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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xp1d2m1eqxm1d2 8601 | 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 8602 | 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 8603* | 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 8604* | 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 8605* | A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.) |
Syntax | cn0 8606 | Extend class notation to include the class of nonnegative integers. |
Definition | df-n0 8607 | Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | elnn0 8608 | Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nnssnn0 8609 | Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0ssre 8610 | Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0sscn 8611 | Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.) |
Theorem | nn0ex 8612 | The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
Theorem | nnnn0 8613 | A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.) |
Theorem | nnnn0i 8614 | A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.) |
Theorem | nn0re 8615 | A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0cn 8616 | A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | nn0rei 8617 | A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
Theorem | nn0cni 8618 | A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
Theorem | dfn2 8619 | 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 8620 | The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Theorem | 0nn0 8621 | 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 1nn0 8622 | 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 2nn0 8623 | 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | 3nn0 8624 | 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 4nn0 8625 | 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | 5nn0 8626 | 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 6nn0 8627 | 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 7nn0 8628 | 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 8nn0 8629 | 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | 9nn0 8630 | 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Theorem | nn0ge0 8631 | 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 8632 | A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0i 8633 | Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0le0eq0 8634 | A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.) |
Theorem | nn0p1gt0 8635 | A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
Theorem | nnnn0addcl 8636 | 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 8637 | A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.) |
Theorem | 0mnnnnn0 8638 | 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 8639 | If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | un0mulcl 8640 | If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcl 8641 | Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0mulcl 8642 | Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
Theorem | nn0addcli 8643 | Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0mulcli 8644 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0p1nn 8645 | 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 8646 | Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | nnm1nn0 8647 | 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 8648 | 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 8649 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.) |
Theorem | elnnnn0b 8650 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.) |
Theorem | elnnnn0c 8651 | The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Theorem | nn0addge1 8652 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2 8653 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge1i 8654 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0addge2i 8655 | A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Theorem | nn0le2xi 8656 | A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0lele2xi 8657 | 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | nn0supp 8658 | Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.) |
Theorem | nnnn0d 8659 | A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0red 8660 | A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0cnd 8661 | A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0ge0d 8662 | A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0addcld 8663 | Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0mulcld 8664 | Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nn0readdcl 8665 | Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.) |
Theorem | nn0ge2m1nn 8666 | 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 8667 | 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 8668 | 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 7470. | ||
Syntax | cxnn0 8669 | The set of extended nonnegative integers. |
NN0* | ||
Definition | df-xnn0 8670 | Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers , see df-xr 7470. If we assumed excluded middle, this would be essentially the same as ℕ_{∞} as defined at df-nninf 6735 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | elxnn0 8671 | An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0ssxnn0 8672 | The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0xnn0 8673 | A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xr 8674 | An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | 0xnn0 8675 | Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | pnf0xnn0 8676 | Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnf 8677 | No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Theorem | nn0xnn0d 8678 | A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | nn0nepnfd 8679 | No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.) |
Theorem | xnn0nemnf 8680 | No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0xrnemnf 8681 | The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Theorem | xnn0nnn0pnf 8682 | An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.) |
NN0* | ||
Syntax | cz 8683 | Extend class notation to include the class of integers. |
Definition | df-z 8684 | 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 8685 | Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | nnnegz 8686 | The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | zre 8687 | An integer is a real. (Contributed by NM, 8-Jan-2002.) |
Theorem | zcn 8688 | An integer is a complex number. (Contributed by NM, 9-May-2004.) |
Theorem | zrei 8689 | An integer is a real number. (Contributed by NM, 14-Jul-2005.) |
Theorem | zssre 8690 | The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.) |
Theorem | zsscn 8691 | The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | zex 8692 | The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | elnnz 8693 | Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.) |
Theorem | 0z 8694 | Zero is an integer. (Contributed by NM, 12-Jan-2002.) |
Theorem | 0zd 8695 | Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | elnn0z 8696 | Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn0nn 8697 | Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Theorem | elznn0 8698 | Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Theorem | elznn 8699 | Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.) |
Theorem | nnssz 8700 | Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.) |
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