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Theorem List for Intuitionistic Logic Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodfap0 11501* 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 )
 
Theoremprodfrecap 11502* 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 ) ) )
 
Theoremprodfdivap 11503* 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 ) ) )
 
4.8.10.2  Non-trivial convergence
 
Theoremntrivcvgap 11504* 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  ~~>  )
 
Theoremntrivcvgap0 11505* 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 )
 )
 
4.8.10.3  Complex products
 
Syntaxcprod 11506 Extend class notation to include complex products.
 class  prod_ k  e.  A  B
 
Definitiondf-proddc 11507* Define the product of a series with an index set of integers  A. This definition takes most of the aspects of df-sumdc 11310 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 ) ) ) )
 
Theoremprodeq1f 11508 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 )
 
Theoremprodeq1 11509* Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
 |-  ( A  =  B  -> 
 prod_ k  e.  A  C  =  prod_ k  e.  B  C )
 
Theoremnfcprod1 11510* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F/_ k A   =>    |-  F/_ k prod_ k  e.  A  B
 
Theoremnfcprod 11511* 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
 
Theoremprodeq2w 11512* 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 )
 
Theoremprodeq2 11513* 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 )
 
Theoremcbvprod 11514* 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
 
Theoremcbvprodv 11515* 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
 
Theoremcbvprodi 11516* 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
 
Theoremprodeq1i 11517* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   =>    |-  prod_ k  e.  A  C  =  prod_ k  e.  B  C
 
Theoremprodeq2i 11518* 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
 
Theoremprodeq12i 11519* 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
 
Theoremprodeq1d 11520* 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 )
 
Theoremprodeq2d 11521* Equality deduction for product. Note that unlike prodeq2dv 11522, 
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 )
 
Theoremprodeq2dv 11522* 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 )
 
Theoremprodeq2sdv 11523* 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 )
 
Theorem2cprodeq2dv 11524* 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 )
 
Theoremprodeq12dv 11525* 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 )
 
Theoremprodeq12rdv 11526* 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 )
 
Theoremprodrbdclem 11527* Lemma for prodrbdc 11530. (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 ) )
 
Theoremfproddccvg 11528* 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 ) )
 
Theoremprodrbdclem2 11529* Lemma for prodrbdc 11530. (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 )
 )
 
Theoremprodrbdc 11530* 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 )
 )
 
Theoremprodmodclem3 11531* Lemma for prodmodc 11534. (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 ) )
 
Theoremprodmodclem2a 11532* Lemma for prodmodc 11534. (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 ) )
 
Theoremprodmodclem2 11533* Lemma for prodmodc 11534. (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 )
 )
 
Theoremprodmodc 11534* 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 ) ) ) )
 
Theoremzproddc 11535* 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 ) ) )
 
Theoremiprodap 11536* 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 ) ) )
 
Theoremzprodap0 11537* 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 )
 
Theoremiprodap0 11538* 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 )
 
4.8.10.4  Finite products
 
Theoremfprodseq 11539* 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 ) )
 
Theoremfprodntrivap 11540* 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 )
 )
 
Theoremprod0 11541 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |- 
 prod_ k  e.  (/)  A  =  1
 
Theoremprod1dc 11542* 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 )
 
Theoremprodfct 11543* 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 )
 
Theoremfprodf1o 11544* 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 )
 
Theoremprodssdc 11545* 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 )
 
Theoremfprodssdc 11546* 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 )
 
Theoremfprodmul 11547* 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 ) )
 
Theoremprodsnf 11548* A product of a singleton is the term. A version of prodsn 11549 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 )
 
Theoremprodsn 11549* 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 )
 
Theoremfprod1 11550* 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 )
 
Theoremclimprod1 11551 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 )
 
Theoremfprodsplitdc 11552* Split a finite product into two parts. New proofs should use fprodsplit 11553 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 ) )
 
Theoremfprodsplit 11553* 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 ) )
 
Theoremfprodm1 11554* 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 ) )
 
Theoremfprod1p 11555* 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 ) )
 
Theoremfprodp1 11556* 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 ) )
 
Theoremfprodm1s 11557* 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 ) )
 
Theoremfprodp1s 11558* 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 ) )
 
Theoremprodsns 11559* 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 )
 
Theoremfprodunsn 11560* Multiply in an additional term in a finite product. See also fprodsplitsn 11589 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 ) )
 
Theoremfprodcl2lem 11561* 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 )
 
Theoremfprodcllem 11562* 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 )
 
Theoremfprodcl 11563* 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 )
 
Theoremfprodrecl 11564* 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 )
 
Theoremfprodzcl 11565* 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 )
 
Theoremfprodnncl 11566* 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 )
 
Theoremfprodrpcl 11567* 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+ )
 
Theoremfprodnn0cl 11568* 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 )
 
Theoremfprodcllemf 11569* Finite product closure lemma. A version of fprodcllem 11562 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 )
 
Theoremfprodreclf 11570* Closure of a finite product of real numbers. A version of fprodrecl 11564 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 )
 
Theoremfprodfac 11571* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  =  prod_ k  e.  (
 1 ... A ) k )
 
Theoremfprodabs 11572* 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 ) )
 
Theoremfprodeq0 11573* 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
 )
 
Theoremfprodshft 11574* 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 )
 
Theoremfprodrev 11575* 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 )
 
Theoremfprodconst 11576* 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 )
 ) )
 
Theoremfprodap0 11577* 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 )
 
Theoremfprod2dlemstep 11578* Lemma for fprod2d 11579- 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 )
 
Theoremfprod2d 11579* Write a double product as a product over a two-dimensional region. Compare fsum2d 11391. (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 )
 
Theoremfprodxp 11580* 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 )
 
Theoremfprodcnv 11581* 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 )
 
Theoremfprodcom2fi 11582* 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 )
 
Theoremfprodcom 11583* 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 )
 
Theoremfprod0diagfz 11584* Two ways to express "the product of  A ( j ,  k ) over the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N. Compare fisum0diag 11397. (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 )
 
Theoremfprodrec 11585* 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 ) )
 
Theoremfproddivap 11586* 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 ) )
 
Theoremfproddivapf 11587* The quotient of two finite products. A version of fproddivap 11586 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 ) )
 
Theoremfprodsplitf 11588* Split a finite product into two parts. A version of fprodsplit 11553 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 ) )
 
Theoremfprodsplitsn 11589* Separate out a term in a finite product. See also fprodunsn 11560 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 ) )
 
Theoremfprodsplit1f 11590* 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 ) )
 
Theoremfprodclf 11591* Closure of a finite product of complex numbers. A version of fprodcl 11563 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 )
 
Theoremfprodap0f 11592* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11577 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 )
 
Theoremfprodge0 11593* 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 )
 
Theoremfprodeq0g 11594* 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 )
 
Theoremfprodge1 11595* 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 )
 
Theoremfprodle 11596* 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 )
 
Theoremfprodmodd 11597* 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 ) )
 
4.9  Elementary trigonometry
 
4.9.1  The exponential, sine, and cosine functions
 
Syntaxce 11598 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 11599 Extend class notation to include Euler's constant  _e = 2.71828....
 class  _e
 
Syntaxcsin 11600 Extend class notation to include the sine function.
 class  sin
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