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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfacndiv 11301 No natural number (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( M  e.  NN0  /\  N  e.  NN )  /\  ( 1  <  N  /\  N  <_  M ) )  ->  -.  ( ( ( ! `
  M )  +  1 )  /  N )  e.  ZZ )
 
Theoremfacwordi 11302 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( ! `  M )  <_  ( ! `  N ) )
 
Theoremfaclbnd 11303 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ ( N  +  1 )
 )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd2 11304 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( ( 2 ^ N )  /  2
 )  <_  ( ! `  N ) )
 
Theoremfaclbnd3 11305 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M ^ N )  <_  ( ( M ^ M )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem1 11306 Lemma for faclbnd4 11310. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)
 |-  N  e.  NN   &    |-  K  e.  NN0   &    |-  M  e.  NN0   =>    |-  ( ( ( ( N  -  1 ) ^ K )  x.  ( M ^ ( N  -  1 ) ) )  <_  ( (
 ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
 ( M  +  K ) ) )  x.  ( ! `  ( N  -  1 ) ) )  ->  ( ( N ^ ( K  +  1 ) )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ (
 ( K  +  1 ) ^ 2 ) )  x.  ( M ^ ( M  +  ( K  +  1
 ) ) ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem2 11307 Lemma for faclbnd4 11310. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 11306 to antecedents. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  K  e.  NN0  /\  N  e.  NN )  ->  (
 ( ( ( N  -  1 ) ^ K )  x.  ( M ^ ( N  -  1 ) ) ) 
 <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  ( N  -  1 ) ) ) 
 ->  ( ( N ^
 ( K  +  1 ) )  x.  ( M ^ N ) ) 
 <_  ( ( ( 2 ^ ( ( K  +  1 ) ^
 2 ) )  x.  ( M ^ ( M  +  ( K  +  1 ) ) ) )  x.  ( ! `  N ) ) ) )
 
Theoremfaclbnd4lem3 11308 Lemma for faclbnd4 11310. The  N  =  0 case. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0 )  ->  ( ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4lem4 11309 Lemma for faclbnd4 11310. Prove the  0  <  N case by induction on  K. (Contributed by NM, 19-Dec-2005.)
 |-  ( ( N  e.  NN  /\  K  e.  NN0  /\  M  e.  NN0 )  ->  ( ( N ^ K )  x.  ( M ^ N ) ) 
 <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd4 11310 Variant of faclbnd5 11311 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  M  e.  NN0 )  ->  (
 ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^
 2 ) )  x.  ( M ^ ( M  +  K )
 ) )  x.  ( ! `  N ) ) )
 
Theoremfaclbnd5 11311 The factorial function grows faster than powers and exponentiations. If we consider  K and  M to be constants, the right-hand side of the inequality is a constant times 
N-factorial. (Contributed by NM, 24-Dec-2005.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  M  e.  NN )  ->  (
 ( N ^ K )  x.  ( M ^ N ) )  < 
 ( ( 2  x.  ( ( 2 ^
 ( K ^ 2
 ) )  x.  ( M ^ ( M  +  K ) ) ) )  x.  ( ! `
  N ) ) )
 
Theoremfaclbnd6 11312 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)
 |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( ! `  N )  x.  (
 ( N  +  1 ) ^ M ) )  <_  ( ! `  ( N  +  M ) ) )
 
Theoremfacubnd 11313 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( ! `  N ) 
 <_  ( N ^ N ) )
 
Theoremfacavg 11314 The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ! `  ( |_ `  ( ( M  +  N )  / 
 2 ) ) ) 
 <_  ( ( ! `  M )  x.  ( ! `  N ) ) )
 
5.6.7  The binomial coefficient operation
 
Syntaxcbc 11315 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
 class  _C
 
Definitiondf-bc 11316* Define the binomial coefficient operation. In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C".  ( N  _C  K ) is read " N choose  K." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  k  <_  n does not hold. (Contributed by NM, 10-Jul-2005.)
 |- 
 _C  =  ( n  e.  NN0 ,  k  e. 
 ZZ  |->  if ( k  e.  ( 0 ... n ) ,  ( ( ! `  n )  /  ( ( ! `  ( n  -  k
 ) )  x.  ( ! `  k ) ) ) ,  0 ) )
 
Theorembcval 11317 Value of the binomial coefficient, 
N choose  K. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when  0  <_  K  <_  N does not hold. See bcval2 11318 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  if ( K  e.  ( 0 ... N ) ,  (
 ( ! `  N )  /  ( ( ! `
  ( N  -  K ) )  x.  ( ! `  K ) ) ) ,  0 ) )
 
Theorembcval2 11318 Value of the binomial coefficient, 
N choose  K, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  =  ( ( ! `
  N )  /  ( ( ! `  ( N  -  K ) )  x.  ( ! `  K ) ) ) )
 
Theorembcval3 11319 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  -.  K  e.  ( 0
 ... N ) ) 
 ->  ( N  _C  K )  =  0 )
 
Theorembcval4 11320 Value of the binomial coefficient, 
N choose  K, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ  /\  ( K  <  0  \/  N  <  K ) )  ->  ( N  _C  K )  =  0 )
 
Theorembcrpcl 11321 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 11335.) (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
 
Theorembccmpl 11322 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)
 |-  ( ( N  e.  NN0  /\  K  e.  ZZ )  ->  ( N  _C  K )  =  ( N  _C  ( N  -  K ) ) )
 
Theorembcn0 11323  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 11324  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 11325 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 11326 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 11327 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 11328 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 11329 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 11330 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 11331 Binomial coefficient:  N choose  2. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  2
 )  =  ( ( N  x.  ( N  -  1 ) ) 
 /  2 ) )
 
Theorembcp1m1 11332 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 11333 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 11334 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 11335 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 11336 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 11337 Extend the definition of a class to include the size function.
 class  #
 
Definitiondf-hash 11338 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 11339 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 11340* The value of the  # function in terms of the mapping  G from  om to  NN0. The proof avoids the use of ax-ac 8085. (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 11341*  `' 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 11342 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 11343 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 11344 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 11345 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 11346 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 11347 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 11348 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 11349 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 11350 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 11351 Extended real closure of the 
# function. (Contributed by Mario Carneiro, 22-Apr-2015.)
 |-  ( A  e.  V  ->  ( # `  A )  e.  RR* )
 
Theoremhashclb 11352 Reverse closure of the  # function. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( # `
  A )  e. 
 NN0 ) )
 
Theoremhasheq0 11353 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 11354 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( A  e.  Fin  ->  ( ( # `  A )  e.  NN  <->  A  =/=  (/) ) )
 
Theoremhash0 11355 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( # `  (/) )  =  0
 
Theoremhashsng 11356 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 11357 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F  Fn  A  ->  ( # `  F )  =  ( # `  A ) )
 
Theoremfseq1hash 11358 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 11359  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 11360 A short expression for the  G function of hashgf1o 11033. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  ( #  |`  om )  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
 
Theoremhashdom 11361 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 11362 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 11363 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 11364 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 11365 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 11366 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 ) ) ) )
 
Theoremhashunsng 11367 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 11368 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 11369 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 11370 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  1o )  =  1
 
Theoremhash2 11371 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  2o )  =  2
 
Theoremhash3 11372 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  3o )  =  3
 
Theoremhash4 11373 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( # `  4o )  =  4
 
Theoremhashssdif 11374 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 11375 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 11376 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 11377 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 11378 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 11379 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_ 
 2 )
 
Theoremhashtplei 11380 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 11381 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 11382 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 11383 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 11384 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 11385 Lemma for hashxp 11386. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  B  e.  Fin   =>    |-  ( A  e.  Fin  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashxp 11386 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 11387 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 11388 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 11389 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 11390* Lemma for hashbc 11391: 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 11391* 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 11392* 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 11393* Lemma for hashf1 11395. (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 11394* Lemma for hashf1 11395. (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 11395* 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 11396* 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 11397 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 11398 Version of isorel 5823 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 11399* Lemma for fz1iso 11400. (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 11400* 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 ) )
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