |- ( 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	prodmolem2a 13723*	Lemma for prodmo 13725. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )   &    |- H = ( j e. NN |-> [_ ( K ` j ) / k ]_ B )   &    |- ( 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	prodmolem2 13724*	Lemma for prodmo 13725. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )   =>    |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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	prodmo 13725*	A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> [_ ( f ` j ) / k ]_ B )   =>    |- ( ph -> E* x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ 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	zprod 13726*	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 /\ 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	iprod 13727*	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	zprodn0 13728*	Non-zero 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 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	iprodn0 13729*	Non-zero 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 )
5.9.12.4  Finite products
Theorem	fprod 13730*	The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.)
|- ( 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. , G ) ` M ) )
Theorem	fprodntriv 13731*	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 13732	A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
|- prod_ k e. (/) A = 1
Theorem	prod1 13733*	Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.)
|- ( ( A C_ ( ZZ>= ` M ) \/ A e. Fin ) -> prod_ k e. A 1 = 1 )
Theorem	prodfc 13734*	A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
|- prod_ j e. A ( ( k e. A |-> B ) ` j ) = prod_ k e. A B
Theorem	fprodf1o 13735*	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	prodss 13736*	Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.)
|- ( 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 /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B C_ ( ZZ>= ` M ) )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	fprodss 13737*	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 /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
Theorem	fprodser 13738*	A finite product expressed in terms of a partial product of an infinite sequence. The recursive definition of a finite product follows from here. (Contributed by Scott Fenton, 14-Dec-2017.)
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )   =>    |- ( ph -> prod_ k e. ( M ... N ) A = ( seq M ( x. , F ) ` N ) )
Theorem	fprodcl2lem 13739*	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 -> A =/= (/) )   =>    |- ( ph -> prod_ k e. A B e. S )
Theorem	fprodcllem 13740*	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 13741*	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 13742*	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 13743*	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 13744*	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 13745*	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 13746*	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	fprodmul 13747*	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	fproddiv 13748*	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	prodsn 13749*	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 13750*	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 13751	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	fprodsplit 13752*	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 13753*	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 13754*	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 13755*	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 13756*	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 13757*	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 13758*	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	fprodfac 13759*	Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k )
Theorem	fprodabs 13760*	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 13761*	Anything 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 13762*	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 13763*	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 13764*	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	fprodn0 13765*	A finite product of non-zero terms is non-zero. (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	fprod2dlem 13766*	Lemma for fprod2d 13767- 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 )   &    |- ( 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 13767*	Write a double product as a product over a two-dimensional region. Compare fsum2d 13568. (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 13768*	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 13769*	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	fprodcom2 13770*	Interchange order of multiplication. Note that B ( j ) and D ( k ) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.)
|- ( ph -> A e. Fin )   &    |- ( ph -> C e. Fin )   &    |- ( ( ph /\ j e. A ) -> B 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 13771*	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	fprod0diag 13772*	Two ways to express "the product of A ( j , k ) over the triangular region M <_ j, M <_ k, j + k <_ N. Compare fsum0diag 13574. (Contributed by Scott Fenton, 2-Feb-2018.)
|- ( ( ph /\ ( j e. ( 0 ... N ) /\ k e. ( 0 ... ( N - j ) ) ) ) -> A e. CC )   =>    |- ( 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 )
5.9.12.5  Infinite products
Theorem	iprodclim 13773*	An infinite product equals the value its sequence converges to. (Contributed by Scott Fenton, 18-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 ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( x. , F ) ~~> B )   =>    |- ( ph -> prod_ k e. Z A = B )
Theorem	iprodclim2 13774*	A converging product converges to its infinite product. (Contributed by Scott Fenton, 18-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 ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   =>    |- ( ph -> seq M ( x. , F ) ~~> prod_ k e. Z A )
Theorem	iprodclim3 13775*	The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that j must not occur in A. (Contributed by Scott Fenton, 18-Dec-2017.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , ( k e. Z |-> A ) ) ~~> y ) )   &    |- ( ph -> F e. dom ~~> )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ( ph /\ j e. Z ) -> ( F ` j ) = prod_ k e. ( M ... j ) A )   =>    |- ( ph -> F ~~> prod_ k e. Z A )
Theorem	iprodcl 13776*	The product of a non-trivially converging infinite sequence is a complex number. (Contributed by Scott Fenton, 18-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 ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   =>    |- ( ph -> prod_ k e. Z A e. CC )
Theorem	iprodrecl 13777*	The product of a non-trivially converging infinite real sequence is a real number. (Contributed by Scott Fenton, 18-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 ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. RR )   =>    |- ( ph -> prod_ k e. Z A e. RR )
Theorem	iprodmul 13778*	Multiplication of infinite sums. (Contributed by Scott Fenton, 18-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 ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. Z ) -> B e. CC )   =>    |- ( ph -> prod_ k e. Z ( A x. B ) = ( prod_ k e. Z A x. prod_ k e. Z B ) )
5.10  Elementary trigonometry
5.10.1  The exponential, sine, and cosine functions
Syntax	ce 13779	Extend class notation to include the exponential function.
class exp
Syntax	ceu 13780	Extend class notation to include Euler's constant = 2.7182818....
class _e
Syntax	csin 13781	Extend class notation to include the sine function.
class sin
Syntax	ccos 13782	Extend class notation to include the cosine function.
class cos
Syntax	ctan 13783	Extend class notation to include the tangent function.
class tan
Syntax	cpi 13784	Extend class notation to include pi = 3.14159....
class pi
Definition	df-ef 13785*	Define the exponential function. (Contributed by NM, 14-Mar-2005.)
|- exp = ( x e. CC |-> sum_ k e. NN0 ( ( x ^ k ) / ( ! ` k ) ) )
Definition	df-e 13786	Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
|- _e = ( exp ` 1 )
Definition	df-sin 13787	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 13788	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 13789	Define the tangent function. We define it this way for cmpt 4495, 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 13790	Define pi = 3.14159..., which is the smallest positive number whose sine is zero. Definition of pi in [Gleason] p. 311. (We use the inverse of less-than, " `' <", to turn supremum into infimum; currently we don't have infimum defined separately.) (Contributed by Paul Chapman, 23-Jan-2008.)
|- pi = sup ( ( RR+ i^i ( `' sin " { 0 } ) ) , RR , `' < )
Theorem	eftcl 13791	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 13792	The terms of the series expansion of the exponential function of a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
|- ( ( A e. RR /\ K e. NN0 ) -> ( ( A ^ K ) / ( ! ` K ) ) e. RR )
Theorem	eftabs 13793	The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
|- ( ( A e. CC /\ K e. NN0 ) -> ( abs ` ( ( A ^ K ) / ( ! ` K ) ) ) = ( ( ( abs ` A ) ^ K ) / ( ! ` K ) ) )
Theorem	eftval 13794*	The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( N e. NN0 -> ( F ` N ) = ( ( A ^ N ) / ( ! ` N ) ) )
Theorem	efcllem 13795*	Lemma for efcl 13800. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 13674 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
Theorem	ef0lem 13796*	The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )
Theorem	efval 13797*	Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
Theorem	esum 13798	Value of Euler's constant _e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
|- _e = sum_ k e. NN0 ( 1 / ( ! ` k ) )
Theorem	eff 13799	Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
|- exp : CC --> CC
Theorem	efcl 13800	Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
|- ( A e. CC -> ( exp ` A ) e. CC )