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Theorem List for Metamath Proof Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement

5.4.5  The Archimedean property

Theoremnnunb 10201* The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)

Theoremarch 10202* 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.)

Theoremnnrecl 10203* There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)

Theorembndndx 10204* A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)

5.4.6  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 10205 Extend class notation to include the class of nonnegative integers.

Definitiondf-n0 10206 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremelnn0 10207 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnnssnn0 10208 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0ssre 10209 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0sscn 10210 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

Theoremnn0ex 10211 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

Theoremnnnn0 10212 A natural number is a nonnegative integer. (Contributed by NM, 9-May-2004.)

Theoremnnnn0i 10213 A natural number is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Theoremnn0re 10214 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)

Theoremnn0cn 10215 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremnn0rei 10216 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)

Theoremnn0cni 10217 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)

Theoremdfn2 10218 The set of natural numbers (positive integers) defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremelnnne0 10219 The natural number property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theorem0nn0 10220 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem1nn0 10221 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem2nn0 10222 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem3nn0 10223 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem4nn0 10224 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem5nn0 10225 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem6nn0 10226 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem7nn0 10227 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem8nn0 10228 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem9nn0 10229 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem10nn0 10230 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theoremnn0ge0 10231 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnn0nlt0 10232 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0i 10233 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0le0eq0 10234 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)

Theoremnnnn0addcl 10235 A natural number plus a nonnegative integer is a natural number. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0nnaddcl 10236 A nonnegative integer plus a natural number is a natural number. (Contributed by NM, 22-Dec-2005.)

Theoremun0addcl 10237 If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremun0mulcl 10238 If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcl 10239 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0mulcl 10240 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcli 10241 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0mulcli 10242 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0p1nn 10243 A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)

Theorempeano2nn0 10244 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnnm1nn0 10245 A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremelnn0nn 10246 The nonnegative integer property expressed in terms of natural numbers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremelnnnn0 10247 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)

Theoremelnnnn0b 10248 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)

Theoremelnnnn0c 10249 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)

Theoremnn0addge1 10250 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2 10251 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge1i 10252 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2i 10253 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0sub 10254 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0le2xi 10255 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0lele2xi 10256 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0supp 10257 Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremnnnn0d 10258 A natural number is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0red 10259 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0cnd 10260 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 10261 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0addcld 10262 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 10263 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0n0n1ge2 10264 A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremnn0n0n1ge2b 10265 A nonnegative integer is neither 0 nor 1 if and only if it is is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)

5.4.7  Integers (as a subset of complex numbers)

Syntaxcz 10266 Extend class notation to include the class of integers.

Definitiondf-z 10267 Define the set of integers, which are the positive and negative natural numbers 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.)

Theoremelz 10268 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Theoremnnnegz 10269 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremzre 10270 An integer is a real. (Contributed by NM, 8-Jan-2002.)

Theoremzcn 10271 An integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremzrei 10272 An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Theoremzssre 10273 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

Theoremzsscn 10274 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremzex 10275 The set of integers exists. See also zexALT 10284. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremelnnz 10276 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 10277 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremelnn0z 10278 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 10279 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 10280 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 10281 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

Theoremelz2 10282* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the natural numbers. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremdfz2 10283 Alternative definition of the integers, based on elz2 10282. (Contributed by Mario Carneiro, 16-May-2014.)

TheoremzexALT 10284 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnnssz 10285 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssz 10286 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

Theoremnnz 10287 A natural number is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0z 10288 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnnzi 10289 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn0zi 10290 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremelnnz1 10291 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznnnlt1 10292 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremnnzrab 10293 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theoremnn0zrab 10294 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theorem1z 10295 One is an integer. (Contributed by NM, 10-May-2004.)

Theorem2z 10296 Two is an integer. (Contributed by NM, 10-May-2004.)

Theoremznegcl 10297 Closure law for negative integers. (Contributed by NM, 9-May-2004.)

Theoremznegclb 10298 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnn0negz 10299 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0negzi 10300 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

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