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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sqge0d 10001 | A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | sqgt0apd 10002 | The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.) |
# | ||
Theorem | leexp2ad 10003 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | leexp2rd 10004 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lt2sqd 10005 | The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | le2sqd 10006 | The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | sq11d 10007 | The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | sq11ap 10008 | Analogue to sq11 9917 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# # | ||
Theorem | sq10 10009 | The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
; ;; | ||
Theorem | sq10e99m1 10010 | The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
; ; | ||
Theorem | 3dec 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.) |
;; ; ; | ||
Theorem | expcanlem 10012 | Lemma for expcan 10013. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.) |
Theorem | expcan 10013 | Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Theorem | expcand 10014 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | nn0le2msqd 10015 | The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Theorem | nn0opthlem1d 10016 | A rather pretty lemma for nn0opth2 10020. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Theorem | nn0opthlem2d 10017 | Lemma for nn0opth2 10020. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Theorem | nn0opthd 10018 | 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 3439 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Theorem | nn0opth2d 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.) |
Theorem | nn0opth2 10020 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10018. (Contributed by NM, 22-Jul-2004.) |
Syntax | cfa 10021 | Extend class notation to include the factorial of nonnegative integers. |
Definition | df-fac 10022 | Define the factorial function on nonnegative integers. For example, because (ex-fac 11084). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.) |
Theorem | facnn 10023 | Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac0 10024 | The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac1 10025 | The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | facp1 10026 | The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
Theorem | fac2 10027 | The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
Theorem | fac3 10028 | The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
Theorem | fac4 10029 | The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
; | ||
Theorem | facnn2 10030 | Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.) |
Theorem | faccl 10031 | Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
Theorem | faccld 10032 | Closure of the factorial function, deduction version of faccl 10031. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | facne0 10033 | The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) |
Theorem | facdiv 10034 | A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.) |
Theorem | facndiv 10035 | 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 10036 | Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) |
Theorem | faclbnd 10037 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Theorem | faclbnd2 10038 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Theorem | faclbnd3 10039 | A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.) |
Theorem | faclbnd6 10040 | Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.) |
Theorem | facubnd 10041 | An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.) |
Theorem | facavg 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.) |
Syntax | cbc 10043 | Extend class notation to include the binomial coefficient operation (combinatorial choose operation). |
Definition | df-bc 10044* |
Define the binomial coefficient operation. For example,
(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". 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 10045 | 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 10046 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
Theorem | bcval2 10046 | 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 10047 | 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 10048 | 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 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.) |
Theorem | bccmpl 10050 | "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 10051 | choose 0 is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bc0k 10052 | The binomial coefficient " 0 choose " is 0 for a positive integer K. Note that (see bcn0 10051). (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Theorem | bcnn 10053 | choose is 1. Remark in [Gleason] p. 296. (Contributed by NM, 17-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcn1 10054 | Binomial coefficient: choose . (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcnp1n 10055 | Binomial coefficient: choose . (Contributed by NM, 20-Jun-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
Theorem | bcm1k 10056 | The proportion of one binomial coefficient to another with decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
Theorem | bcp1n 10057 | The proportion of one binomial coefficient to another with increased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.) |
Theorem | bcp1nk 10058 | The proportion of one binomial coefficient to another with and increased by 1. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Theorem | ibcval5 10059 | 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 Jim Kingdon, 6-Nov-2021.) |
Theorem | bcn2 10060 | Binomial coefficient: choose . (Contributed by Mario Carneiro, 22-May-2014.) |
Theorem | bcp1m1 10061 | Compute the binomial coefficient of over (Contributed by Scott Fenton, 11-May-2014.) (Revised by Mario Carneiro, 22-May-2014.) |
Theorem | bcpasc 10062 | 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 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.) |
Theorem | bccl2 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.) |
Theorem | bcn2m1 10065 | 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 10066 | 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 10067 | The number of permutations of objects from a collection of objects is a positive integer. (Contributed by Jason Orendorff, 24-Jan-2007.) |
Theorem | bcnm1 10068 | The binomial coefficent of is . (Contributed by Scott Fenton, 16-May-2014.) |
Theorem | 4bc3eq4 10069 | The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.) |
Theorem | 4bc2eq6 10070 | The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
Syntax | chash 10071 | Extend the definition of a class to include the set size function. |
♯ | ||
Definition | df-ihash 10072* |
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 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 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 10073* | The ordinal size of an infinite set is . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Theorem | hashinfom 10074 | The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.) |
♯ | ||
Theorem | hashennnuni 10075* | The ordinal size of a set equinumerous to an element of is that element of . (Contributed by Jim Kingdon, 20-Feb-2022.) |
Theorem | hashennn 10076* | The size of a set equinumerous to an element of . (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ frec | ||
Theorem | hashcl 10077 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
♯ | ||
Theorem | hashfiv01gt1 10078 | The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ ♯ ♯ | ||
Theorem | hashfz1 10079 | The set has elements. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
♯ | ||
Theorem | hashen 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.) |
♯ ♯ | ||
Theorem | hasheqf1o 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.) |
♯ ♯ | ||
Theorem | fiinfnf1o 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 . (Contributed by Alexander van der Vekens, 25-Dec-2017.) |
Theorem | focdmex 10083 | The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.) |
Theorem | fihasheqf1oi 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.) |
♯ ♯ | ||
Theorem | fihashf1rn 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.) |
♯ ♯ | ||
Theorem | fihasheqf1od 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.) |
♯ ♯ | ||
Theorem | fz1eqb 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.) |
Theorem | filtinf 10088 | The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.) |
♯ ♯ | ||
Theorem | isfinite4im 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.) |
♯ | ||
Theorem | fihasheq0 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.) |
♯ | ||
Theorem | fihashneq0 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.) |
♯ | ||
Theorem | hashnncl 10092 | Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.) |
♯ | ||
Theorem | hash0 10093 | The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.) |
♯ | ||
Theorem | hashsng 10094 | The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
♯ | ||
Theorem | fihashen1 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.) |
♯ | ||
Theorem | fihashfn 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.) |
♯ ♯ | ||
Theorem | fseq1hash 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.) |
♯ | ||
Theorem | omgadd 10098 | Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.) |
frec | ||
Theorem | fihashdom 10099 | Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.) |
♯ ♯ | ||
Theorem | hashunlem 10100 | Lemma for hashun 10101. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.) |
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