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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nn0nnaddcl 9001 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 9002 | 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 9003 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 9004 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 9005 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 9006 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 9007 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 9008 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 9009 | 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 9010 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 9011 | 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 9012 | 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 9013 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 9014 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 9015 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 9016 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 9017 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 9018 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 9019 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 9020 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 9021 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 9022 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 9023 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 9024 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 9025 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 9026 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 9027 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 9028 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 9029 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 9030 | 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 9031 | 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 9032 | 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 7797. | ||
Syntax | cxnn0 9033 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 9034 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7797. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7000 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 9035 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0ssxnn0 9036 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0xnn0 9037 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xr 9038 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | 0xnn0 9039 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | pnf0xnn0 9040 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnf 9041 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Theorem | nn0xnn0d 9042 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnfd 9043 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
Theorem | xnn0nemnf 9044 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xrnemnf 9045 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0nnn0pnf 9046 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Syntax | cz 9047 | Extend class notation to include the class of integers. |
Definition | df-z 9048 | 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 9049 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | nnnegz 9050 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | zre 9051 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
Theorem | zcn 9052 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | zrei 9053 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
Theorem | zssre 9054 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
Theorem | zsscn 9055 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | zex 9056 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | elnnz 9057 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | 0z 9058 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | 0zd 9059 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elnn0z 9060 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn0nn 9061 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Theorem | elznn0 9062 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn 9063 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
Theorem | nnssz 9064 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
Theorem | nn0ssz 9065 | Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnz 9066 | A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nn0z 9067 | A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnzi 9068 | A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nn0zi 9069 | A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | elnnz1 9070 | Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nnzrab 9071 | Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | nn0zrab 9072 | Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.) |
Theorem | 1z 9073 | One is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 1zzd 9074 | 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
Theorem | 2z 9075 | Two is an integer. (Contributed by NM, 10-May-2004.) |
Theorem | 3z 9076 | 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4z 9077 | 4 is an integer. (Contributed by BJ, 26-Mar-2020.) |
Theorem | znegcl 9078 | Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
Theorem | neg1z 9079 | -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
Theorem | znegclb 9080 | A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Theorem | nn0negz 9081 | The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Theorem | nn0negzi 9082 | The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | peano2z 9083 | Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
Theorem | zaddcllempos 9084 | Lemma for zaddcl 9087. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | peano2zm 9085 | "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.) |
Theorem | zaddcllemneg 9086 | Lemma for zaddcl 9087. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zaddcl 9087 | Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | zsubcl 9088 | Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
Theorem | ztri3or0 9089 | Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | ztri3or 9090 | Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zletric 9091 | Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.) |
Theorem | zlelttric 9092 | Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.) |
Theorem | zltnle 9093 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Theorem | zleloe 9094 | Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.) |
Theorem | znnnlt1 9095 | An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.) |
Theorem | zletr 9096 | Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
Theorem | zrevaddcl 9097 | Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.) |
Theorem | znnsub 9098 | The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8752.) (Contributed by NM, 11-May-2004.) |
Theorem | nzadd 9099 | 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.) |
Theorem | zmulcl 9100 | Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.) |
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