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

Theorem9nn0 10201 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem10nn0 10202 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theoremnn0ge0 10203 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnn0nlt0 10204 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0i 10205 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0le0eq0 10206 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)

Theoremnnnn0addcl 10207 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 10208 A nonnegative integer plus a natural number is a natural number. (Contributed by NM, 22-Dec-2005.)

Theoremun0addcl 10209 If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremun0mulcl 10210 If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcl 10211 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0mulcl 10212 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcli 10213 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0mulcli 10214 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0p1nn 10215 A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)

Theorempeano2nn0 10216 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnnm1nn0 10217 A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremelnn0nn 10218 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 10219 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)

Theoremelnnnn0b 10220 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)

Theoremelnnnn0c 10221 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)

Theoremnn0addge1 10222 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2 10223 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge1i 10224 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2i 10225 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0sub 10226 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0le2xi 10227 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0lele2xi 10228 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0supp 10229 Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremnnnn0d 10230 A natural number is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0red 10231 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0cnd 10232 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 10233 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0addcld 10234 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 10235 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0n0n1ge2 10236 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 10237 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 10238 Extend class notation to include the class of integers.

Definitiondf-z 10239 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 10240 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Theoremnnnegz 10241 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremzre 10242 An integer is a real. (Contributed by NM, 8-Jan-2002.)

Theoremzcn 10243 An integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremzrei 10244 An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Theoremzssre 10245 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

Theoremzsscn 10246 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremzex 10247 The set of integers exists. See also zexALT 10256. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremelnnz 10248 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 10249 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremelnn0z 10250 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 10251 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 10252 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 10253 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

Theoremelz2 10254* 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 10255 Alternative definition of the integers, based on elz2 10254. (Contributed by Mario Carneiro, 16-May-2014.)

TheoremzexALT 10256 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 10257 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssz 10258 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

Theoremnnz 10259 A natural number is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0z 10260 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnnzi 10261 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn0zi 10262 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremelnnz1 10263 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznnnlt1 10264 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremnnzrab 10265 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theoremnn0zrab 10266 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theorem1z 10267 One is an integer. (Contributed by NM, 10-May-2004.)

Theorem2z 10268 Two is an integer. (Contributed by NM, 10-May-2004.)

Theoremznegcl 10269 Closure law for negative integers. (Contributed by NM, 9-May-2004.)

Theoremznegclb 10270 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnn0negz 10271 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0negzi 10272 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremzaddcl 10273 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theorempeano2z 10274 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)

Theoremzsubcl 10275 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)

Theorempeano2zm 10276 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)

Theoremzrevaddcl 10277 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)

Theoremznnsub 10278 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9994.) (Contributed by NM, 11-May-2004.)

Theoremznn0sub 10279 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 10226.) (Contributed by NM, 14-Jul-2005.)

Theoremzmulcl 10280 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 10281 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 10282 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 10283 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 10284 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremnnleltp1 10285 Natural number ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 10286 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 10287 Closure of addition of natural numbers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 10288 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 10289 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 10290 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 10291 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremnn0lt10b 10292 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lem1lt 10293 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 10294 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 10295 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremzdiv 10296* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 10297 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 10298 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 10299* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 10300* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

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