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

5.5.3  Supremum on the extended reals

Theoremxrsupexmnf 10501* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Theoremxrinfmexpnf 10502* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupsslem 10503* Lemma for xrsupss 10505. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmsslem 10504* Lemma for xrinfmss 10506. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupss 10505* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss 10506* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss2 10507* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremxrub 10508* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxr 10509* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)

Theoremsupxr2 10510* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxrcl 10511 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)

Theoremsupxrun 10512 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

Theoreminfmxrcl 10513 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by Mario Carneiro, 16-Mar-2014.)

Theoremsupxrmnf 10514 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrpnf 10515 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrunb1 10516* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrunb2 10517* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrbnd1 10518* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremsupxrbnd2 10519* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremxrsup0 10520 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrub 10521 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)

Theoremsupxrlub 10522* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)

Theoremsupxrleub 10523* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsupxrre 10524* The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)

Theoremsupxrbnd 10525 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrgtmnf 10526 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

Theoremsupxrre1 10527 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrre2 10528 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrss 10529 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoreminfmxrlb 10530 A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoreminfmxrgelb 10531* The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoreminfmxrre 10532* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.)

Theoremxrinfm0 10533 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.)

5.5.4  Real number intervals

Syntaxcioo 10534 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 10535 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 10536 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 10537 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 10538* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 10539* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 10540* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 10541* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 10542* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 10543* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 10544* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremixxex 10545* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 10546* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 10547* Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxssixx 10548* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 10549* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxun 10550* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxin 10551* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxss1 10552* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 10553* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 10554* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxub 10555* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremixxlb 10556* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremiooex 10557 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooval 10558* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremioo0 10559 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremioon0 10560 An open interval of extended reals is nonempty iff the lower argument is less than the upper argument. (Contributed by NM, 2-Mar-2007.)

Theoremndmioo 10561 The open interval function's value is empty outside of its domain. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremiooid 10562 An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo3g 10563 Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioore 10564 A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremlbioo 10565 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremubioo 10566 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremiooval2 10567* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooin 10568 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooss1 10569 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiooss2 10570 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiocval 10571* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremicoval 10572* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccval 10573* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1 10574 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo2 10575 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremelioc1 10576 Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelico1 10577 Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelicc1 10578 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccid 10579 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremico0 10580 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremioc0 10581 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremicc0 10582 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremubioc1 10583 The upper bound belongs to an open-below, closed-above interval. See ubicc2 10631. (Contributed by FL, 29-May-2014.)

Theoremlbico1 10584 The lower bound belongs to a closed-below, open-above interval. See lbicc2 10630. (Contributed by FL, 29-May-2014.)

Theoremiccleub 10585 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeffrey Hankins, 14-Jul-2009.)

Theoremelioo5 10586 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooxr 10587 A non-empty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooord 10588 Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremelioo4g 10589 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremioossre 10590 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)

Theoremelioc2 10591 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelico2 10592 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2 10593 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2i 10594 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremelicc4 10595 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)

Theoremiccss 10596 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiccssioo 10597 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremiccss2 10598 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremiccssico 10599 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccssioo2 10600 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

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