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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqge0d 10001 A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A ^ 2
 ) )
 
Theoremsqgt0apd 10002 The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  0  <  ( A ^ 2
 ) )
 
Theoremleexp2ad 10003 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1 
 <_  A )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   =>    |-  ( ph  ->  ( A ^ M )  <_  ( A ^ N ) )
 
Theoremleexp2rd 10004 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  1 )   =>    |-  ( ph  ->  ( A ^ N )  <_  ( A ^ M ) )
 
Theoremlt2sqd 10005 The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A ^
 2 )  <  ( B ^ 2 ) ) )
 
Theoremle2sqd 10006 The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A ^
 2 )  <_  ( B ^ 2 ) ) )
 
Theoremsq11d 10007 The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   &    |-  ( ph  ->  ( A ^ 2 )  =  ( B ^
 2 ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsq11ap 10008 Analogue to sq11 9917 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  ->  ( ( A ^
 2 ) #  ( B ^ 2 )  <->  A #  B )
 )
 
Theoremsq10 10009 The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  (; 1 0 ^ 2
 )  = ;; 1 0 0
 
Theoremsq10e99m1 10010 The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  (; 1 0 ^ 2
 )  =  (; 9 9  +  1 )
 
Theorem3dec 10011 A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.)
 |-  A  e.  NN0   &    |-  B  e.  NN0   =>    |- ;; A B C  =  ( ( ( (; 1
 0 ^ 2 )  x.  A )  +  (; 1 0  x.  B ) )  +  C )
 
Theoremexpcanlem 10012 Lemma for expcan 10013. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  1  <  A )   =>    |-  ( ph  ->  (
 ( A ^ M )  <_  ( A ^ N )  ->  M  <_  N ) )
 
Theoremexpcan 10013 Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
 |-  ( ( ( A  e.  RR  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  1  <  A )  ->  (
 ( A ^ M )  =  ( A ^ N )  <->  M  =  N ) )
 
Theoremexpcand 10014 Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  ( A ^ M )  =  ( A ^ N ) )   =>    |-  ( ph  ->  M  =  N )
 
3.6.7  Ordered pair theorem for nonnegative integers
 
Theoremnn0le2msqd 10015 The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  x.  A )  <_  ( B  x.  B ) ) )
 
Theoremnn0opthlem1d 10016 A rather pretty lemma for nn0opth2 10020. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   =>    |-  ( ph  ->  ( A  <  C  <->  ( ( A  x.  A )  +  ( 2  x.  A ) )  <  ( C  x.  C ) ) )
 
Theoremnn0opthlem2d 10017 Lemma for nn0opth2 10020. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  ->  (
 ( C  x.  C )  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) ) )
 
Theoremnn0opthd 10018 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers  A and  B by  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3439 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( (
 ( C  +  D )  x.  ( C  +  D ) )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremnn0opth2d 10019 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10018. (Contributed by Jim Kingdon, 31-Oct-2021.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   &    |-  ( ph  ->  C  e.  NN0 )   &    |-  ( ph  ->  D  e.  NN0 )   =>    |-  ( ph  ->  (
 ( ( ( A  +  B ) ^
 2 )  +  B )  =  ( (
 ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremnn0opth2 10020 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10018. (Contributed by NM, 22-Jul-2004.)
 |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  ->  (
 ( ( ( A  +  B ) ^
 2 )  +  B )  =  ( (
 ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
3.6.8  Factorial function
 
Syntaxcfa 10021 Extend class notation to include the factorial of nonnegative integers.
 class  !
 
Definitiondf-fac 10022 Define the factorial function on nonnegative integers. For example,  ( ! `  5 )  =  1 2 0 because  1  x.  2  x.  3  x.  4  x.  5  =  1 2 0 (ex-fac 11084). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)
 |-  !  =  ( { <. 0 ,  1 >. }  u.  seq 1 (  x.  ,  _I  ,  CC ) )
 
Theoremfacnn 10023 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  (  seq 1 (  x.  ,  _I  ,  CC ) `  N ) )
 
Theoremfac0 10024 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  0
 )  =  1
 
Theoremfac1 10025 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( ! `  1
 )  =  1
 
Theoremfacp1 10026 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)
 |-  ( N  e.  NN0  ->  ( ! `  ( N  +  1 ) )  =  ( ( ! `
  N )  x.  ( N  +  1 ) ) )
 
Theoremfac2 10027 The factorial of 2. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  2
 )  =  2
 
Theoremfac3 10028 The factorial of 3. (Contributed by NM, 17-Mar-2005.)
 |-  ( ! `  3
 )  =  6
 
Theoremfac4 10029 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ! `  4
 )  = ; 2 4
 
Theoremfacnn2 10030 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN  ->  ( ! `  N )  =  ( ( ! `  ( N  -  1 ) )  x.  N ) )
 
Theoremfaccl 10031 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  e.  NN )
 
Theoremfaccld 10032 Closure of the factorial function, deduction version of faccl 10031. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( ! `  N )  e. 
 NN )
 
Theoremfacne0 10033 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)
 |-  ( N  e.  NN0  ->  ( ! `  N )  =/=  0 )
 
Theoremfacdiv 10034 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN  /\  N  <_  M )  ->  ( ( ! `  M )  /  N )  e.  NN )
 
Theoremfacndiv 10035 No positive integer (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 10036 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( ! `  M )  <_  ( ! `  N ) )
 
Theoremfaclbnd 10037 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 10038 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( ( 2 ^ N )  /  2
 )  <_  ( ! `  N ) )
 
Theoremfaclbnd3 10039 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 ) ) )
 
Theoremfaclbnd6 10040 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 10041 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)
 |-  ( N  e.  NN0  ->  ( ! `  N ) 
 <_  ( N ^ N ) )
 
Theoremfacavg 10042 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 ) ) )
 
3.6.9  The binomial coefficient operation
 
Syntaxcbc 10043 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).
 class  _C
 
Definitiondf-bc 10044* Define the binomial coefficient operation. For example,  ( 5  _C  3 )  =  1 0 (ex-bc 11085).

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 10045 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 10046 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 10046 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 10047 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 10048 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 10049 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10064.) (Contributed by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  RR+ )
 
Theorembccmpl 10050 "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 10051  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 )
 
Theorembc0k 10052 The binomial coefficient " 0 choose  K " is 0 for a positive integer K. Note that  ( 0  _C  0 )  =  1 (see bcn0 10051). (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( K  e.  NN  ->  ( 0  _C  K )  =  0 )
 
Theorembcnn 10053  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 10054 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 10055 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 10056 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 10057 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 10058 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
 ) ) ) )
 
Theoremibcval5 10059 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 nonpositive  K. (Contributed by Jim Kingdon, 6-Nov-2021.)
 |-  ( ( N  e.  NN0  /\  K  e.  NN )  ->  ( N  _C  K )  =  ( (  seq ( ( N  -  K )  +  1
 ) (  x.  ,  _I  ,  CC ) `  N )  /  ( ! `  K ) ) )
 
Theorembcn2 10060 Binomial coefficient:  N choose  2. (Contributed by Mario Carneiro, 22-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  2
 )  =  ( ( N  x.  ( N  -  1 ) ) 
 /  2 ) )
 
Theorembcp1m1 10061 Compute the binomial coefficient 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 10062 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 10063 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 10064 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  _C  K )  e.  NN )
 
Theorembcn2m1 10065 Compute the binomial coefficient " N choose 2 " from " ( N  -  1 ) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)
 |-  ( N  e.  NN  ->  ( ( N  -  1 )  +  (
 ( N  -  1
 )  _C  2 )
 )  =  ( N  _C  2 ) )
 
Theorembcn2p1 10066 Compute the binomial coefficient " ( N  +  1
) choose 2 " from " N choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)
 |-  ( N  e.  NN0  ->  ( N  +  ( N  _C  2 ) )  =  ( ( N  +  1 )  _C  2 ) )
 
Theorempermnn 10067 The number of permutations of  N  -  R objects from a collection of  N objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)
 |-  ( R  e.  (
 0 ... N )  ->  ( ( ! `  N )  /  ( ! `  R ) )  e.  NN )
 
Theorembcnm1 10068 The binomial coefficent of  ( N  -  1 ) is  N. (Contributed by Scott Fenton, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  _C  ( N  -  1 ) )  =  N )
 
Theorem4bc3eq4 10069 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)
 |-  ( 4  _C  3
 )  =  4
 
Theorem4bc2eq6 10070 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)
 |-  ( 4  _C  2
 )  =  6
 
3.6.10  The ` # ` (set size) function
 
Syntaxchash 10071 Extend the definition of a class to include the set size function.
 class
 
Definitiondf-ihash 10072* Define the set size function ♯, which gives the cardinality of a finite set as a member of 
NN0, and assigns all infinite sets the value +oo. For example,  ( `  {
0 ,  1 ,  2 } )  =  3.

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 7992). We adopt the former notation from Corollary 8.2.4 of [AczelRathjen], p. 80 (although that work only defines it for finite sets).

This definition (in terms of  U. and 
~<_) is not taken directly from the literature, but for finite sets should be equivalent to the conventional definition that the size of a finite set is the unique natural number which is equinumerous to the given set. (Contributed by Jim Kingdon, 19-Feb-2022.)

 |- =  ( (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )  u.  { <. om , +oo >. } )  o.  ( x  e.  _V  |->  U.
 { y  e.  ( om  u.  { om }
 )  |  y  ~<_  x } ) )
 
Theoremhashinfuni 10073* The ordinal size of an infinite set is  om. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  U.
 { y  e.  ( om  u.  { om }
 )  |  y  ~<_  A }  =  om )
 
Theoremhashinfom 10074 The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  ( `  A )  = +oo )
 
Theoremhashennnuni 10075* The ordinal size of a set equinumerous to an element of  om is that element of  om. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( ( N  e.  om 
 /\  N  ~~  A )  ->  U. { y  e.  ( om  u.  { om } )  |  y  ~<_  A }  =  N )
 
Theoremhashennn 10076* The size of a set equinumerous to an element of  om. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( N  e.  om 
 /\  N  ~~  A )  ->  ( `  A )  =  (frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 ) `  N ) )
 
Theoremhashcl 10077 Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)
 |-  ( A  e.  Fin  ->  ( `  A )  e. 
 NN0 )
 
Theoremhashfiv01gt1 10078 The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( M  e.  Fin  ->  ( ( `  M )  =  0  \/  ( `  M )  =  1  \/  1  <  ( `  M ) ) )
 
Theoremhashfz1 10079 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 10080 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 ) )
 
Theoremhasheqf1o 10081* The size of two finite sets is equal if and only if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  =  ( `  B ) 
 <-> 
 E. f  f : A -1-1-onto-> B ) )
 
Theoremfiinfnf1o 10082* There is no bijection between a finite set and an infinite set. By infnfi 6556 the theorem would also hold if "infinite" were expressed as  om  ~<_  B. (Contributed by Alexander van der Vekens, 25-Dec-2017.)
 |-  ( ( A  e.  Fin  /\  -.  B  e.  Fin )  ->  -.  E. f  f : A -1-1-onto-> B )
 
Theoremfocdmex 10083 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
 |-  ( ( A  e.  V  /\  F : A -onto-> B )  ->  B  e.  _V )
 
Theoremfihasheqf1oi 10084 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-onto-> B )  ->  ( `  A )  =  ( `  B ) )
 
Theoremfihashf1rn 10085 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )
 
Theoremfihasheqf1od 10086 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A -1-1-onto-> B )   =>    |-  ( ph  ->  ( `  A )  =  ( `  B ) )
 
Theoremfz1eqb 10087 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 )
 )
 
Theoremfiltinf 10088 The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  om  ~<_  B )  ->  ( `  A )  < 
 ( `  B ) )
 
Theoremisfinite4im 10089 A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 1 ... ( `  A ) )  ~~  A )
 
Theoremfihasheq0 10090 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  0  <->  A  =  (/) ) )
 
Theoremfihashneq0 10091 Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6546. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 0  <  ( `  A )  <->  A  =/=  (/) ) )
 
Theoremhashnncl 10092 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  e.  NN  <->  A  =/=  (/) ) )
 
Theoremhash0 10093 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( `  (/) )  =  0
 
Theoremhashsng 10094 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 )
 
Theoremfihashen1 10095 A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  1  <->  A  ~~  1o )
 )
 
Theoremfihashfn 10096 A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( `  F )  =  ( `  A )
 )
 
Theoremfseq1hash 10097 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 )
 
Theoremomgadd 10098 Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `  A )  +  ( G `  B ) ) )
 
Theoremfihashdom 10099 Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  <_  ( `  B )  <->  A  ~<_  B ) )
 
Theoremhashunlem 10100 Lemma for hashun 10101. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  M  e.  om )   &    |-  ( ph  ->  A 
 ~~  N )   &    |-  ( ph  ->  B  ~~  M )   =>    |-  ( ph  ->  ( A  u.  B )  ~~  ( N  +o  M ) )
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