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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement

5.6.5  Ordered pair theorem for nonnegative integers

Theoremnn0le2msqi 11501 The square function on nonnegative integers is monotonic. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem1 11502 A rather pretty lemma for nn0opthi 11504. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0opthlem2 11503 Lemma for nn0opthi 11504. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.)

Theoremnn0opthi 11504 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers and by . 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 3780 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)

Theoremnn0opth2i 11505 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 11504. (Contributed by NM, 22-Jul-2004.)

Theoremnn0opth2 11506 An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 11504. (Contributed by NM, 22-Jul-2004.)

5.6.6  Factorial function

Syntaxcfa 11507 Extend class notation to include the factorial of nonnegative integers.

Definitiondf-fac 11508 Define the factorial function on nonnegative integers. For example, ; because ; (fac4 11515). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.)

Theoremfacnn 11509 Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac0 11510 The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac1 11511 The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfacp1 11512 The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremfac2 11513 The factorial of 2. (Contributed by NM, 17-Mar-2005.)

Theoremfac3 11514 The factorial of 3. (Contributed by NM, 17-Mar-2005.)

Theoremfac4 11515 The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.)
;

Theoremfacnn2 11516 Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.)

Theoremfaccl 11517 Closure of the factorial function. (Contributed by NM, 2-Dec-2004.)

Theoremfacne0 11518 The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.)

Theoremfacdiv 11519 A natural number divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.)

Theoremfacndiv 11520 No natural number (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.)

Theoremfacwordi 11521 Ordering property of factorial. (Contributed by NM, 9-Dec-2005.)

Theoremfaclbnd 11522 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd2 11523 A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.)

Theoremfaclbnd3 11524 A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.)

Theoremfaclbnd4lem1 11525 Lemma for faclbnd4 11529. Prepare the induction step. (Contributed by NM, 20-Dec-2005.)

Theoremfaclbnd4lem2 11526 Lemma for faclbnd4 11529. Use the weak deduction theorem to convert the hypotheses of faclbnd4lem1 11525 to antecedents. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd4lem3 11527 Lemma for faclbnd4 11529. The case. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd4lem4 11528 Lemma for faclbnd4 11529. Prove the case by induction on . (Contributed by NM, 19-Dec-2005.)

Theoremfaclbnd4 11529 Variant of faclbnd5 11530 providing a non-strict lower bound. (Contributed by NM, 23-Dec-2005.)

Theoremfaclbnd5 11530 The factorial function grows faster than powers and exponentiations. If we consider and to be constants, the right-hand side of the inequality is a constant times -factorial. (Contributed by NM, 24-Dec-2005.)

Theoremfaclbnd6 11531 Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.)

Theoremfacubnd 11532 An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremfacavg 11533 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.)

5.6.7  The binomial coefficient operation

Syntaxcbc 11534 Extend class notation to include the binomial coefficient operation (combinatorial choose operation).

Definitiondf-bc 11535* 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". 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 11536 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 11537 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval2 11537 Value of the binomial coefficient, choose , in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theorembcval3 11538 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 11539 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 11540 Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 11555.) (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembccmpl 11541 "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.)

Theorembcn0 11542 choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembc0k 11543 The binomial coefficient " 0 choose " is 0 for a positive integer K. Note that (see bcn0 11542). (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theorembcnn 11544 choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcn1 11545 Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcnp1n 11546 Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.)

Theorembcm1k 11547 The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1n 11548 The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembcp1nk 11549 The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorembcval5 11550 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 non-positive . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcn2 11551 Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.)

Theorembcp1m1 11552 Compute the binomial coefficent of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Theorembcpasc 11553 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 11554 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 11555 A binomial coefficient, in its standard domain, is a natural number. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.)

Theorembcn2m1 11556 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 11557 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 11558 The number of permutations of objects from a collection of objects is a natural number. (Contributed by Jason Orendorff, 24-Jan-2007.)

5.6.8  The ` # ` (finite set size) function

Syntaxchash 11559 Extend the definition of a class to include the size function.

Definitiondf-hash 11560 Define the function, which gives the cardinality of a finite set as a member of , and assigns all infinite sets the value . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremhashkf 11561 The finite part of the size function maps all finite sets to their cardinality, as members of . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashgval 11562* The value of the function in terms of the mapping from to . The proof avoids the use of ax-ac 8286. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremhashginv 11563* maps the size function's value to . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashinf 11564 The value of the function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.)

Theoremhashbnd 11565 If has size bounded by an integer , then is finite. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremhashf 11566 The size function maps all finite sets to their cardinality, as members of , and infinite sets to . (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashnn0pnf 11567 The value of the hash function for a set is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremhashnnn0genn0 11568 If the size of a set is not a nonnegative integer, it is greater than or equal to any nonnegative integer. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremhashnemnf 11569 The size of a set is never minus infinity. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Theoremhashv01gt1 11570 The size of a set is either 0 or 1 or greater than 1. (Contributed by Alexander van der Vekens, 29-Dec-2017.)

Theoremhashfz1 11571 The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashen 11572 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.)

Theoremhasheni 11573 Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.)

Theoremhasheqf1o 11574* 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 11575* There is no bijection between a finite set and an infinite set. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

Theoremhasheqf1oi 11576* The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Alexander van der Vekens, 25-Dec-2017.)

Theoremhashf1rn 11577 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 Alexander van der Vekens, 12-Jan-2018.)

Theoremfz1eqb 11578 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.)

Theoremhashcard 11579 The size function of the cardinality function. (Contributed by Mario Carneiro, 19-Sep-2013.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremhashcl 11580 Closure of the function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.)

Theoremhashxrcl 11581 Extended real closure of the function. (Contributed by Mario Carneiro, 22-Apr-2015.)

Theoremhashclb 11582 Reverse closure of the function. (Contributed by Mario Carneiro, 15-Jan-2015.)

Theoremhashvnfin 11583 A set of finite size is a finite set. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremhashnfinnn0 11584 The size of an infinite set is not a nonnegative integer. (Contributed by Alexander van der Vekens, 21-Dec-2017.) (Proof shortened by Alexander van der Vekens, 18-Jan-2018.)

Theoremhasheq0 11585 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.)

Theoremhashnncl 11586 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theoremhash0 11587 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

Theoremhashsng 11588 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremhashrabrsn 11589* The size of a restricted class abstraction restricted to a singleton is a nonnegative integer. (Contributed by Alexander van der Vekens, 22-Dec-2017.)

Theoremhashfn 11590 A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfseq1hash 11591 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.)

Theoremhashgadd 11592 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashgval2 11593 A short expression for the function of hashgf1o 11251. (Contributed by Mario Carneiro, 24-Jan-2015.)

Theoremhashdom 11594 Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremhashdomi 11595 Non-strict order relation of the function on the full cardinal poset. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theoremhashsdom 11596 Strict dominance relation for the size function. (Contributed by Mario Carneiro, 18-Aug-2014.)

Theoremhashun 11597 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.)

Theoremhashun2 11598 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.)

Theoremhashun3 11599 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.)

Theoremhashinfxadd 11600 The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)

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