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Theorem List for Metamath Proof Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeluzelz 10501 A member of a set of upper integers is an integer. (Contributed by NM, 6-Sep-2005.)

Theoremeluzelre 10502 A member of a set of upper integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)

Theoremeluzle 10503 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)

Theoremeluz 10504 Membership in a set of upper integers. (Contributed by NM, 2-Oct-2005.)

Theoremuzid 10505 Membership of the least member in a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremuzn0 10506 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)

Theoremuztrn 10507 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)

Theoremuztrn2 10508 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremuzneg 10509 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)

Theoremuzssz 10510 A set of upper integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremuzss 10511 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)

Theoremuztric 10512 Totality of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)

Theoremuz11 10513 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)

Theoremeluzp1m1 10514 Membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1l 10515 Strict ordering implied by membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1p1 10516 Membership in the next set of upper integers. (Contributed by NM, 5-Oct-2005.)

Theoremeluzaddi 10517 Membership in a later set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)

Theoremeluzsubi 10518 Membership in an earlier set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)

Theoremeluzadd 10519 Membership in a later set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeluzsub 10520 Membership in an earlier set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremuzm1 10521 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremuznn0sub 10522 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)

Theoremuzin 10523 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremuzp1 10524 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnn0uz 10525 Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremnnuz 10526 Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremelnnuz 10527 A natural number expressed as a member of a set of upper integers. (Contributed by NM, 6-Jun-2006.)

Theoremelnn0uz 10528 A nonnegative integer expressed as a member a set of upper integers. (Contributed by NM, 6-Jun-2006.)

Theoremuznnssnn 10529 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremraluz 10530* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremraluz2 10531* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremrexuz 10532* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremrexuz2 10533* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theorem2rexuz 10534* Double existential quantification in a set of upper integers. (Contributed by NM, 3-Nov-2005.)

Theorempeano2uz 10535 Second Peano postulate for a set of upper integers. (Contributed by NM, 7-Sep-2005.)

Theorempeano2uzs 10536 Second Peano postulate for a set of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theorempeano2uzr 10537 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)

Theoremuzaddcl 10538 Addition closure law for a set of upper integers. (Contributed by NM, 4-Jun-2006.)

Theoremuzind4 10539* 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 10540* Induction on the set of upper integers that starts at an integer . The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction hypothesis. Either uzind4 10539 or uzind4ALT 10540 may be used; see comment for nnind 10023. (Contributed by NM, 7-Sep-2005.)

Theoremuzind4s 10541* 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 10542* 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 10541 when and must be distinct in . (Contributed by NM, 16-Nov-2005.)

Theoremuzind4i 10543* 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 10544* Well-ordering principle: any non-empty subset of a set of upper integers has the least element. (Contributed by NM, 8-Oct-2005.)

TheoremuzwoOLD 10545* Well-ordering principle: any non-empty subset of the upper integers has the least element. (Contributed by NM, 8-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremuzwo2 10546* Well-ordering principle: any non-empty subset of upper integers has a unique least element. (Contributed by NM, 8-Oct-2005.)

Theoremnnwo 10547* 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 10548* 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 10549* Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)

Theoremindstr 10550* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)

Theoremeluznn0 10551 Membership in a nonegative set of upper integers implies membership in . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremeluz2b1 10552 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b2 10553 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b3 10554 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremuz2m1nn 10555 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)

Theorem1nuz2 10556 1 is not in . (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremelnn1uz2 10557 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremuz2mulcl 10558 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremindstr2 10559* 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 10560 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 10561 The infimum of the set of natural numbers is one. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoremnn0infm 10562 The infimum of the set of nonnegative integers is zero. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoreminfmssuzle 10563 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 10564 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)

Theoremublbneg 10565* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremeqreznegel 10566* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegn0 10567* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsupminf 10568* 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 10569* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremzsupss 10570* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9073.) (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsuprzcl2 10571* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 10354 avoids ax-pre-sup 9073.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsuprzub 10572* 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 10573* 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 10574* Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 10546 allows the lower bound to be any real number. See also nnwo 10547 and nnwos 10549. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)

Theoremzmin 10575* 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 10576* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)

Theoremzbtwnre 10577* 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 10578* 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 10579 Extend class notation to include the class of rationals.

Definitiondf-q 10580 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10581 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

Theoremelq 10581* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)

Theoremqmulz 10582* 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 10583 The ratio of an integer and a natural number is a rational number. (Contributed by NM, 12-Jan-2002.)

Theoremqre 10584 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)

Theoremzq 10585 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)

Theoremzssq 10586 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssq 10587 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremnnssq 10588 The natural numbers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremqssre 10589 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

Theoremqsscn 10590 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremqex 10591 The set of rational numbers exists. See also qexALT 10594. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnq 10592 A natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremqcn 10593 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)

TheoremqexALT 10594 The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremqnegcl 10596 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)

Theoremqmulcl 10597 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)

Theoremqsubcl 10598 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremqreccl 10599 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqdivcl 10600 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)

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