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

Theoremrexuz 10201* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremrexuz2 10202* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theorem2rexuz 10203* Double existential quantification in a set of upper integers. (Contributed by NM, 3-Nov-2005.)

Theorempeano2uz 10204 Second Peano postulate for a set of upper integers. (Contributed by NM, 7-Sep-2005.)

Theorempeano2uzs 10205 Second Peano postulate for a set of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theorempeano2uzr 10206 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)

Theoremuzaddcl 10207 Addition closure law for a set of upper integers. (Contributed by NM, 4-Jun-2006.)

Theoremuzind4 10208* Induction on the set of upper integers that starts 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, 7-Sep-2005.)

Theoremuzind4ALT 10209* Alternate version of uzind4 10208 with different hypothesis order for easier use with the Metamath Proof Assistant, since "assign last" will assign the substitutions first. (This may or may not be kept permanenently, or it may replace uzind4 10208- I haven't decided yet. --NM) (Contributed by NM, 7-Sep-2005.)

Theoremuzind4s 10210* Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. (Contributed by NM, 4-Nov-2005.)

Theoremuzind4s2 10211* Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. Use this instead of uzind4s 10210 when and must be distinct in . (Contributed by NM, 16-Nov-2005.)

Theoremuzind4i 10212* Induction on the upper integers that start at . The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 4-Sep-2005.)

Theoremuzwo 10213* Well-ordering principle: any non-empty subset of a set of upper integers has a least element. (Contributed by NM, 8-Oct-2005.)

TheoremuzwoOLD 10214* Well-ordering principle: any non-empty subset of the upper integers has a least element. (Contributed by NM, 8-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremuzwo2 10215* Well-ordering principle: any non-empty subset of upper integers has a unique least element. (Contributed by NM, 8-Oct-2005.)

Theoremnnwo 10216* Well-ordering principle: any non-empty set of natural numbers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)

Theoremnnwof 10217* Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows and to be present in as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnnwos 10218* Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)

Theoremindstr 10219* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)

Theoremeluznn0 10220 Membership in a nonegative set of upper integers implies membership in . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremeluz2b1 10221 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b2 10222 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b3 10223 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremuz2m1nn 10224 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)

Theorem1nuz2 10225 1 is not in . (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremelnn1uz2 10226 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremuz2mulcl 10227 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremindstr2 10228* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremuzinfmi 10229 Extract the lower bound of a set of upper integers as its infimum. Note that the " " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)

Theoremnninfm 10230 The infimum of the set of natural numbers is one. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoremnn0infm 10231 The infimum of the set of nonnegative integers is zero. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoreminfmssuzle 10232 The infimum of a subset of a set of upper integers is less than or equal to all members of the subset. Note that the " " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.)

Theoreminfmssuzcl 10233 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)

Theoremublbneg 10234* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremeqreznegel 10235* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegn0 10236* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsupminf 10237* The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremlbzbi 10238* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremzsupss 10239* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 8748.) (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsuprzcl2 10240* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 10023 avoids ax-pre-sup 8748.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsuprzub 10241* The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremuzsupss 10242* Any bounded subset of upper integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)

5.4.10  Well-ordering principle for bounded-below sets of integers

Theoremuzwo3 10243* Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 10215 allows the lower bound to be any real number. See also nnwo 10216 and nnwos 10218. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)

Theoremzmin 10244* There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theoremzmax 10245* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)

Theoremzbtwnre 10246* There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)

Theoremrebtwnz 10247* There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)

5.4.11  Rational numbers (as a subset of complex numbers)

Syntaxcq 10248 Extend class notation to include the class of rationals.

Definitiondf-q 10249 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10250 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

Theoremelq 10250* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)

Theoremqmulz 10251* If is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)

Theoremznq 10252 The ratio of an integer and a natural number is a rational number. (Contributed by NM, 12-Jan-2002.)

Theoremqre 10253 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)

Theoremzq 10254 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)

Theoremzssq 10255 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssq 10256 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremnnssq 10257 The natural numbers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremqssre 10258 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

Theoremqsscn 10259 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremqex 10260 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnq 10261 A natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremqcn 10262 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)

TheoremqexALT 10263 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.)

Theoremqnegcl 10265 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)

Theoremqmulcl 10266 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)

Theoremqsubcl 10267 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremqreccl 10268 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqdivcl 10269 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqrevaddcl 10270 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremnnrecq 10271 The reciprocal of a natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremirradd 10272 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)

Theoremirrmul 10273 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 10274* Lemma for rpnnen1 10279. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem2 10275* Lemma for rpnnen1 10279. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem3 10276* Lemma for rpnnen1 10279. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem4 10277* Lemma for rpnnen1 10279. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem5 10278* Lemma for rpnnen1 10279. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1 10279* One half of rpnnen 12432, 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 10280 The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.)

Theoremcnref1o 10281* 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 8676), 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 10282 The set of complex numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.)

Theoremxrex 10283 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)

Theoremaddex 10284 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremmulex 10285 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 10286 Extend class notation to include the class of positive reals.

Definitiondf-rp 10287 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremelrp 10288 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)

Theoremelrpii 10289 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)

Theorem1rp 10290 1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)

Theorem2rp 10291 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpre 10292 A positive real is a real. (Contributed by NM, 27-Oct-2007.)

Theoremrpxr 10293 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremrpcn 10294 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)

Theoremnnrp 10295 A natural number is a positive real. (Contributed by NM, 28-Nov-2008.)

Theoremrpssre 10296 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)

Theoremrpgt0 10297 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)

Theoremrpge0 10298 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)

Theoremrpregt0 10299 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrprege0 10300 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)

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