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

Theoremnn0cnd 10201 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 10202 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0addcld 10203 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 10204 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0n0n1ge2 10205 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 10206 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 10207 Extend class notation to include the class of integers.

Definitiondf-z 10208 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 10209 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Theoremnnnegz 10210 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremzre 10211 An integer is a real. (Contributed by NM, 8-Jan-2002.)

Theoremzcn 10212 An integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremzrei 10213 An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Theoremzssre 10214 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

Theoremzsscn 10215 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremzex 10216 The set of integers exists. See also zexALT 10225. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremelnnz 10217 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 10218 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremelnn0z 10219 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 10220 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 10221 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 10222 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

Theoremelz2 10223* 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 10224 Alternative definition of the integers, based on elz2 10223. (Contributed by Mario Carneiro, 16-May-2014.)

TheoremzexALT 10225 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 10226 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssz 10227 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

Theoremnnz 10228 A natural number is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0z 10229 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnnzi 10230 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn0zi 10231 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremelnnz1 10232 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznnnlt1 10233 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremnnzrab 10234 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theoremnn0zrab 10235 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theorem1z 10236 One is an integer. (Contributed by NM, 10-May-2004.)

Theorem2z 10237 Two is an integer. (Contributed by NM, 10-May-2004.)

Theoremznegcl 10238 Closure law for negative integers. (Contributed by NM, 9-May-2004.)

Theoremznegclb 10239 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnn0negz 10240 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0negzi 10241 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremzaddcl 10242 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theorempeano2z 10243 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)

Theoremzsubcl 10244 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)

Theorempeano2zm 10245 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)

Theoremzrevaddcl 10246 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)

Theoremznnsub 10247 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9963.) (Contributed by NM, 11-May-2004.)

Theoremznn0sub 10248 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 10195.) (Contributed by NM, 14-Jul-2005.)

Theoremzmulcl 10249 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 10250 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 10251 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 10252 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 10253 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremnnleltp1 10254 Natural number ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 10255 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 10256 Closure of addition of natural numbers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 10257 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 10258 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 10259 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 10260 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremnn0lt10b 10261 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lem1lt 10262 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 10263 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 10264 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremzdiv 10265* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 10266 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 10267 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 10268* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 10269* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 10270 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 10271 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 10272 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 10273 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theoremsuprzcl 10274* The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremprime 10275* Two ways to express " is a prime number (or 1)." See also isprm 13001. (Contributed by NM, 4-May-2005.)

Theoremmsqznn 10276 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 10277 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneo 10278 A natural number is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoi 10279 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 10280 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 10281 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 10282* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzi 10283* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5uzti 10284* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theoremdfuzi 10285* An expression for the upper integers that start at that is analogous to df-nn 9926 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 10286* Induction on the upper integers that start at . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 5-Jul-2005.)

Theoremuzind2 10287* Induction on the upper integers that start after an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 25-Jul-2005.)

Theoremuzind3 10288* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 26-Jul-2005.)

TheoremuzindOLD 10289* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list. Anyone is welcome to try. (Contributed by NM, 11-May-2004.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremuzind3OLD 10290* Induction on the set of upper integers that starts at . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 9-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnn0ind 10291* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 13-May-2004.)

Theoremnn0indALT 10292* Principle of Mathematical Induction (inference schema) on nonnegative integers. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction hypothesis. Either nn0ind 10291 or nn0indALT 10292 may be used; see comment for nnind 9943. (Contributed by NM, 28-Nov-2005.)

Theoremfzind 10293* Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremfnn0ind 10294* Induction on the integers from to inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0ind-raph 10295* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)

Theoremzindd 10296* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)

Theorembtwnz 10297* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 10298 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 10299 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 10300 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

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