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Type | Label | Description |
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Statement | ||
Theorem | iap0 9101 | The imaginary unit is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
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Theorem | 2muliap0 9102 | is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
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Theorem | 2muline0 9103 | . See also 2muliap0 9102. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | halfcl 9104 | Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.) |
Theorem | rehalfcl 9105 | Real closure of half. (Contributed by NM, 1-Jan-2006.) |
Theorem | half0 9106 | Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.) |
Theorem | 2halves 9107 | Two halves make a whole. (Contributed by NM, 11-Apr-2005.) |
Theorem | halfpos2 9108 | A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.) |
Theorem | halfpos 9109 | A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | halfnneg2 9110 | A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.) |
Theorem | halfaddsubcl 9111 | Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
Theorem | halfaddsub 9112 | Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.) |
Theorem | lt2halves 9113 | A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.) |
Theorem | addltmul 9114 | Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.) |
Theorem | nominpos 9115* | There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.) |
Theorem | avglt1 9116 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avglt2 9117 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avgle1 9118 | Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.) |
Theorem | avgle2 9119 | Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | 2timesd 9120 | Two times a number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | times2d 9121 | A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | halfcld 9122 | Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | 2halvesd 9123 | Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | rehalfcld 9124 | Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2halvesd 9125 | A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | rehalfcli 9126 | Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.) |
Theorem | add1p1 9127 | Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.) |
Theorem | sub1m1 9128 | Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.) |
Theorem | cnm2m1cnm3 9129 | 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 9130 | 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 9131 | 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 9132* | 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 9133* | 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 9134* | A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.) |
Syntax | cn0 9135 | Extend class notation to include the class of nonnegative integers. |
Definition | df-n0 9136 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | elnn0 9137 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nnssnn0 9138 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0ssre 9139 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0sscn 9140 | Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.) |
Theorem | nn0ex 9141 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
Theorem | nnnn0 9142 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnnn0i 9143 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
Theorem | nn0re 9144 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0cn 9145 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0rei 9146 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
Theorem | nn0cni 9147 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
Theorem | dfn2 9148 | 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 9149 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Theorem | 0nn0 9150 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 1nn0 9151 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 2nn0 9152 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 3nn0 9153 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 4nn0 9154 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 5nn0 9155 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 6nn0 9156 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 7nn0 9157 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 8nn0 9158 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 9nn0 9159 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | nn0ge0 9160 | 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 9161 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0i 9162 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0le0eq0 9163 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
Theorem | nn0p1gt0 9164 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Theorem | nnnn0addcl 9165 | 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 9166 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 9167 | 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 9168 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 9169 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 9170 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 9171 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 9172 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 9173 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 9174 | 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 9175 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 9176 | 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 9177 | 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 9178 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 9179 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 9180 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 9181 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 9182 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 9183 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 9184 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 9185 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 9186 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 9187 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 9188 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 9189 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 9190 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 9191 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 9192 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 9193 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 9194 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 9195 | 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 9196 | 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 9197 | 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 7958. | ||
Syntax | cxnn0 9198 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 9199 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7958. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7097 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 9200 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* |
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