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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | facnn 10501 | Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac0 10502 | The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac1 10503 | The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | facp1 10504 | The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac2 10505 | The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
Theorem | fac3 10506 | The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
Theorem | fac4 10507 | The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
; | ||
Theorem | facnn2 10508 | Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.) |
Theorem | faccl 10509 | Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
Theorem | faccld 10510 | Closure of the factorial function, deduction version of faccl 10509. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | facne0 10511 | The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) |
Theorem | facdiv 10512 | A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.) |
Theorem | facndiv 10513 | No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.) |
Theorem | facwordi 10514 | Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) |
Theorem | faclbnd 10515 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Theorem | faclbnd2 10516 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Theorem | faclbnd3 10517 | A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.) |
Theorem | faclbnd6 10518 | Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.) |
Theorem | facubnd 10519 | An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.) |
Theorem | facavg 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.) |
Syntax | cbc 10521 | Extend class notation to include the binomial coefficient operation (combinatorial choose operation). |
Definition | df-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.) |
Theorem | bcval 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.) |
Theorem | bcval2 10524 | Value of the binomial coefficient, choose , in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Theorem | bcval3 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.) |
Theorem | bcval4 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.) |
Theorem | bcrpcl 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.) |
Theorem | bccmpl 10528 | "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.) |
Theorem | bcn0 10529 | choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bc0k 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.) |
Theorem | bcnn 10531 | choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcn1 10532 | Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcnp1n 10533 | Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcm1k 10534 | The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
Theorem | bcp1n 10535 | The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
Theorem | bcp1nk 10536 | The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | bcval5 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.) |
Theorem | bcn2 10538 | Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.) |
Theorem | bcp1m1 10539 | Compute the binomial coefficient of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
Theorem | bcpasc 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.) |
Theorem | bccl 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.) |
Theorem | bccl2 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.) |
Theorem | bcn2m1 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.) |
Theorem | bcn2p1 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.) |
Theorem | permnn 10545 | The number of permutations of objects from a collection of objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.) |
Theorem | bcnm1 10546 | The binomial coefficent of is . (Contributed by Scott Fenton, 16-May-2014.) |
Theorem | 4bc3eq4 10547 | The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.) |
Theorem | 4bc2eq6 10548 | The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
Syntax | chash 10549 | Extend the definition of a class to include the set size function. |
♯ | ||
Definition | df-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 | ||
Theorem | hashinfuni 10551* | The ordinal size of an infinite set is . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Theorem | hashinfom 10552 | The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.) |
♯ | ||
Theorem | hashennnuni 10553* | The ordinal size of a set equinumerous to an element of is that element of . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Theorem | hashennn 10554* | The size of a set equinumerous to an element of . (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ frec | ||
Theorem | hashcl 10555 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
♯ | ||
Theorem | hashfiv01gt1 10556 | The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ ♯ ♯ | ||
Theorem | hashfz1 10557 | The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
♯ | ||
Theorem | hashen 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.) |
♯ ♯ | ||
Theorem | hasheqf1o 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.) |
♯ ♯ | ||
Theorem | fiinfnf1o 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.) |
Theorem | focdmex 10561 | The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.) |
Theorem | fihasheqf1oi 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.) |
♯ ♯ | ||
Theorem | fihashf1rn 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.) |
♯ ♯ | ||
Theorem | fihasheqf1od 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.) |
♯ ♯ | ||
Theorem | fz1eqb 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.) |
Theorem | filtinf 10566 | The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ ♯ | ||
Theorem | isfinite4im 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.) |
♯ | ||
Theorem | fihasheq0 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.) |
♯ | ||
Theorem | fihashneq0 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.) |
♯ | ||
Theorem | hashnncl 10570 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
♯ | ||
Theorem | hash0 10571 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
♯ | ||
Theorem | hashsng 10572 | The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
♯ | ||
Theorem | fihashen1 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.) |
♯ | ||
Theorem | fihashfn 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.) |
♯ ♯ | ||
Theorem | fseq1hash 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.) |
♯ | ||
Theorem | omgadd 10576 | Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.) |
frec | ||
Theorem | fihashdom 10577 | Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.) |
♯ ♯ | ||
Theorem | hashunlem 10578 | Lemma for hashun 10579. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.) |
Theorem | hashun 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.) |
♯ ♯ ♯ | ||
Theorem | 1elfz0hash 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.) |
♯ | ||
Theorem | hashunsng 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.) |
♯ ♯ | ||
Theorem | hashprg 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.) |
♯ | ||
Theorem | prhash2ex 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.) |
♯ | ||
Theorem | hashp1i 10584 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
♯ ♯ | ||
Theorem | hash1 10585 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
♯ | ||
Theorem | hash2 10586 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
♯ | ||
Theorem | hash3 10587 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
♯ | ||
Theorem | hash4 10588 | Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.) |
♯ | ||
Theorem | pr0hash2ex 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.) |
♯ | ||
Theorem | fihashss 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.) |
♯ ♯ | ||
Theorem | fiprsshashgt1 10591 | The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.) |
♯ | ||
Theorem | fihashssdif 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.) |
♯ ♯ ♯ | ||
Theorem | hashdifsn 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.) |
♯ ♯ | ||
Theorem | hashdifpr 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.) |
♯ ♯ | ||
Theorem | hashfz 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.) |
♯ | ||
Theorem | hashfzo 10596 | Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
♯..^ | ||
Theorem | hashfzo0 10597 | Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
♯..^ | ||
Theorem | hashfzp1 10598 | Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.) |
♯ | ||
Theorem | hashfz0 10599 | Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.) |
♯ | ||
Theorem | hashxp 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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