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Type | Label | Description |
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Statement | ||
Theorem | sqdivapd 10701 | Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.) |
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Theorem | expdivapd 10702 | Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.) |
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Theorem | mulexpd 10703 | Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | 0expd 10704 | Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | reexpcld 10705 | Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | expge0d 10706 | A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | expge1d 10707 | A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | sqoddm1div8 10708 | A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) |
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Theorem | nnsqcld 10709 | The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | nnexpcld 10710 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | nn0expcld 10711 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | rpexpcld 10712 | Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | reexpclzapd 10713 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.) |
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Theorem | resqcld 10714 | Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | sqge0d 10715 | A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | sqgt0apd 10716 | The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.) |
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Theorem | leexp2ad 10717 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | leexp2rd 10718 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lt2sqd 10719 | The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | le2sqd 10720 | The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | sq11d 10721 | The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | sq11ap 10722 | Analogue to sq11 10627 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.) |
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Theorem | zzlesq 10723 | An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.) |
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Theorem | nn0ltexp2 10724 | Special case of ltexp2 14837 which we use here because we haven't yet defined df-rpcxp 14757 which is used in the current proof of ltexp2 14837. (Contributed by Jim Kingdon, 7-Oct-2024.) |
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Theorem | nn0leexp2 10725 | Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.) |
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Theorem | mulsubdivbinom2ap 10726 | The square of a binomial with factor minus a number divided by a number apart from zero. (Contributed by AV, 19-Jul-2021.) |
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Theorem | sq10 10727 | The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
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Theorem | sq10e99m1 10728 | The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
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Theorem | 3dec 10729 | 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.) |
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Theorem | expcanlem 10730 | Lemma for expcan 10731. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.) |
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Theorem | expcan 10731 | Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
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Theorem | expcand 10732 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | apexp1 10733 | Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.) |
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Theorem | nn0le2msqd 10734 | The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.) |
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Theorem | nn0opthlem1d 10735 | A rather pretty lemma for nn0opth2 10739. (Contributed by Jim Kingdon, 31-Oct-2021.) |
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Theorem | nn0opthlem2d 10736 | Lemma for nn0opth2 10739. (Contributed by Jim Kingdon, 31-Oct-2021.) |
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Theorem | nn0opthd 10737 |
An ordered pair theorem for nonnegative integers. Theorem 17.3 of
[Quine] p. 124. We can represent an
ordered pair of nonnegative
integers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nn0opth2d 10738 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10737. (Contributed by Jim Kingdon, 31-Oct-2021.) |
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Theorem | nn0opth2 10739 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10737. (Contributed by NM, 22-Jul-2004.) |
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Syntax | cfa 10740 | Extend class notation to include the factorial of nonnegative integers. |
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Definition | df-fac 10741 |
Define the factorial function on nonnegative integers. For example,
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Theorem | facnn 10742 | Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
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Theorem | fac0 10743 | The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
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Theorem | fac1 10744 | The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
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Theorem | facp1 10745 | The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
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Theorem | fac2 10746 | The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
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Theorem | fac3 10747 | The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
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Theorem | fac4 10748 | The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
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Theorem | facnn2 10749 | Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.) |
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Theorem | faccl 10750 | Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
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Theorem | faccld 10751 | Closure of the factorial function, deduction version of faccl 10750. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | facne0 10752 | The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) |
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Theorem | facdiv 10753 | A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.) |
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Theorem | facndiv 10754 | No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.) |
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Theorem | facwordi 10755 | Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) |
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Theorem | faclbnd 10756 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
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Theorem | faclbnd2 10757 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
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Theorem | faclbnd3 10758 | A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.) |
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Theorem | faclbnd6 10759 | Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.) |
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Theorem | facubnd 10760 | An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.) |
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Theorem | facavg 10761 | 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.) |
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Syntax | cbc 10762 | Extend class notation to include the binomial coefficient operation (combinatorial choose operation). |
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Definition | df-bc 10763* |
Define the binomial coefficient operation. For example,
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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". |
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Theorem | bcval 10764 |
Value of the binomial coefficient, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bcval2 10765 |
Value of the binomial coefficient, ![]() ![]() |
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Theorem | bcval3 10766 |
Value of the binomial coefficient, ![]() ![]() |
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Theorem | bcval4 10767 |
Value of the binomial coefficient, ![]() ![]() |
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Theorem | bcrpcl 10768 | Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10783.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
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Theorem | bccmpl 10769 | "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.) |
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Theorem | bcn0 10770 |
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Theorem | bc0k 10771 |
The binomial coefficient " 0 choose ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bcnn 10772 |
![]() ![]() |
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Theorem | bcn1 10773 |
Binomial coefficient: ![]() ![]() |
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Theorem | bcnp1n 10774 |
Binomial coefficient: ![]() ![]() ![]() ![]() |
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Theorem | bcm1k 10775 |
The proportion of one binomial coefficient to another with ![]() |
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Theorem | bcp1n 10776 |
The proportion of one binomial coefficient to another with ![]() |
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Theorem | bcp1nk 10777 |
The proportion of one binomial coefficient to another with ![]() ![]() |
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Theorem | bcval5 10778 |
Write out the top and bottom parts of the binomial coefficient
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Theorem | bcn2 10779 |
Binomial coefficient: ![]() ![]() |
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Theorem | bcp1m1 10780 |
Compute the binomial coefficient of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bcpasc 10781 |
Pascal's rule for the binomial coefficient, generalized to all integers
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Theorem | bccl 10782 | A binomial coefficient, in its extended domain, is a nonnegative integer. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 9-Nov-2013.) |
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Theorem | bccl2 10783 | A binomial coefficient, in its standard domain, is a positive integer. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 10-Mar-2014.) |
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Theorem | bcn2m1 10784 |
Compute the binomial coefficient "![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bcn2p1 10785 |
Compute the binomial coefficient "![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | permnn 10786 |
The number of permutations of ![]() ![]() ![]() ![]() |
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Theorem | bcnm1 10787 |
The binomial coefficent of ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 4bc3eq4 10788 | The value of four choose three. (Contributed by Scott Fenton, 11-Jun-2016.) |
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Theorem | 4bc2eq6 10789 | The value of four choose two. (Contributed by Scott Fenton, 9-Jan-2017.) |
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Syntax | chash 10790 | Extend the definition of a class to include the set size function. |
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Definition | df-ihash 10791* |
Define the set size function ♯, which gives the cardinality of a
finite set as a member of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Note that we use the sharp sign (♯) for this function and we use the different character octothorpe (#) for the apartness relation (see df-ap 8570). 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 |
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Theorem | hashinfuni 10792* |
The ordinal size of an infinite set is ![]() |
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Theorem | hashinfom 10793 | The value of the ♯ function on an infinite set. (Contributed by Jim Kingdon, 20-Feb-2022.) |
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Theorem | hashennnuni 10794* |
The ordinal size of a set equinumerous to an element of ![]() ![]() |
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Theorem | hashennn 10795* |
The size of a set equinumerous to an element of ![]() |
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Theorem | hashcl 10796 | Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
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Theorem | hashfiv01gt1 10797 | The size of a finite set is either 0 or 1 or greater than 1. (Contributed by Jim Kingdon, 21-Feb-2022.) |
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Theorem | hashfz1 10798 |
The set ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hashen 10799 | 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.) |
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Theorem | hasheqf1o 10800* | 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.) |
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