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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2muliap0 8701 | is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
# | ||
Theorem | 2muline0 8702 | . See also 2muliap0 8701. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | halfcl 8703 | Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.) |
Theorem | rehalfcl 8704 | Real closure of half. (Contributed by NM, 1-Jan-2006.) |
Theorem | half0 8705 | Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.) |
Theorem | 2halves 8706 | Two halves make a whole. (Contributed by NM, 11-Apr-2005.) |
Theorem | halfpos2 8707 | A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.) |
Theorem | halfpos 8708 | A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | halfnneg2 8709 | A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.) |
Theorem | halfaddsubcl 8710 | Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
Theorem | halfaddsub 8711 | Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
Theorem | lt2halves 8712 | A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.) |
Theorem | addltmul 8713 | Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.) |
Theorem | nominpos 8714* | There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.) |
Theorem | avglt1 8715 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avglt2 8716 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avgle1 8717 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avgle2 8718 | Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | 2timesd 8719 | Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | times2d 8720 | A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | halfcld 8721 | Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | 2halvesd 8722 | Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | rehalfcld 8723 | Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2halvesd 8724 | A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | rehalfcli 8725 | Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.) |
Theorem | add1p1 8726 | Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
Theorem | sub1m1 8727 | Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.) |
Theorem | cnm2m1cnm3 8728 | Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Theorem | xp1d2m1eqxm1d2 8729 | A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.) |
Theorem | div4p1lem1div2 8730 | An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.) |
Theorem | arch 8731* | Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.) |
Theorem | nnrecl 8732* | There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.) |
Theorem | bndndx 8733* | A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.) |
Syntax | cn0 8734 | Extend class notation to include the class of nonnegative integers. |
Definition | df-n0 8735 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | elnn0 8736 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nnssnn0 8737 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0ssre 8738 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0sscn 8739 | Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.) |
Theorem | nn0ex 8740 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
Theorem | nnnn0 8741 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnnn0i 8742 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
Theorem | nn0re 8743 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0cn 8744 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0rei 8745 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
Theorem | nn0cni 8746 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
Theorem | dfn2 8747 | The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
Theorem | elnnne0 8748 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Theorem | 0nn0 8749 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 1nn0 8750 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 2nn0 8751 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 3nn0 8752 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 4nn0 8753 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 5nn0 8754 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 6nn0 8755 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 7nn0 8756 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 8nn0 8757 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 9nn0 8758 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | nn0ge0 8759 | A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nn0nlt0 8760 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0i 8761 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0le0eq0 8762 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
Theorem | nn0p1gt0 8763 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Theorem | nnnn0addcl 8764 | A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nn0nnaddcl 8765 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 8766 | The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.) |
Theorem | un0addcl 8767 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 8768 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 8769 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 8770 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 8771 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 8772 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 8773 | A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | peano2nn0 8774 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 8775 | A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | elnn0nn 8776 | The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | elnnnn0 8777 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 8778 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 8779 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 8780 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 8781 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 8782 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 8783 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 8784 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 8785 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 8786 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 8787 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 8788 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 8789 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 8790 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 8791 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 8792 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 8793 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 8794 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.) |
Theorem | nn0ge2m1nn0 8795 | If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
Theorem | nn0nndivcl 8796 | Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers , see df-xr 7587. | ||
Syntax | cxnn0 8797 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 8798 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7587. If we assumed excluded middle, this would be essentially the same as ℕ_{∞} as defined at df-nninf 6852 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 8799 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0ssxnn0 8800 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
NN0* |
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