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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembcn0 11201  N choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  0
 )  =  1 )
 
Theorembcnn 11202  N choose  N is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  N )  =  1 )
 
Theorembcn1 11203 Binomial coefficient:  N choose  1. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( N  _C  1
 )  =  N )
 
Theorembcnp1n 11204 Binomial coefficient:  N  +  1 choose  N. (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  N )  =  ( N  +  1 ) )
 
Theorembcm1k 11205 The proportion of one binomial coefficient to another with  K decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 1 ... N )  ->  ( N  _C  K )  =  ( ( N  _C  ( K  -  1 ) )  x.  ( ( N  -  ( K  -  1
 ) )  /  K ) ) )
 
Theorembcp1n 11206 The proportion of one binomial coefficient to another with  N increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  K )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  (
 ( N  +  1 )  -  K ) ) ) )
 
Theorembcp1nk 11207 The proportion of one binomial coefficient to another with  N and  K increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( ( N  +  1 )  _C  ( K  +  1 )
 )  =  ( ( N  _C  K )  x.  ( ( N  +  1 )  /  ( K  +  1
 ) ) ) )
 
Theorembcval5 11208 Write out the top and bottom parts of the binomial coefficient  ( N  _C  K )  =  ( N  x.  ( N  -  1 )  x. 
...  x.  ( ( N  -  K )  +  1 ) )  /  K ! explicitly. In this form, it is valid even for  N  <  K, although it is no longer valid for non-positive  K. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN )  ->  ( N  _C  K )  =  ( (  seq  ( ( N  -  K )  +  1
 ) (  x.  ,  _I  ) `  N ) 
 /  ( ! `  K ) ) )
 
Theorembcn2 11209 Binomial coefficient:  N choose  2. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  2
 )  =  ( ( N  x.  ( N  -  1 ) ) 
 /  2 ) )
 
Theorembcp1m1 11210 Compute the binomial coefficent of  ( N  +  1 ) over  ( N  - 
1 ) (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( ( N  +  1 )  _C  ( N  -  1 ) )  =  ( ( ( N  +  1 )  x.  N )  / 
 2 ) )
 
Theorembcpasc 11211 Pascal's rule for the binomial coefficient, generalized to all integers  K. Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( ( N  _C  K )  +  ( N  _C  ( K  -  1 ) ) )  =  ( ( N  +  1 )  _C  K ) )
 
Theorembccl 11212 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  e.  NN0 )
 
Theorembccl2 11213 A binomial coefficient, in its standard domain, is a natural number. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  NN )
 
Theorempermnn 11214 The number of permutations of  N  -  R objects from a collection of  N objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)
 |-  ( R  e.  (
 0 ... N )  ->  ( ( ! `  N )  /  ( ! `  R ) )  e.  NN )
 
5.6.8  The ` # ` (finite set size) function
 
Syntaxchash 11215 Extend the definition of a class to include the size function.
 class  #
 
Definitiondf-hash 11216 Define the  # function, which gives the cardinality of a finite set as a member of  NN0, and assigns all infinite sets the value  +oo. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  #  =  ( (
 ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )  o.  card )  u.  ( ( _V  \  Fin )  X.  {  +oo } )
 )
 
Theoremhashkf 11217 The finite part of the size function maps all finite sets to their cardinality, as members of  NN0. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   &    |-  K  =  ( G  o.  card )   =>    |-  K : Fin --> NN0
 
Theoremhashgval 11218* The value of the  # function in terms of the mapping  G from  om to  NN0. The proof avoids the use of ax-ac 7969. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( A  e.  Fin  ->  ( G `  ( card `  A ) )  =  ( # `
  A ) )
 
Theoremhashginv 11219*  `' G maps the size function's value to  card. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( A  e.  Fin  ->  ( `' G `  ( # `  A ) )  =  ( card `  A )
 )
 
Theoremhashinf 11220 The value of the  # function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( # `  A )  =  +oo )
 
Theoremhashbnd 11221 If  A has size bounded by an integer  B, then  A is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  NN0  /\  ( # `  A )  <_  B )  ->  A  e.  Fin )
 
Theoremhashf 11222 The size function maps all finite sets to their cardinality, as members of  NN0, and infinite sets to  +oo. (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.)
 |-  # : _V --> ( NN0  u. 
 {  +oo } )
 
Theoremhashfz1 11223 The set  ( 1 ... N ) has  N elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( N  e.  NN0  ->  ( # `  ( 1
 ... N ) )  =  N )
 
Theoremhashen 11224 Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( # `  A )  =  ( # `  B ) 
 <->  A  ~~  B ) )
 
Theoremhasheni 11225 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)
 |-  ( A  ~~  B  ->  ( # `  A )  =  ( # `  B ) )
 
Theoremfz1eqb 11226 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ( 1 ...
 M )  =  ( 1 ... N )  <->  M  =  N )
 )
 
Theoremhashcard 11227 The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  ( A  e.  Fin  ->  ( # `  ( card `  A ) )  =  ( # `  A ) )
 
Theoremhashcl 11228 Closure of the  # function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
 |-  ( A  e.  Fin  ->  ( # `  A )  e.  NN0 )
 
Theoremhashxrcl 11229 Extended real closure of the 
# function. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( A  e.  V  ->  ( # `  A )  e.  RR* )
 
Theoremhashclb 11230 Reverse closure of the  # function. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( # `
  A )  e. 
 NN0 ) )
 
Theoremhasheq0 11231 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)
 |-  ( A  e.  V  ->  ( ( # `  A )  =  0  <->  A  =  (/) ) )
 
Theoremhashnncl 11232 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( A  e.  Fin  ->  ( ( # `  A )  e.  NN  <->  A  =/=  (/) ) )
 
Theoremhash0 11233 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( # `  (/) )  =  0
 
Theoremhashsng 11234 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
 )
 
Theoremhashfn 11235 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F  Fn  A  ->  ( # `  F )  =  ( # `  A ) )
 
Theoremfseq1hash 11236 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 11237  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 11238 A short expression for the  G function of hashgf1o 10911. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( #  |`  om )  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
 
Theoremhashdom 11239 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 11240 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 11241 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 11242 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 11243 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 ) ) )
 
Theoremhashunsng 11244 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 11245 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 ) )
 
Theoremhashp1i 11246 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 11247 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  1o )  =  1
 
Theoremhash2 11248 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  2o )  =  2
 
Theoremhash3 11249 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  3o )  =  3
 
Theoremhash4 11250 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  4o )  =  4
 
Theoremhashssdif 11251 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 11252 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 ) ) ) )
 
Theoremhashsnlei 11253 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 )
 
Theoremhashunlei 11254 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 11255 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 11256 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_ 
 2 )
 
Theoremhashtplei 11257 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A ,  B ,  C }  e.  Fin  /\  ( # `  { A ,  B ,  C }
 )  <_  3 )
 
Theoremhashfz 11258 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 11259 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 11260 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 11261 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 11262 Lemma for hashxp 11263. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  B  e.  Fin   =>    |-  ( A  e.  Fin  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashxp 11263 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 11264 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 11265 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 11266 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 11267* Lemma for hashbc 11268: 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 11268* 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 11269* 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 11270* Lemma for hashf1 11272. (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 11271* Lemma for hashf1 11272. (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 11272* 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 11273* 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 11274 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 11275 Version of isorel 5675 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 11276* Lemma for fz1iso 11277. (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 11277* 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 11278* 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 11279* 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 ) ) )
 
5.6.9  Words over a set
 
Syntaxcword 11280 Syntax for the Word operator.
 class Word  S
 
Syntaxcconcat 11281 Syntax for the concatenation operator.
 class concat
 
Syntaxcs1 11282 Syntax for the singleton word constructor.
 class  <" A ">
 
Syntaxcsubstr 11283 Syntax for the word slicing operator.
 class substr
 
Syntaxcsplice 11284 Syntax for the word splicing operator.
 class splice
 
Syntaxcreverse 11285 Syntax for the word reverse operator.
 class reverse
 
Definitiondf-word 11286* Define the class of words over a set. A word is an 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 11287* 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 11288 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 11289* 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 11290* 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 11291* 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 11292* 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 11293* 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 11294 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 11295 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (/)  e. Word  S
 
Theoremwrdf 11296 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 11297 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  ( W  e. Word  S  ->  W  e.  Fin )
 
Theoremlencl 11298 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 11299 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 11300 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 )
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