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Theorem List for Metamath Proof Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhash0 11601 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( # `  (/) )  =  0
 
Theoremhashsng 11602 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |-  ( A  e.  V  ->  ( # `  { A } )  =  1
 )
 
Theoremhashrabrsn 11603* The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  ( # `  { x  e.  { A }  |  ph
 } )  e.  NN0
 
Theoremhashfn 11604 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F  Fn  A  ->  ( # `  F )  =  ( # `  A ) )
 
Theoremfseq1hash 11605 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( # `  F )  =  N )
 
Theoremhashgadd 11606  G maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  (
 ( A  e.  om  /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
  A )  +  ( G `  B ) ) )
 
Theoremhashgval2 11607 A short expression for the  G function of hashgf1o 11265. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( #  |`  om )  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
 
Theoremhashdom 11608 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  V ) 
 ->  ( ( # `  A )  <_  ( # `  B ) 
 <->  A  ~<_  B ) )
 
Theoremhashdomi 11609 Non-strict order relation of the  # function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( A  ~<_  B  ->  ( # `  A )  <_  ( # `  B ) )
 
Theoremhashsdom 11610 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  A )  <  ( # `  B ) 
 <->  A  ~<  B )
 )
 
Theoremhashun 11611 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( # `
  ( A  u.  B ) )  =  ( ( # `  A )  +  ( # `  B ) ) )
 
Theoremhashun2 11612 The size of the union of finite sets is less than or equal to the sum of their sizes. (Contributed by Mario Carneiro, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 27-Jul-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B ) ) 
 <_  ( ( # `  A )  +  ( # `  B ) ) )
 
Theoremhashun3 11613 The size of the union of finite sets is the sum of their sizes minus the size of the intersection. (Contributed by Mario Carneiro, 6-Aug-2017.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  u.  B ) )  =  ( ( ( # `  A )  +  ( # `  B ) )  -  ( # `  ( A  i^i  B ) ) ) )
 
Theoremhashinfxadd 11614 The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( # `  A )  e/  NN0 )  ->  (
 ( # `  A ) + e ( # `  B ) )  = 
 +oo )
 
Theoremhashunx 11615 The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 11611. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  ( A  i^i  B )  =  (/) )  ->  ( # `  ( A  u.  B ) )  =  ( ( # `  A ) + e
 ( # `  B ) ) )
 
Theoremhashge0 11616 The cardinality of a set is greater than or equal to zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( A  e.  V  ->  0  <_  ( # `  A ) )
 
Theoremhashgt0 11617 The cardinality of a non-empty set is greater than zero. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  ( ( A  e.  V  /\  A  =/=  (/) )  -> 
 0  <  ( # `  A ) )
 
Theoremhashge1 11618 The cardinality of a non-empty set is greater or equal to one. (Contributed by Thierry Arnoux, 20-Jun-2017.)
 |-  ( ( A  e.  V  /\  A  =/=  (/) )  -> 
 1  <_  ( # `  A ) )
 
Theoremhashnn0n0nn 11619 If a nonnegative integer is the size of a set which contains at least one element, this integer is a positive integer. (Contributed by Alexander van der Vekens, 9-Jan-2018.)
 |-  ( ( ( V  e.  W  /\  Y  e.  NN0 )  /\  (
 ( # `  V )  =  Y  /\  N  e.  V ) )  ->  Y  e.  NN )
 
Theoremhashunsng 11620 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( B  e.  V  ->  ( ( A  e.  Fin  /\  -.  B  e.  A )  ->  ( # `  ( A  u.  { B }
 ) )  =  ( ( # `  A )  +  1 )
 ) )
 
Theoremhashprg 11621 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.)
 |-  ( ( A  e.  V  /\  B  e.  V )  ->  ( A  =/=  B  <-> 
 ( # `  { A ,  B } )  =  2 ) )
 
Theoremelprchashprn2 11622 If one element of an unordered pair is not a set, the size of the unordered pair is not 2. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  ( -.  M  e.  _V 
 ->  -.  ( # `  { M ,  N } )  =  2 )
 
Theoremhashprb 11623 The size of an unordered pair is 2 if and only if its elements are different sets. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
 |-  ( ( M  e.  _V 
 /\  N  e.  _V  /\  M  =/=  N )  <-> 
 ( # `  { M ,  N } )  =  2 )
 
Theoremhashle00 11624 If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( V  e.  W  ->  ( ( # `  V )  <_  0  <->  V  =  (/) ) )
 
Theoremhashgt0elex 11625* If the size of a set is greater than zero, the set must contain at least one element. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( V  e.  W  /\  0  <  ( # `
  V ) ) 
 ->  E. x  x  e.  V )
 
Theoremhashgt0elexb 11626* The size of a set is greater than zero if and only if the set contains at least one element. (Contributed by Alexander van der Vekens, 18-Jan-2018.)
 |-  ( V  e.  W  ->  ( 0  <  ( # `
  V )  <->  E. x  x  e.  V ) )
 
Theoremhashp1i 11627 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  A  e.  om   &    |-  B  =  suc  A   &    |-  ( # `  A )  =  M   &    |-  ( M  +  1 )  =  N   =>    |-  ( # `
  B )  =  N
 
Theoremhash1 11628 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  1o )  =  1
 
Theoremhash2 11629 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  2o )  =  2
 
Theoremhash3 11630 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  3o )  =  3
 
Theoremhash4 11631 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  4o )  =  4
 
Theoremhashssdif 11632 The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  ( # `  ( A  \  B ) )  =  ( ( # `  A )  -  ( # `
  B ) ) )
 
Theoremhashdif 11633 The size of the difference of a finite set and another set is the first set's size minus that of the intersection of both. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  ( A  e.  Fin  ->  ( # `  ( A 
 \  B ) )  =  ( ( # `  A )  -  ( # `
  ( A  i^i  B ) ) ) )
 
Theoremhashdifsn 11634 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( # `  ( A  \  { B }
 ) )  =  ( ( # `  A )  -  1 ) )
 
Theoremhashsnlei 11635 Get an upper bound on a concretely specified finite set. Base case: singleton set. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A }  e.  Fin  /\  ( # `  { A } )  <_  1 )
 
Theoremhash1snb 11636* The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.)
 |-  ( V  e.  W  ->  ( ( # `  V )  =  1  <->  E. a  V  =  { a } )
 )
 
Theoremhashgt12el 11637* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  ( ( V  e.  W  /\  1  <  ( # `
  V ) ) 
 ->  E. a  e.  V  E. b  e.  V  a  =/=  b )
 
Theoremhashgt12el2 11638* In a set with more than one element are two different elements. (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  ( ( V  e.  W  /\  1  <  ( # `
  V )  /\  A  e.  V )  ->  E. b  e.  V  A  =/=  b )
 
Theoremhashunlei 11639 Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  C  =  ( A  u.  B )   &    |-  ( A  e.  Fin  /\  ( # `
  A )  <_  K )   &    |-  ( B  e.  Fin  /\  ( # `  B )  <_  M )   &    |-  K  e.  NN0   &    |-  M  e.  NN0   &    |-  ( K  +  M )  =  N   =>    |-  ( C  e.  Fin  /\  ( # `  C )  <_  N )
 
Theoremhashsslei 11640 Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  B  C_  A   &    |-  ( A  e.  Fin  /\  ( # `  A )  <_  N )   &    |-  N  e.  NN0   =>    |-  ( B  e.  Fin  /\  ( # `  B )  <_  N )
 
Theoremhashprlei 11641 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_ 
 2 )
 
Theoremhash2pr 11642* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b  V  =  { a ,  b } )
 
Theoremhash2prde 11643* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( ( V  e.  W  /\  ( # `  V )  =  2 )  ->  E. a E. b
 ( a  =/=  b  /\  V  =  { a ,  b } ) )
 
Theoremhash2prb 11644* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)
 |-  ( V  e.  W  ->  ( ( # `  V )  =  2  <->  E. a E. b
 ( a  =/=  b  /\  V  =  { a ,  b } ) ) )
 
Theoremhashtplei 11645 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A ,  B ,  C }  e.  Fin  /\  ( # `  { A ,  B ,  C }
 )  <_  3 )
 
Theoremhashtpg 11646 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V  /\  C  e.  _V )  ->  ( ( A  =/=  B 
 /\  B  =/=  C  /\  C  =/=  A )  <-> 
 ( # `  { A ,  B ,  C }
 )  =  3 ) )
 
Theoremhashfz 11647 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( # `  ( A
 ... B ) )  =  ( ( B  -  A )  +  1 ) )
 
Theoremfzsdom2 11648 Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.)
 |-  ( ( ( B  e.  ( ZZ>= `  A )  /\  C  e.  ZZ )  /\  B  <  C )  ->  ( A ... B )  ~<  ( A ... C ) )
 
Theoremhashfzo 11649 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( # `  ( A..^ B ) )  =  ( B  -  A ) )
 
Theoremhashfzo0 11650 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  NN0  ->  ( # `  ( 0..^ B ) )  =  B )
 
Theoremhashxplem 11651 Lemma for hashxp 11652. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  B  e.  Fin   =>    |-  ( A  e.  Fin  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashxp 11652 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashmap 11653 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  ^m  B ) )  =  ( ( # `  A ) ^ ( # `
  B ) ) )
 
Theoremhashpw 11654 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)
 |-  ( A  e.  Fin  ->  ( # `  ~P A )  =  ( 2 ^ ( # `  A ) ) )
 
Theoremhashfun 11655 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( F  e.  Fin  ->  ( Fun  F  <->  ( # `  F )  =  ( # `  dom  F ) ) )
 
Theoremhashbclem 11656* Lemma for hashbc 11657: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  A. j  e. 
 ZZ  ( ( # `  A )  _C  j
 )  =  ( # ` 
 { x  e.  ~P A  |  ( # `  x )  =  j }
 ) )   &    |-  ( ph  ->  K  e.  ZZ )   =>    |-  ( ph  ->  ( ( # `  ( A  u.  { z }
 ) )  _C  K )  =  ( # `  { x  e.  ~P ( A  u.  { z } )  |  ( # `  x )  =  K }
 ) )
 
Theoremhashbc 11657* The binomial coefficient counts the number of subsets of a finite set of a given size. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A )  _C  K )  =  ( # `  { x  e.  ~P A  |  ( # `  x )  =  K } ) )
 
Theoremhashfacen 11658* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  { f  |  f : A -1-1-onto-> C }  ~~  { f  |  f : B -1-1-onto-> D } )
 
Theoremhashf1lem1 11659* Lemma for hashf1 11661. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  ( ( # `  A )  +  1 )  <_  ( # `  B ) )   &    |-  ( ph  ->  F : A -1-1-> B )   =>    |-  ( ph  ->  { f  |  ( ( f  |`  A )  =  F  /\  f : ( A  u.  { z }
 ) -1-1-> B ) }  ~~  ( B  \  ran  F ) )
 
Theoremhashf1lem2 11660* Lemma for hashf1 11661. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  ( ( # `  A )  +  1 )  <_  ( # `  B ) )   =>    |-  ( ph  ->  ( # `
  { f  |  f : ( A  u.  { z }
 ) -1-1-> B } )  =  ( ( ( # `  B )  -  ( # `
  A ) )  x.  ( # `  { f  |  f : A -1-1-> B } ) ) )
 
Theoremhashf1 11661* The permutation number  |  A  |  !  x.  (  |  B  |  _C  |  A  | 
)  =  |  B  |  !  /  (  |  B  |  -  |  A  | 
) ! counts the number of injections from  A to  B. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  { f  |  f : A -1-1-> B } )  =  (
 ( ! `  ( # `
  A ) )  x.  ( ( # `  B )  _C  ( # `
  A ) ) ) )
 
Theoremhashfac 11662* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)
 |-  ( A  e.  Fin  ->  ( # `  { f  |  f : A -1-1-onto-> A } )  =  ( ! `  ( # `
  A ) ) )
 
Theoremleiso 11663 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( A  C_  RR*  /\  B  C_  RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B ) 
 <->  F  Isom  <_  ,  <_  ( A ,  B ) ) )
 
Theoremleisorel 11664 Version of isorel 6005 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A 
 C_  RR*  /\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `
  D ) ) )
 
Theoremfz1isolem 11665* Lemma for fz1iso 11666. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  G  =  ( rec ( ( n  e. 
 _V  |->  ( n  +  1 ) ) ,  1 )  |`  om )   &    |-  B  =  ( NN  i^i  ( `'  <  " { ( ( # `  A )  +  1 ) } )
 )   &    |-  C  =  ( om  i^i  ( `' G `  ( ( # `  A )  +  1 )
 ) )   &    |-  O  = OrdIso ( R ,  A )   =>    |-  (
 ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
 1 ... ( # `  A ) ) ,  A ) )
 
Theoremfz1iso 11666* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
 1 ... ( # `  A ) ) ,  A ) )
 
Theoremseqcoll 11667* The function  F contains a sparse set of non-zero values to be summed. The function  G is an order isomorphism from the set of non-zero values of  F to a 1-based finite sequence, and  H collects these non-zero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( # `
  A ) ) ,  A ) )   &    |-  ( ph  ->  N  e.  ( 1 ... ( # `
  A ) ) )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( G `  ( # `  A ) ) ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
 \  A ) ) 
 ->  ( F `  k
 )  =  Z )   &    |-  ( ( ph  /\  n  e.  ( 1 ... ( # `
  A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
  n ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  ( G `  N ) )  =  (  seq  1
 (  .+  ,  H ) `  N ) )
 
Theoremseqcoll2 11668* The function  F contains a sparse set of non-zero values to be summed. The function  G is an order isomorphism from the set of non-zero values of  F to a 1-based finite sequence, and  H collects these non-zero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( # `
  A ) ) ,  A ) )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ( M ... N )  \  A ) )  ->  ( F `  k )  =  Z )   &    |-  ( ( ph  /\  n  e.  ( 1 ... ( # `
  A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
  n ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  1 (  .+  ,  H ) `  ( # `  A ) ) )
 
5.6.8.1  Finite induction on the size of the first component of a binary relation
 
Theorembrfi1indlem 11669 Lemma for brfi1ind 11671: The size of a set is the size of this set with one element removed, increased by 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |-  ( ( V  e.  W  /\  N  e.  V  /\  Y  e.  NN0 )  ->  ( ( # `  V )  =  ( Y  +  1 )  ->  ( # `  ( V 
 \  { N }
 ) )  =  Y ) )
 
Theorembrfi1uzind 11670* Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with  L  =  0 (see brfi1ind 11671) or  L  =  1. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |- 
 Rel  G   &    |-  F  e.  U   &    |-  L  e.  NN0   &    |-  ( ( v  =  V  /\  e  =  E )  ->  ( ps 
 <-> 
 ph ) )   &    |-  (
 ( v  =  w 
 /\  e  =  f )  ->  ( ps  <->  th ) )   &    |-  ( ( v G e  /\  n  e.  v )  ->  (
 v  \  { n } ) G F )   &    |-  ( ( w  =  ( v  \  { n } )  /\  f  =  F )  ->  ( th 
 <->  ch ) )   &    |-  (
 ( v G e 
 /\  ( # `  v
 )  =  L ) 
 ->  ps )   &    |-  ( ( ( ( y  +  1 )  e.  NN0  /\  (
 v G e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  ch )  ->  ps )   =>    |-  (
 ( V G E  /\  V  e.  Fin  /\  L  <_  ( # `  V ) )  ->  ph )
 
Theorembrfi1ind 11671* Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, e.g. usgrafis 21382. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |- 
 Rel  G   &    |-  F  e.  U   &    |-  (
 ( v  =  V  /\  e  =  E )  ->  ( ps  <->  ph ) )   &    |-  (
 ( v  =  w 
 /\  e  =  f )  ->  ( ps  <->  th ) )   &    |-  ( ( v G e  /\  n  e.  v )  ->  (
 v  \  { n } ) G F )   &    |-  ( ( w  =  ( v  \  { n } )  /\  f  =  F )  ->  ( th 
 <->  ch ) )   &    |-  (
 ( v G e 
 /\  ( # `  v
 )  =  0 ) 
 ->  ps )   &    |-  ( ( ( ( y  +  1 )  e.  NN0  /\  (
 v G e  /\  ( # `  v )  =  ( y  +  1 )  /\  n  e.  v ) )  /\  ch )  ->  ps )   =>    |-  (
 ( V G E  /\  V  e.  Fin )  -> 
 ph )
 
5.6.9  Words over a set
 
Syntaxcword 11672 Syntax for the Word operator.
 class Word  S
 
Syntaxcconcat 11673 Syntax for the concatenation operator.
 class concat
 
Syntaxcs1 11674 Syntax for the singleton word constructor.
 class  <" A ">
 
Syntaxcsubstr 11675 Syntax for the subword operator.
 class substr
 
Syntaxcsplice 11676 Syntax for the word splicing operator.
 class splice
 
Syntaxcreverse 11677 Syntax for the word reverse operator.
 class reverse
 
Definitiondf-word 11678* Define the class of words over a set. A word is a finite sequence of symbols from a set. The domain is forced so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |- Word  S  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }
 
Definitiondf-concat 11679* Define the concatenation operator which combines two words. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
 |- concat  =  ( s  e.  _V ,  t  e.  _V  |->  ( x  e.  (
 0..^ ( ( # `  s )  +  ( # `
  t ) ) )  |->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
  s ) ) ) ) ) )
 
Definitiondf-s1 11680 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
 
Definitiondf-substr 11681* Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |- substr  =  ( s  e.  _V ,  b  e.  ( ZZ  X.  ZZ )  |->  if ( ( ( 1st `  b )..^ ( 2nd `  b ) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b
 )  -  ( 1st `  b ) ) ) 
 |->  ( s `  ( x  +  ( 1st `  b ) ) ) ) ,  (/) ) )
 
Definitiondf-splice 11682* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |- splice  =  ( s  e.  _V ,  b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
 ) ) >. ) concat  ( 2nd `  b ) ) concat 
 ( s substr  <. ( 2nd `  ( 1st `  b
 ) ) ,  ( # `
  s ) >. ) ) )
 
Definitiondf-reverse 11683* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |- reverse  =  ( s  e.  _V  |->  ( x  e.  (
 0..^ ( # `  s
 ) )  |->  ( s `
  ( ( ( # `  s )  -  1 )  -  x ) ) ) )
 
Theoremiswrd 11684* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( W  e. Word  S  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S )
 
Theoremwrdval 11685* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e.  V  -> Word 
 S  =  U_ l  e.  NN0  ( S  ^m  ( 0..^ l ) ) )
 
Theoremiswrdi 11686 A one-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( W : ( 0..^ L ) --> S  ->  W  e. Word  S )
 
Theoremwrd0 11687 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (/)  e. Word  S
 
Theoremwrdf 11688 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( W  e. Word  S  ->  W : ( 0..^ ( # `  W ) ) --> S )
 
Theoremwrdfin 11689 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  ( W  e. Word  S  ->  W  e.  Fin )
 
Theoremlencl 11690 The length of a word is a nonnegative integer. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( W  e. Word  S  ->  ( # `  W )  e.  NN0 )
 
Theoremlennncl 11691 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( W  e. Word  S 
 /\  W  =/=  (/) )  ->  ( # `  W )  e.  NN )
 
Theoremsswrd 11692 The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  C_  T  -> Word 
 S  C_ Word  T )
 
Theoremwrdeq 11693 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  =  T  -> Word 
 S  = Word  T )
 
Theoremwrdexg 11694 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e.  V  -> Word 
 S  e.  _V )
 
Theoremnfwrd 11695 Hypothesis builder for Word  S. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  F/_ x S   =>    |-  F/_ xWord  S
 
Theoremccatfn 11696 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- concat  Fn  ( _V  X.  _V )
 
Theoremccatfval 11697* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( ( # `  S )  +  ( # `  T ) ) )  |->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `
  x ) ,  ( T `  ( x  -  ( # `  S ) ) ) ) ) )
 
Theoremccatcl 11698 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
 
Theoremccatlen 11699 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S )  +  ( # `  T ) ) )
 
Theoremccatval1 11700 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B  /\  I  e.  (
 0..^ ( # `  S ) ) )  ->  ( ( S concat  T ) `  I )  =  ( S `  I
 ) )
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