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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodfclim1 24401 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 )
 
18.7.7  Non-trivial convergence
 
Theoremntrivcvg 24402* 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 24403* 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 24404* 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 24405* 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 24406* Lemma for ntrivcvgmul 24407. (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 24407* 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 ) )
 
18.7.8  Complex products
 
Syntaxcprod 24408 Extend class notation to include complex products.
 class  prod_ k  e.  A B
 
Definitiondf-prod 24409* Define the product of a series with an index set of integers  A. This definition takes most of the aspects of df-sum 12206 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 24410 A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  prod_ k  e.  A B  e.  _V
 
Theoremprodeq1f 24411 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 24412* 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 24413* Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  F/_ k A   =>    |-  F/_ k prod_ k  e.  A B
 
Theoremnfcprod 24414* 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 24415* 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 24416* 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 24417* 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 24418* 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 24419* 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 24420* 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 24421* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
 |-  A  =  B   =>    |- 
 prod_ k  e.  A C  =  prod_ k  e.  B C
 
Theoremprodeq2i 24422* 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 24423* 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 24424* 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 24425* Equality deduction for sum. Note that unlike prodeq2dv 24426, 
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 24426* 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 24427* 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 24428* 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 24429* 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 24430* 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 24431* 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 24432* Lemma for prodrb 24435. (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 24433* 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 24434* Lemma for prodrb 24435. (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 24435* 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 24436* Lemma for prodmo 24439. (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 24437* Lemma for prodmo 24439. (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 24438* Lemma for prodmo 24439. (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 24439* 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 24440* 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 24441* 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 24442* 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 24443* 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 )
 
18.7.9  Finite products
 
Theoremfprod 24444* 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 24445* 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 24446 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
 |-  prod_ k  e.  (/) A  =  1
 
Theoremprod1 24447* 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 24448* 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 24449* 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 24450* 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 24451* 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 24452* 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 24453* 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 24454* 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 24455* 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 24456* 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 24457* 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 24458* 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 24459* 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 24460* 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 24461* 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 ) )
 
Theoremprodsn 24462* 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 24463* 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 24464 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 24465* 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 24466* 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 24467* 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 24468* Multiply in the last term in a finite sum. (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 ) )
 
Theoremfprodfac 24469* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  = 
 prod_ k  e.  (
 1 ... A ) k )
 
18.7.10  Infinite products
 
Theoremiprodclim 24470* 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 24471* 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 24472* 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 24473* 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 24474* 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 24475* 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 ) )
 
18.7.11  Euler's Gamma Function
 
Syntaxcgamma 24476 Declare the syntax for the gamma function.
 class Γ
 
Definitiondf-gamma 24477* Define Euler's gamma function. This is a function over the complex plane except for the non-positive integers that obeys  (Γ N ) = ( !  ( N  - 
1 ) ) over the naturals. The usual definition is in terms of the analytic continuation of an improper integral, but here we prefer to use Euler's original definition as an infinite product, as this requires no implicit continuation operations. (Contributed by Scott Fenton, 25-Dec-2017.)
 |- Γ  =  ( z  e.  ( CC  \  ( ZZ  \  NN ) )  |->  ( ( 1  /  z )  x.  prod_ k  e.  NN ( ( exp `  (
 ( log `  ( 1  +  ( 1  /  k
 ) ) )  x.  z ) )  /  ( 1  +  (
 z  /  k )
 ) ) ) )
 
Theoremgammaval 24478* The value of the gamma function. (Contributed by Scott Fenton, 25-Dec-2017.)
 |-  ( A  e.  ( CC  \  ( ZZ  \  NN ) )  ->  (Γ `  A )  =  ( (
 1  /  A )  x.  prod_ k  e.  NN ( ( exp `  (
 ( log `  ( 1  +  ( 1  /  k
 ) ) )  x.  A ) )  /  ( 1  +  ( A  /  k ) ) ) ) )
 
Theoremfngamma 24479 Functionhood and domain of the gamma function. (Contributed by Scott Fenton, 25-Dec-2017.)
 |- Γ  Fn  ( CC  \  ( ZZ  \  NN ) )
 
Theoremgamma1 24480 Gamma of one is one. (Contributed by Scott Fenton, 25-Dec-2017.)
 |-  (Γ `  1 )  =  1
 
18.7.12  Factorial limits
 
Theoremfaclimlem1 24481* Lemma for faclim 24484. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( M  e.  NN0  ->  seq  1
 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( M  /  n ) )  x.  ( 1  +  ( 1  /  n ) ) )  /  ( 1  +  (
 ( M  +  1 )  /  n ) ) ) ) )  =  ( x  e. 
 NN  |->  ( ( M  +  1 )  x.  ( ( x  +  1 )  /  ( x  +  ( M  +  1 ) ) ) ) ) )
 
Theoremfaclimlem2 24482* Lemma for faclim 24484. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( M  e.  NN0  ->  seq  1
 (  x.  ,  ( n  e.  NN  |->  ( ( ( 1  +  ( M  /  n ) )  x.  ( 1  +  ( 1  /  n ) ) )  /  ( 1  +  (
 ( M  +  1 )  /  n ) ) ) ) )  ~~>  ( M  +  1
 ) )
 
Theoremfaclimlem3 24483 Lemma for faclim 24484. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  (
 ( M  e.  NN0  /\  B  e.  NN )  ->  ( ( ( 1  +  ( 1  /  B ) ) ^
 ( M  +  1 ) )  /  (
 1  +  ( ( M  +  1 ) 
 /  B ) ) )  =  ( ( ( ( 1  +  ( 1  /  B ) ) ^ M )  /  ( 1  +  ( M  /  B ) ) )  x.  ( ( ( 1  +  ( M  /  B ) )  x.  ( 1  +  (
 1  /  B )
 ) )  /  (
 1  +  ( ( M  +  1 ) 
 /  B ) ) ) ) )
 
Theoremfaclim 24484* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( 1  +  ( 1  /  n ) ) ^ A )  /  (
 1  +  ( A 
 /  n ) ) ) )   =>    |-  ( A  e.  NN0  ->  seq  1 (  x.  ,  F )  ~~>  ( ! `  A ) )
 
Theoremiprodfac 24485* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
 |-  ( A  e.  NN0  ->  ( ! `  A )  = 
 prod_ k  e.  NN ( ( ( 1  +  ( 1  /  k ) ) ^ A )  /  (
 1  +  ( A 
 /  k ) ) ) )
 
Theoremfaclim2 24486* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
 |-  F  =  ( n  e.  NN  |->  ( ( ( ! `
  n )  x.  ( ( n  +  1 ) ^ M ) )  /  ( ! `  ( n  +  M ) ) ) )   =>    |-  ( M  e.  NN0  ->  F 
 ~~>  1 )
 
18.7.13  Greatest common divisor and divisibility
 
Theorempdivsq 24487 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( P  e.  Prime  /\  M  e.  ZZ )  ->  ( P  ||  M  <->  P 
 ||  ( M ^
 2 ) ) )
 
Theoremdvdspw 24488 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N ) ) )
 
Theoremgcd32 24489 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  C )  =  ( ( A 
 gcd  C )  gcd  B ) )
 
Theoremgcdabsorb 24490 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  gcd  B )  =  ( A  gcd  B ) )
 
18.7.14  Properties of relationships
 
Theorembrtp 24491 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
 
Theoremdftr6 24492 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
 |-  A  e.  _V   =>    |-  ( Tr  A  <->  A  e.  ( _V  \  ran  ( (  _E  o.  _E  )  \  _E  ) ) )
 
Theoremcoep 24493* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A (  _E  o.  R ) B  <->  E. x  e.  B  A R x )
 
Theoremcoepr 24494* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
 
Theoremdffr5 24495 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
 |-  ( R  Fr  A  <->  ( ~P A  \  { (/) } )  C_  ran  (  _E  \  (  _E  o.  `' R ) ) )
 
Theoremdfso2 24496 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Or  A  <->  ( R  Po  A  /\  ( A  X.  A )  C_  ( R  u.  (  _I  u.  `' R ) ) ) )
 
Theoremdfpo2 24497 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
 |-  ( R  Po  A  <->  ( ( R  i^i  (  _I  |`  A ) )  =  (/)  /\  (
 ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  R ) )
 
Theorembr8 24498* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  (
 g  =  G  ->  ( ze  <->  si ) )   &    |-  ( h  =  H  ->  (
 si 
 <->  rh ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  E. g  e.  P  E. h  e.  P  ( p  = 
 <. <. a ,  b >. ,  <. c ,  d >.
 >.  /\  q  =  <. <.
 e ,  f >. , 
 <. g ,  h >. >.  /\  ph ) }   =>    |-  ( ( ( X  e.  S  /\  A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q  /\  E  e.  Q )  /\  ( F  e.  Q  /\  G  e.  Q  /\  H  e.  Q )
 )  ->  ( <. <. A ,  B >. , 
 <. C ,  D >. >. R <. <. E ,  F >. ,  <. G ,  H >.
 >. 
 <->  rh ) )
 
Theorembr6 24499* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  (
 e  =  E  ->  ( ta  <->  et ) )   &    |-  (
 f  =  F  ->  ( et  <->  ze ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  E. e  e.  P  E. f  e.  P  ( p  = 
 <. a ,  <. b ,  c >. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ph ) }   =>    |-  (
 ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q  /\  C  e.  Q )  /\  ( D  e.  Q  /\  E  e.  Q  /\  F  e.  Q )
 )  ->  ( <. A ,  <. B ,  C >.
 >. R <. D ,  <. E ,  F >. >.  <->  ze ) )
 
Theorembr4 24500* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
 |-  (
 a  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 b  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 c  =  C  ->  ( ch  <->  th ) )   &    |-  (
 d  =  D  ->  ( th  <->  ta ) )   &    |-  ( x  =  X  ->  P  =  Q )   &    |-  R  =  { <. p ,  q >.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  = 
 <. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) }   =>    |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q ) )  ->  ( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
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