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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfacnn 10501 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 10502 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 10503 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 10504 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 10505 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 10506 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

Theoremfac4 10507 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
;

Theoremfacnn2 10508 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

Theoremfaccl 10509 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)

Theoremfaccld 10510 Closure of the factorial function, deduction version of faccl 10509. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Theoremfacne0 10511 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)

Theoremfacdiv 10512 A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)

Theoremfacndiv 10513 No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)

Theoremfacwordi 10514 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)

Theoremfaclbnd 10515 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd2 10516 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd3 10517 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)

Theoremfaclbnd6 10518 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)

Theoremfacubnd 10519 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremfacavg 10520 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.)

4.6.9  The binomial coefficient operation

Syntaxcbc 10521 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).

Definitiondf-bc 10522* Define the binomial coefficient operation. For example, (ex-bc 13095).

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". is read " choose ." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when does not hold. (Contributed by NM, 10-Jul-2005.)

Theorembcval 10523 Value of the binomial coefficient, choose . Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when does not hold. See bcval2 10524 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval2 10524 Value of the binomial coefficient, choose , in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval3 10525 Value of the binomial coefficient, choose , outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcval4 10526 Value of the binomial coefficient, choose , outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcrpcl 10527 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10542.) (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembccmpl 10528 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)

Theorembcn0 10529 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembc0k 10530 The binomial coefficient " 0 choose " is 0 for a positive integer K. Note that (see bcn0 10529). (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theorembcnn 10531 choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcn1 10532 Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnp1n 10533 Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcm1k 10534 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1n 10535 The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1nk 10536 The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorembcval5 10537 Write out the top and bottom parts of the binomial coefficient explicitly. In this form, it is valid even for , although it is no longer valid for nonpositive . (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)

Theorembcn2 10538 Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcp1m1 10539 Compute the binomial coefficient of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembcpasc 10540 Pascal's rule for the binomial coefficient, generalized to all integers . Equation 2 of [Gleason] p. 295. (Contributed by NM, 13-Jul-2005.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembccl 10541 A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theorembccl2 10542 A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembcn2m1 10543 Compute the binomial coefficient " choose 2 " from " choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018.)

Theorembcn2p1 10544 Compute the binomial coefficient " choose 2 " from " choose 2 ": N + ( N 2 ) = ( (N+1) 2 ). (Contributed by Alexander van der Vekens, 8-Jan-2018.)

Theorempermnn 10545 The number of permutations of objects from a collection of objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.)

Theorembcnm1 10546 The binomial coefficent of is . (Contributed by Scott Fenton, 16-May-2014.)

Theorem4bc3eq4 10547 The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.)

Theorem4bc2eq6 10548 The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.)

4.6.10  The ` # ` (set size) function

Syntaxchash 10549 Extend the definition of a class to include the set size function.

Definitiondf-ihash 10550* Define the set size function ♯, which gives the cardinality of a finite set as a member of , and assigns all infinite sets the value . For example, .

Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8364). 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 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

Theoremhashinfuni 10551* The ordinal size of an infinite set is . (Contributed by Jim Kingdon, 20-Feb-2022.)

Theoremhashinfom 10552 The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.)

Theoremhashennnuni 10553* The ordinal size of a set equinumerous to an element of is that element of . (Contributed by Jim Kingdon, 20-Feb-2022.)

Theoremhashennn 10554* The size of a set equinumerous to an element of . (Contributed by Jim Kingdon, 21-Feb-2022.)
frec

Theoremhashcl 10555 Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashfiv01gt1 10556 The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.)

Theoremhashfz1 10557 The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashen 10558 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.)

Theoremhasheqf1o 10559* 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.)

Theoremfiinfnf1o 10560* There is no bijection between a finite set and an infinite set. By infnfi 6793 the theorem would also hold if "infinite" were expressed as . (Contributed by Alexander van der Vekens, 25-Dec-2017.)

Theoremfocdmex 10561 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)

Theoremfihasheqf1oi 10562 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.)

Theoremfihashf1rn 10563 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.)

Theoremfihasheqf1od 10564 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.)

Theoremfz1eqb 10565 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.)

Theoremfiltinf 10566 The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)

Theoremisfinite4im 10567 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.)

Theoremfihasheq0 10568 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.)

Theoremfihashneq0 10569 Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6783. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

Theoremhashnncl 10570 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremhash0 10571 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremhashsng 10572 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremfihashen1 10573 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.)

Theoremfihashfn 10574 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.)

Theoremfseq1hash 10575 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.)

Theoremomgadd 10576 Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
frec

Theoremfihashdom 10577 Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)

Theoremhashunlem 10578 Lemma for hashun 10579. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)

Theoremhashun 10579 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.)

Theorem1elfz0hash 10580 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)

Theoremhashunsng 10581 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.)

Theoremhashprg 10582 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)

Theoremprhash2ex 10583 There is (at least) one set with two different elements: the unordered pair containing and . In contrast to pr0hash2ex 10589, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)

Theoremhashp1i 10584 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash1 10585 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash2 10586 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash3 10587 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremhash4 10588 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theorempr0hash2ex 10589 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)

Theoremfihashss 10590 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)

Theoremfiprsshashgt1 10591 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)

Theoremfihashssdif 10592 The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)

Theoremhashdifsn 10593 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.)

Theoremhashdifpr 10594 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)

Theoremhashfz 10595 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.)

Theoremhashfzo 10596 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashfzo0 10597 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremhashfzp1 10598 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)

Theoremhashfz0 10599 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

Theoremhashxp 10600 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

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