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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodeq2w 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 )
 
Theoremprodeq2ii 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 )
 
Theoremprodeq2 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 )
 
Theoremcbvprod 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
 
Theoremcbvprodv 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
 
Theoremcbvprodi 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
 
Theoremprodeq1i 13707* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   =>    |-  prod_ k  e.  A  C  =  prod_ k  e.  B  C
 
Theoremprodeq2i 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
 
Theoremprodeq12i 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
 
Theoremprodeq1d 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 )
 
Theoremprodeq2d 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 )
 
Theoremprodeq2dv 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 )
 
Theoremprodeq2sdv 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 )
 
Theorem2cprodeq2dv 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 )
 
Theoremprodeq12dv 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 )
 
Theoremprodeq12rdv 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 )
 
Theoremprod2id 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 )
 
Theoremprodrblem 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 ) )
 
Theoremfprodcvg 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 ) )
 
Theoremprodrblem2 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 )
 )
 
Theoremprodrb 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 )
 )
 
Theoremprodmolem3 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 ) )
 
Theoremprodmolem2a 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 ) )
 
Theoremprodmolem2 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 )
 )
 
Theoremprodmo 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 ) ) ) )
 
Theoremzprod 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 ) ) )
 
Theoremiprod 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 ) ) )
 
Theoremzprodn0 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 )
 
Theoremiprodn0 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
 
Theoremfprod 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 ) )
 
Theoremfprodntriv 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 )
 )
 
Theoremprod0 13732 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |- 
 prod_ k  e.  (/)  A  =  1
 
Theoremprod1 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 )
 
Theoremprodfc 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
 
Theoremfprodf1o 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 )
 
Theoremprodss 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 )
 
Theoremfprodss 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 )
 
Theoremfprodser 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 ) )
 
Theoremfprodcl2lem 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 )
 
Theoremfprodcllem 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 )
 
Theoremfprodcl 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 )
 
Theoremfprodrecl 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 )
 
Theoremfprodzcl 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 )
 
Theoremfprodnncl 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 )
 
Theoremfprodrpcl 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+ )
 
Theoremfprodnn0cl 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 )
 
Theoremfprodmul 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 ) )
 
Theoremfproddiv 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 ) )
 
Theoremprodsn 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 )
 
Theoremfprod1 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 )
 
Theoremclimprod1 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 )
 
Theoremfprodsplit 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 ) )
 
Theoremfprodm1 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 ) )
 
Theoremfprod1p 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 ) )
 
Theoremfprodp1 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 ) )
 
Theoremfprodm1s 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 ) )
 
Theoremfprodp1s 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 ) )
 
Theoremprodsns 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 )
 
Theoremfprodfac 13759* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  =  prod_ k  e.  (
 1 ... A ) k )
 
Theoremfprodabs 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 ) )
 
Theoremfprodeq0 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
 )
 
Theoremfprodshft 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 )
 
Theoremfprodrev 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 )
 
Theoremfprodconst 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 ) ) )
 
Theoremfprodn0 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
 )
 
Theoremfprod2dlem 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 )
 
Theoremfprod2d 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 )
 
Theoremfprodxp 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 )
 
Theoremfprodcnv 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 )
 
Theoremfprodcom2 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 )
 
Theoremfprodcom 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 )
 
Theoremfprod0diag 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
 
Theoremiprodclim 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 )
 
Theoremiprodclim2 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 )
 
Theoremiprodclim3 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 )
 
Theoremiprodcl 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 )
 
Theoremiprodrecl 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 )
 
Theoremiprodmul 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
 
Syntaxce 13779 Extend class notation to include the exponential function.
 class  exp
 
Syntaxceu 13780 Extend class notation to include Euler's constant = 2.7182818....
 class  _e
 
Syntaxcsin 13781 Extend class notation to include the sine function.
 class  sin
 
Syntaxccos 13782 Extend class notation to include the cosine function.
 class  cos
 
Syntaxctan 13783 Extend class notation to include the tangent function.
 class  tan
 
Syntaxcpi 13784 Extend class notation to include pi = 3.14159....
 class  pi
 
Definitiondf-ef 13785* Define the exponential function. (Contributed by NM, 14-Mar-2005.)
 |- 
 exp  =  ( x  e.  CC  |->  sum_ k  e.  NN0  ( ( x ^
 k )  /  ( ! `  k ) ) )
 
Definitiondf-e 13786 Define Euler's constant 2.71828.... (Contributed by NM, 14-Mar-2005.)
 |-  _e  =  ( exp `  1 )
 
Definitiondf-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 ) ) )
 
Definitiondf-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 ) )
 
Definitiondf-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 ) ) )
 
Definitiondf-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 ,  `'  <  )
 
Theoremeftcl 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 )
 
Theoremreeftcl 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 )
 
Theoremeftabs 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 ) ) )
 
Theoremeftval 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 ) ) )
 
Theoremefcllem 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  ~~>  )
 
Theoremef0lem 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 )
 
Theoremefval 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 ) ) )
 
Theoremesum 13798 Value of Euler's constant  _e = 2.71828... (Contributed by Steve Rodriguez, 5-Mar-2006.)
 |-  _e  =  sum_ k  e.  NN0  ( 1  /  ( ! `  k ) )
 
Theoremeff 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
 
Theoremefcl 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 )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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