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

Theoremzq 10201 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)

Theoremzssq 10202 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssq 10203 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremnnssq 10204 The natural numbers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremqssre 10205 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

Theoremqsscn 10206 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremqex 10207 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnq 10208 A natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremqcn 10209 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)

TheoremqexALT 10210 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.)

Theoremqnegcl 10212 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)

Theoremqmulcl 10213 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)

Theoremqsubcl 10214 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremqreccl 10215 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqdivcl 10216 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqrevaddcl 10217 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremnnrecq 10218 The reciprocal of a natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremirradd 10219 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)

Theoremirrmul 10220 The product of an irrational with a nonzero rational is irrational. (Contributed by NM, 7-Nov-2008.)

5.4.12  Existence of the set of complex numbers

Theoremrpnnen1lem1 10221* Lemma for rpnnen1 10226. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem2 10222* Lemma for rpnnen1 10226. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem3 10223* Lemma for rpnnen1 10226. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem4 10224* Lemma for rpnnen1 10226. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem5 10225* Lemma for rpnnen1 10226. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1 10226* One half of rpnnen 12379, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number to the sequence such that is the largest rational number with denominator that is strictly less than . In this manner, we get a monotonically increasing sequence that converges to , and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.)

TheoremreexALT 10227 The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.)

Theoremcnref1o 10228* There is a natural one-to-one mapping from to , where we map to . In our construction of the complex numbers, this is in fact our definition of (see df-c 8623), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)

TheoremcnexALT 10229 The set of complex numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.)

Theoremxrex 10230 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)

Theoremaddex 10231 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremmulex 10232 The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

5.5  Order sets

5.5.1  Positive reals (as a subset of complex numbers)

Syntaxcrp 10233 Extend class notation to include the class of positive reals.

Definitiondf-rp 10234 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremelrp 10235 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)

Theoremelrpii 10236 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)

Theorem1rp 10237 1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)

Theorem2rp 10238 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpre 10239 A positive real is a real. (Contributed by NM, 27-Oct-2007.)

Theoremrpxr 10240 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremrpcn 10241 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)

Theoremnnrp 10242 A natural number is a positive real. (Contributed by NM, 28-Nov-2008.)

Theoremrpssre 10243 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)

Theoremrpgt0 10244 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)

Theoremrpge0 10245 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)

Theoremrpregt0 10246 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrprege0 10247 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpne0 10248 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)

Theoremrprene0 10249 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)

Theoremrpcnne0 10250 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)

Theoremralrp 10251 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Theoremrexrp 10252 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrpaddcl 10253 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpmulcl 10254 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpdivcl 10255 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)

Theoremrpreccl 10256 Closure law for reciprocation of positive reals. (Contributed by Jeffrey Hankins, 23-Nov-2008.)

Theoremrphalfcl 10257 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpgecl 10258 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalflt 10259 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrerpdivcl 10260 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)

Theoremge0p1rp 10261 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremrpneg 10262 Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.)

Theorem0nrp 10263 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremltsubrp 10264 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)

Theoremltaddrp 10265 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)

Theoremdifrp 10266 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 10267 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 10268 A natural number is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpred 10269 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 10270 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 10271 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 10272 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 10273 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpne0d 10274 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpregt0d 10275 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprege0d 10276 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprene0d 10277 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnne0d 10278 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpreccld 10279 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprecred 10280 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalfcld 10281 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreclt1d 10282 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt1d 10283 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpaddcld 10284 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpmulcld 10285 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpdivcld 10286 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrecd 10287 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerecd 10288 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrec1d 10289 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerec2d 10290 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2ad 10291 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv2d 10292 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2d 10293 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivdivd 10294 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0p1rpd 10295 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrerpdivcld 10296 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltsubrpd 10297 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrpd 10298 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrp2d 10299 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt11d 10300 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

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