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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfprodrev 25301* 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 25302* 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 25303* 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 25304* Lemma for fprod2d 25305- 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 25305* Write a double product as a product over a two dimensional region. Compare fsum2d 12555. (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 25306* 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 25307* 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 25308* 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 25309* 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 25310* Two ways to express "the product of  A ( j ,  k ) over the the triangular region  M  <_  j,  M  <_  k,  j  +  k  <_  N. Compare fsum0diag 12561. (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 )
 
19.7.10  Infinite products
 
Theoremiprodclim 25311* 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 25312* 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 25313* 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 25314* 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 25315* 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 25316* 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 ) )
 
Theoremiprodefisumlem 25317 Lemma for iprodefisum 25318. (Contributed by Scott Fenton, 11-Feb-2018.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F : Z --> CC )   =>    |-  ( ph  ->  seq 
 M (  x.  ,  ( exp  o.  F ) )  =  ( exp 
 o.  seq  M (  +  ,  F )
 ) )
 
Theoremiprodefisum 25318* Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ( ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )   &    |-  ( ( ph  /\  k  e.  Z ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  seq  M (  +  ,  F )  e.  dom  ~~>  )   =>    |-  ( ph  ->  prod_ k  e.  Z ( exp `  B )  =  ( exp ` 
 sum_ k  e.  Z  B ) )
 
Theoremiprodgam 25319* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
 |-  ( ph  ->  A  e.  ( CC  \  ( ZZ  \  NN ) ) )   =>    |-  ( ph  ->  (
 _G `  A )  =  ( prod_ k  e.  NN ( ( ( 1  +  ( 1  /  k ) )  ^ c  A )  /  (
 1  +  ( A 
 /  k ) ) )  /  A ) )
 
19.7.11  Falling and Rising Factorial
 
Syntaxcfallfac 25320 Declare the syntax for the falling factorial.
 class FallFac
 
Syntaxcrisefac 25321 Declare the syntax for the rising factorial.
 class RiseFac
 
Definitiondf-risefac 25322* Define the rising factorial function. This is the function  ( A  x.  ( A  +  1
)  x.  ... ( A  +  N )
) for complex  A and non-negative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- RiseFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  +  k
 ) )
 
Definitiondf-fallfac 25323* Define the falling factorial function. This is the function  ( A  x.  ( A  -  1
)  x.  ... ( A  -  N )
) for complex  A and non-negative integers  N. (Contributed by Scott Fenton, 5-Jan-2018.)
 |- FallFac  =  ( x  e.  CC ,  n  e.  NN0  |->  prod_ k  e.  ( 0 ... ( n  -  1 ) ) ( x  -  k
 ) )
 
Theoremrisefacval 25324* The value of the rising factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  +  k ) )
 
Theoremfallfacval 25325* The value of the falling factorial function. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 0 ... ( N  -  1 ) ) ( A  -  k ) )
 
Theoremrisefacval2 25326* One-based value of rising factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  +  ( k  -  1 ) ) )
 
Theoremfallfacval2 25327* One-based value of falling factorial. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  =  prod_ k  e.  (
 1 ... N ) ( A  -  ( k  -  1 ) ) )
 
Theoremfallfacval3 25328* A product representation of falling factorial when  A is a non-negative integer. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  prod_ k  e.  ( ( A  -  ( N  -  1 ) ) ... A ) k )
 
Theoremrisefaccllem 25329* Lemma for rising factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_ 
 CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  +  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  S )
 
Theoremfallfaccllem 25330* Lemma for falling factorial closure laws. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  S  C_ 
 CC   &    |-  1  e.  S   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  x.  y )  e.  S )   &    |-  ( ( A  e.  S  /\  k  e.  NN0 )  ->  ( A  -  k )  e.  S )   =>    |-  ( ( A  e.  S  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  S )
 
Theoremrisefaccl 25331 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  CC )
 
Theoremfallfaccl 25332 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  CC )
 
Theoremrerisefaccl 25333 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR )
 
Theoremrefallfaccl 25334 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  RR )
 
Theoremnnrisefaccl 25335 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  NN  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN )
 
Theoremzrisefaccl 25336 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  ZZ )
 
Theoremzfallfaccl 25337 Closure law for falling factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN0 )  ->  ( A FallFac  N )  e.  ZZ )
 
Theoremnn0risefaccl 25338 Closure law for rising factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  NN0  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  NN0 )
 
Theoremrprisefaccl 25339 Closure law for rising factorial. (Contributed by Scott Fenton, 9-Jan-2018.)
 |-  (
 ( A  e.  RR+  /\  N  e.  NN0 )  ->  ( A RiseFac  N )  e.  RR+ )
 
Theoremrisefallfac 25340 A relationship between rising and falling factorials. (Contributed by Scott Fenton, 15-Jan-2018.)
 |-  (
 ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X RiseFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X FallFac  N ) ) )
 
Theoremfallrisefac 25341 A relationship between falling and rising factorials. (Contributed by Scott Fenton, 17-Jan-2018.)
 |-  (
 ( X  e.  CC  /\  N  e.  NN0 )  ->  ( X FallFac  N )  =  ( ( -u 1 ^ N )  x.  ( -u X RiseFac  N ) ) )
 
Theoremrisefall0lem 25342 Lemma for risefac0 25343 and fallfac0 25344. Show a particular set of finite integers is empty. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 0 ... ( 0  -  1 ) )  =  (/)
 
Theoremrisefac0 25343 The value of the rising factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  0 )  =  1 )
 
Theoremfallfac0 25344 The value of the falling factorial when  N  =  0. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  0 )  =  1 )
 
Theoremrisefacp1 25345 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A RiseFac  ( N  +  1 ) )  =  ( ( A RiseFac  N )  x.  ( A  +  N )
 ) )
 
Theoremfallfacp1 25346 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN0 )  ->  ( A FallFac  ( N  +  1 ) )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefacp1d 25347 The value of the rising factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A RiseFac  ( N  +  1 ) )  =  ( ( A RiseFac  N )  x.  ( A  +  N ) ) )
 
Theoremfallfacp1d 25348 The value of the falling factorial at a successor. (Contributed by Scott Fenton, 19-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( A FallFac  ( N  +  1 ) )  =  ( ( A FallFac  N )  x.  ( A  -  N ) ) )
 
Theoremrisefac1 25349 The value of rising factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A RiseFac  1 )  =  A )
 
Theoremfallfac1 25350 The value of falling factorial at one. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( A  e.  CC  ->  ( A FallFac  1 )  =  A )
 
Theoremrisefacfac 25351 Relate rising factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  (
 1 RiseFac  N )  =  ( ! `  N ) )
 
Theoremfallfacfwd 25352 The forward difference of a falling factorial. (Contributed by Scott Fenton, 21-Jan-2018.)
 |-  (
 ( A  e.  CC  /\  N  e.  NN )  ->  ( ( ( A  +  1 ) FallFac  N )  -  ( A FallFac  N ) )  =  ( N  x.  ( A FallFac  ( N  -  1 ) ) ) )
 
Theorem0fallfac 25353 The value of the zero falling factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 FallFac  N )  =  0 )
 
Theorem0risefac 25354 The value of the zero rising factorial at natural  N. (Contributed by Scott Fenton, 17-Feb-2018.)
 |-  ( N  e.  NN  ->  ( 0 RiseFac  N )  =  0 )
 
Theorembinomfallfaclem1 25355 Lemma for binomfallfac 25357. Closure law. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  K  e.  ( 0 ... N ) )  ->  ( ( N  _C  K )  x.  ( ( A FallFac  ( N  -  K ) )  x.  ( B FallFac  ( K  +  1 ) ) ) )  e.  CC )
 
Theorembinomfallfaclem2 25356* Lemma for binomfallfac 25357. Inductive step. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ps  ->  ( ( A  +  B ) FallFac  N )  =  sum_ k  e.  ( 0 ...
 N ) ( ( N  _C  k )  x.  ( ( A FallFac  ( N  -  k
 ) )  x.  ( B FallFac  k ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( A  +  B ) FallFac  ( N  +  1 )
 )  =  sum_ k  e.  ( 0 ... ( N  +  1 )
 ) ( ( ( N  +  1 )  _C  k )  x.  ( ( A FallFac  (
 ( N  +  1 )  -  k ) )  x.  ( B FallFac  k ) ) ) )
 
Theorembinomfallfac 25357* A version of the binomial theorem using falling factorials instead of exponentials. (Contributed by Scott Fenton, 13-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) FallFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A FallFac  ( N  -  k ) )  x.  ( B FallFac  k )
 ) ) )
 
Theorembinomrisefac 25358* A version of the binomial theorem using rising factorials instead of exponentials. (Contributed by Scott Fenton, 16-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  N  e.  NN0 )  ->  ( ( A  +  B ) RiseFac  N )  = 
 sum_ k  e.  (
 0 ... N ) ( ( N  _C  k
 )  x.  ( ( A RiseFac  ( N  -  k ) )  x.  ( B RiseFac  k )
 ) ) )
 
Theoremfallfacval4 25359 Represent the falling factorial via factorials when the first argument is a natural. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( N  e.  ( 0 ... A )  ->  ( A FallFac  N )  =  ( ( ! `  A )  /  ( ! `  ( A  -  N ) ) ) )
 
Theorembcfallfac 25360 Binomial coefficient in terms of falling factorials. (Contributed by Scott Fenton, 20-Mar-2018.)
 |-  ( K  e.  ( 0 ... N )  ->  ( N  _C  K )  =  ( ( N FallFac  K ) 
 /  ( ! `  K ) ) )
 
Theoremfallfacfac 25361 Relate falling factorial to factorial. (Contributed by Scott Fenton, 5-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N FallFac  N )  =  ( ! `  N ) )
 
19.7.12  Factorial limits
 
Theoremfaclimlem1 25362* Lemma for faclim 25365. 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 25363* Lemma for faclim 25365. 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 25364 Lemma for faclim 25365. 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 25365* 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 25366* 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 25367* 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 )
 
19.7.13  Greatest common divisor and divisibility
 
Theorempdivsq 25368 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 25369 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 25370 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 25371 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 ) )
 
19.7.14  Properties of relationships
 
Theorembrtp 25372 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 25373 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 25374* 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 25375* 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 25376 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 25377 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 25378 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 25379* 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 25380* 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 25381* 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 ) )
 
Theoremdfres3 25382 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  ran  A )
 )
 
Theoremcnvco1 25383 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( `' A  o.  B )  =  ( `' B  o.  A )
 
Theoremcnvco2 25384 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
 |-  `' ( A  o.  `' B )  =  ( B  o.  `' A )
 
Theoremeldm3 25385 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  dom  B  <->  ( B  |`  { A } )  =/=  (/) )
 
Theoremelrn3 25386 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
 |-  ( A  e.  ran  B  <->  ( B  i^i  ( _V  X.  { A } ) )  =/=  (/) )
 
Theorempocnv 25387 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  Po  A  ->  `' R  Po  A )
 
Theoremsocnv 25388 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
 |-  ( R  Or  A  ->  `' R  Or  A )
 
19.7.15  Properties of functions and mappings
 
Theoremfunpsstri 25389 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
 |-  (
 ( Fun  H  /\  ( F  C_  H  /\  G  C_  H )  /\  ( dom  F  C_  dom  G  \/  dom  G  C_  dom  F ) )  ->  ( F 
 C.  G  \/  F  =  G  \/  G  C.  F ) )
 
Theoremfundmpss 25390 If a class  F is a proper subset of a function  G, then  dom  F  C.  dom  G. (Contributed by Scott Fenton, 20-Apr-2011.)
 |-  ( Fun  G  ->  ( F  C.  G  ->  dom  F  C.  dom 
 G ) )
 
Theoremfvresval 25391 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
 |-  (
 ( ( F  |`  B ) `
  A )  =  ( F `  A )  \/  ( ( F  |`  B ) `  A )  =  (/) )
 
Theoremmptrel 25392 The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
 |-  Rel  ( x  e.  A  |->  B )
 
Theoremfunsseq 25393 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( Fun  F  /\  Fun 
 G  /\  dom  F  =  dom  G )  ->  ( F  =  G  <->  F  C_  G ) )
 
Theoremfununiq 25394 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( Fun  F  ->  ( ( A F B  /\  A F C ) 
 ->  B  =  C ) )
 
Theoremfunbreq 25395 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( ( Fun  F  /\  A F B ) 
 ->  ( A F C  <->  B  =  C ) )
 
Theoremmpteq12d 25396 An equality inference for the maps to notation. Compare mpteq12dv 4287. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/ x ph   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
 
Theoremfprb 25397* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  =/=  B  ->  ( F : { A ,  B } --> R  <->  E. x  e.  R  E. y  e.  R  F  =  { <. A ,  x >. ,  <. B ,  y >. } ) )
 
Theorembr1steq 25398 Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 1st C  <->  C  =  A )
 
Theorembr2ndeq 25399 Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( <. A ,  B >. 2nd C  <->  C  =  B )
 
Theoremdfdm5 25400 Definition of domain in terms of 
1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
 |-  dom  A  =  ( ( 1st  |`  ( _V  X.  _V ) ) " A )
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