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

Theoremuzinfmi 10501 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 10502 The infimum of the set of natural numbers is one. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoremnn0infm 10503 The infimum of the set of nonnegative integers is zero. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoreminfmssuzle 10504 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 10505 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)

Theoremublbneg 10506* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremeqreznegel 10507* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegn0 10508* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsupminf 10509* 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 10510* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremzsupss 10511* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 9015.) (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsuprzcl2 10512* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 10295 avoids ax-pre-sup 9015.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsuprzub 10513* 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 10514* 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 10515* Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 10487 allows the lower bound to be any real number. See also nnwo 10488 and nnwos 10490. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)

Theoremzmin 10516* 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 10517* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)

Theoremzbtwnre 10518* 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 10519* 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 10520 Extend class notation to include the class of rationals.

Definitiondf-q 10521 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10522 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

Theoremelq 10522* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)

Theoremqmulz 10523* 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 10524 The ratio of an integer and a natural number is a rational number. (Contributed by NM, 12-Jan-2002.)

Theoremqre 10525 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)

Theoremzq 10526 An integer is a rational number. (Contributed by NM, 9-Jan-2002.)

Theoremzssq 10527 The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssq 10528 The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremnnssq 10529 The natural numbers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)

Theoremqssre 10530 The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)

Theoremqsscn 10531 The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremqex 10532 The set of rational numbers exists. See also qexALT 10535. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnq 10533 A natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremqcn 10534 A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)

TheoremqexALT 10535 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 10537 Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)

Theoremqmulcl 10538 Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)

Theoremqsubcl 10539 Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremqreccl 10540 Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqdivcl 10541 Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)

Theoremqrevaddcl 10542 Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)

Theoremnnrecq 10543 The reciprocal of a natural number is rational. (Contributed by NM, 17-Nov-2004.)

Theoremirradd 10544 The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)

Theoremirrmul 10545 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 10546* Lemma for rpnnen1 10551. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem2 10547* Lemma for rpnnen1 10551. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem3 10548* Lemma for rpnnen1 10551. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem4 10549* Lemma for rpnnen1 10551. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1lem5 10550* Lemma for rpnnen1 10551. (Contributed by Mario Carneiro, 12-May-2013.)

Theoremrpnnen1 10551* One half of rpnnen 12767, 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 10552 The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcnref1o 10553* 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 8943), 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 10554 The set of complex numbers exists. See also ax-cnex 8993. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremxrex 10555 The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)

Theoremaddex 10556 The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremmulex 10557 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 10558 Extend class notation to include the class of positive reals.

Definitiondf-rp 10559 Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremelrp 10560 Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)

Theoremelrpii 10561 Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)

Theorem1rp 10562 1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)

Theorem2rp 10563 2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpre 10564 A positive real is a real. (Contributed by NM, 27-Oct-2007.)

Theoremrpxr 10565 A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremrpcn 10566 A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)

Theoremnnrp 10567 A natural number is a positive real. (Contributed by NM, 28-Nov-2008.)

Theoremrpssre 10568 The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)

Theoremrpgt0 10569 A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)

Theoremrpge0 10570 A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)

Theoremrpregt0 10571 A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremrprege0 10572 A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpne0 10573 A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)

Theoremrprene0 10574 A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)

Theoremrpcnne0 10575 A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)

Theoremralrp 10576 Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)

Theoremrexrp 10577 Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrpaddcl 10578 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpmulcl 10579 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremrpdivcl 10580 Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)

Theoremrpreccl 10581 Closure law for reciprocation of positive reals. (Contributed by Jeffrey Hankins, 23-Nov-2008.)

Theoremrphalfcl 10582 Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremrpgecl 10583 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalflt 10584 Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremrerpdivcl 10585 Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)

Theoremge0p1rp 10586 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremrpneg 10587 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 10588 Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)

Theoremltsubrp 10589 Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)

Theoremltaddrp 10590 Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)

Theoremdifrp 10591 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 10592 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 10593 A natural number is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpred 10594 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 10595 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 10596 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 10597 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 10598 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpne0d 10599 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpregt0d 10600 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

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