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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nnnn0addcl 8801 | 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 8802 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 8803 | 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 8804 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 8805 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 8806 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 8807 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 8808 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 8809 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 8810 | 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 8811 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 8812 | 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 8813 | 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 8814 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 8815 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 8816 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 8817 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 8818 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 8819 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 8820 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 8821 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 8822 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 8823 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 8824 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 8825 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 8826 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 8827 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 8828 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 8829 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 8830 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 8831 | 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 8832 | 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 8833 | 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 7623. | ||
Syntax | cxnn0 8834 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 8835 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7623. If we assumed excluded middle, this would be essentially the same as ℕ_{∞} as defined at df-nninf 6878 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 8836 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0ssxnn0 8837 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0xnn0 8838 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xr 8839 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | 0xnn0 8840 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | pnf0xnn0 8841 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnf 8842 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Theorem | nn0xnn0d 8843 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnfd 8844 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
Theorem | xnn0nemnf 8845 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xrnemnf 8846 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0nnn0pnf 8847 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Syntax | cz 8848 | Extend class notation to include the class of integers. |
Definition | df-z 8849 | Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.) |
Theorem | elz 8850 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | nnnegz 8851 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | zre 8852 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
Theorem | zcn 8853 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | zrei 8854 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
Theorem | zssre 8855 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
Theorem | zsscn 8856 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | zex 8857 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | elnnz 8858 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | 0z 8859 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | 0zd 8860 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elnn0z 8861 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn0nn 8862 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Theorem | elznn0 8863 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn 8864 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
Theorem | nnssz 8865 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
Theorem | nn0ssz 8866 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnz 8867 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nn0z 8868 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnzi 8869 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nn0zi 8870 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | elnnz1 8871 | Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nnzrab 8872 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | nn0zrab 8873 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | 1z 8874 | One is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 1zzd 8875 | 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | 2z 8876 | Two is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 3z 8877 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4z 8878 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
Theorem | znegcl 8879 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
Theorem | neg1z 8880 | -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
Theorem | znegclb 8881 | A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Theorem | nn0negz 8882 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nn0negzi 8883 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | peano2z 8884 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
Theorem | zaddcllempos 8885 | Lemma for zaddcl 8888. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | peano2zm 8886 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
Theorem | zaddcllemneg 8887 | Lemma for zaddcl 8888. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zaddcl 8888 | Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | zsubcl 8889 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
Theorem | ztri3or0 8890 | Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | ztri3or 8891 | Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zletric 8892 | Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.) |
Theorem | zlelttric 8893 | Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.) |
Theorem | zltnle 8894 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zleloe 8895 | Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.) |
Theorem | znnnlt1 8896 | An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.) |
Theorem | zletr 8897 | Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
Theorem | zrevaddcl 8898 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
Theorem | znnsub 8899 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8559.) (Contributed by NM, 11-May-2004.) |
Theorem | nzadd 8900 | The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.) |
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