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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupfz 25201 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  <  )  =  N )
 
Theoreminffz 25202 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  sup (
 ( M ... N ) ,  ZZ ,  `'  <  )  =  M )
 
Theorembcnm1 25203 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theoremfz0n 25204 The sequence  ( 0 ... ( N  -  1 ) ) is empty iff  N is zero. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  (
 ( 0 ... ( N  -  1 ) )  =  (/)  <->  N  =  0
 ) )
 
Theorem4bc3eq4 25205 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  (
 4  _C  3 )  =  4
 
Theorem4bc2eq6 25206 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 4  _C  2 )  =  6
 
Theoremhalfthird 25207 Half minus a third. (Contributed by Scott Fenton, 8-Jul-2015.)
 |-  (
 ( 1  /  2
 )  -  ( 1 
 /  3 ) )  =  ( 1  / 
 6 )
 
Theorem5recm6rec 25208 One fifth minus one sixth. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  (
 ( 1  /  5
 )  -  ( 1 
 /  6 ) )  =  ( 1  / ; 3 0 )
 
Theorempnpncand 25209 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  ( B  -  C ) )  +  ( C  -  B ) )  =  A )
 
Theoremshftvalg 25210 Value of a sequence shifted by  A. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  (
 ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A ) ) )
 
Theorempossumd 25211 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 0  <  ( A  +  B )  <->  -u B  <  A ) )
 
Theoremfzp1nel 25212 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  -.  ( N  +  1
 )  e.  ( M
 ... N )
 
Theoremdivcnvshft 25213* Limit of a ratio function. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( A  /  (
 k  +  B ) ) )   =>    |-  ( ph  ->  F  ~~>  0 )
 
Theoremdivcnvlin 25214* Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( ( k  +  A )  /  (
 k  +  B ) ) )   =>    |-  ( ph  ->  F  ~~>  1 )
 
Theoremmuls1d 25215 Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  -  1 ) )  =  ( ( A  x.  B )  -  A ) )
 
Theoremclimlec3 25216* Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  ~~>  A )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( F `  k
 )  e.  RR )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  <_  B )   =>    |-  ( ph  ->  A  <_  B )
 
19.7.6  Product sequences
 
Theoremprodf 25217* An infinite product of complex terms is a function from an upper set of integers to  CC. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  seq  M (  x.  ,  F ) : Z --> CC )
 
Theoremclim2prod 25218* The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq  ( N  +  1 ) (  x.  ,  F )  ~~>  A )   =>    |-  ( ph  ->  seq 
 M (  x.  ,  F )  ~~>  ( (  seq  M (  x.  ,  F ) `  N )  x.  A ) )
 
Theoremclim2div 25219* The limit of an infinite product with an initial segment removed. (Contributed by Scott Fenton, 20-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ph  ->  seq 
 M (  x.  ,  F )  ~~>  A )   &    |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
  N )  =/=  0 )   =>    |-  ( ph  ->  seq  ( N  +  1 )
 (  x.  ,  F ) 
 ~~>  ( A  /  (  seq  M (  x.  ,  F ) `  N ) ) )
 
Theoremprodfmul 25220* The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( H `
  k )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  H ) `  N )  =  ( (  seq  M (  x.  ,  F ) `  N )  x.  (  seq  M (  x.  ,  G ) `
  N ) ) )
 
Theoremprodf1 25221 The value of the partial products in a one-valued infinite product. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( N  e.  Z  ->  ( 
 seq  M (  x.  ,  ( Z  X.  { 1 } ) ) `  N )  =  1
 )
 
Theoremprodf1f 25222 A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  seq 
 M (  x.  ,  ( Z  X.  { 1 } ) )  =  ( Z  X.  {
 1 } ) )
 
Theoremprodfclim1 25223 The constant one product converges to one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  seq 
 M (  x.  ,  ( Z  X.  { 1 } ) )  ~~>  1 )
 
Theoremprodfn0 25224* No term of a non-zero infinite product is zero. (Contributed by Scott Fenton, 14-Jan-2018.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =/=  0 )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
  N )  =/=  0 )
 
Theoremprodfrec 25225* The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  =/=  0 )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( G `
  k )  =  ( 1  /  ( F `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  G ) `  N )  =  ( 1  /  (  seq  M (  x.  ,  F ) `
  N ) ) )
 
Theoremprodfdiv 25226* The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  ( ph  ->  N  e.  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M ... N ) )  ->  ( F `
  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M ... N ) )  ->  ( G `
  k )  =/=  0 )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( H `  k
 )  =  ( ( F `  k ) 
 /  ( G `  k ) ) )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  H ) `  N )  =  ( (  seq  M (  x.  ,  F ) `
  N )  /  (  seq  M (  x. 
 ,  G ) `  N ) ) )
 
19.7.7  Non-trivial convergence
 
Theoremntrivcvg 25227* 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  ~~>  )
 
Theoremntrivcvgn0 25228* A product that converges to a non-zero value 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 ) )
 
Theoremntrivcvgfvn0 25229* Any value of a product sequence that converges to a non-zero value is itself non-zero. (Contributed by Scott Fenton, 20-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  X )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  (  seq  M (  x.  ,  F ) `  N )  =/=  0 )
 
Theoremntrivcvgtail 25230* A tail of a non-trivially convergent sequence converges non-trivially. (Contributed by Scott Fenton, 20-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  X )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   =>    |-  ( ph  ->  (
 (  ~~>  `  seq  N (  x.  ,  F ) )  =/=  0  /\  seq 
 N (  x.  ,  F )  ~~>  (  ~~>  `  seq  N (  x.  ,  F ) ) ) )
 
Theoremntrivcvgmullem 25231* Lemma for ntrivcvgmul 25232. (Contributed by Scott Fenton, 19-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ph  ->  P  e.  Z )   &    |-  ( ph  ->  X  =/=  0 )   &    |-  ( ph  ->  Y  =/=  0
 )   &    |-  ( ph  ->  seq  N (  x.  ,  F )  ~~>  X )   &    |-  ( ph  ->  seq 
 P (  x.  ,  G )  ~~>  Y )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( F `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( G `  k
 )  e.  CC )   &    |-  ( ph  ->  N  <_  P )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  E. q  e.  Z  E. w ( w  =/=  0  /\  seq  q (  x.  ,  H )  ~~>  w ) )
 
Theoremntrivcvgmul 25232* The product of two non-trivially converging products converges non-trivially. (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  ->  E. m  e.  Z  E. z ( z  =/=  0  /\  seq  m (  x.  ,  G )  ~~>  z ) )   &    |-  (
 ( ph  /\  k  e.  Z )  ->  ( G `  k )  e. 
 CC )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  ( H `  k
 )  =  ( ( F `  k )  x.  ( G `  k ) ) )   =>    |-  ( ph  ->  E. p  e.  Z  E. w ( w  =/=  0  /\  seq 
 p (  x.  ,  H )  ~~>  w ) )
 
19.7.8  Complex products
 
Syntaxcprod 25233 Extend class notation to include complex products.
 class  prod_ k  e.  A B
 
Definitiondf-prod 25234* Define the product of a series with an index set of integers  A. This definition takes most of the aspects of df-sum 12482 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a non-zero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  prod_ k  e.  A B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  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  |->  [_ (
 f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Theoremprodex 25235 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  prod_ k  e.  A B  e.  _V
 
Theoremprodeq1f 25236 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 25237* 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 25238* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F/_ k A   =>    |-  F/_ k prod_ k  e.  A B
 
Theoremnfcprod 25239* 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 25240* 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 25241* 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 25242* 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 25243* 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 25244* 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 25245* 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 25246* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   =>    |- 
 prod_ k  e.  A C  =  prod_ k  e.  B C
 
Theoremprodeq2i 25247* 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 25248* 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 25249* 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 25250* Equality deduction for sum. Note that unlike prodeq2dv 25251, 
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 25251* 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 25252* 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 25253* 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 25254* 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 25255* 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 25256* 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 25257* Lemma for prodrb 25260. (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 25258* 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 25259* Lemma for prodrb 25260. (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 25260* 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 25261* Lemma for prodmo 25264. (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 25262* Lemma for prodmo 25264. (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 25263* Lemma for prodmo 25264. (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 25264* 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 25265* 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 25266* 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 25267* 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 25268* 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 )
 
19.7.9  Finite products
 
Theoremfprod 25269* 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 25270* 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 25271 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  prod_ k  e.  (/) A  =  1
 
Theoremprod1 25272* 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 25273* 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 25274* 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 25275* 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 25276* 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 25277* 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 25278* 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 25279* 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 25280* 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 25281* 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 25282* 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 25283* Closure of a finite product of natural numbers. (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 25284* 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 25285* Closure of a finite product of non-negative 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 25286* 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 25287* 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 25288* 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 25289* 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 25290 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 25291* 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 25292* 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 25293* 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 25294* 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 25295* 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 25296* 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 25297* 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 25298* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  = 
 prod_ k  e.  (
 1 ... A ) k )
 
Theoremfprodabs 25299* 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 ) )
 
Theoremfprodefsum 25300* Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  N  e.  Z )   &    |-  ( ( ph  /\  k  e.  Z )  ->  A  e.  CC )   =>    |-  ( ph  ->  prod_ k  e.  ( M ... N ) ( exp `  A )  =  ( exp ` 
 sum_ k  e.  ( M ... N ) A ) )
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