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Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiccid 10701 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremico0 10702 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremioc0 10703 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremicc0 10704 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremubioc1 10705 The upper bound belongs to an open-below, closed-above interval. See ubicc2 10753. (Contributed by FL, 29-May-2014.)

Theoremlbico1 10706 The lower bound belongs to a closed-below, open-above interval. See lbicc2 10752. (Contributed by FL, 29-May-2014.)

Theoremiccleub 10707 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeffrey Hankins, 14-Jul-2009.)

Theoremelioo5 10708 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooxr 10709 A non-empty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooord 10710 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 10711 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremioossre 10712 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)

Theoremelioc2 10713 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelico2 10714 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2 10715 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2i 10716 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremelicc4 10717 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)

Theoremiccss 10718 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 10719 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremiccss2 10720 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 10721 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccssioo2 10722 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremiccssico2 10723 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremioomax 10724 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)

Theoremiccmax 10725 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremioopos 10726 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)

Theoremioorp 10727 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremiooshf 10728 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)

Theoremiocssre 10729 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)

Theoremicossre 10730 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremiccssre 10731 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)

Theoremiccssxr 10732 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)

Theoremiocssxr 10733 An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)

Theoremicossxr 10734 A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)

Theoremioossicc 10735 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)

Theoremiccsupr 10736* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9714). (Contributed by Paul Chapman, 21-Jan-2008.)

Theoremelioopnf 10737 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremelioomnf 10738 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremelicopnf 10739 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrepos 10740 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)

Theoremioof 10741 The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremiccf 10742 The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremunirnioo 10743 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Theoremdfioo2 10744* Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremioorebas 10745 Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremelrege0 10746 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremelxrge0 10747 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremge0addcl 10748 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0mulcl 10749 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0xaddcl 10750 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremge0xmulcl 10751 The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremlbicc2 10752 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremubicc2 10753 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)

Theorem0elunit 10754 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theorem1elunit 10755 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremiooneg 10756 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremiccneg 10757 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremicoshft 10758 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremicoshftf1o 10759* Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremicoun 10760 The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremicodisj 10761 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremsnunioo 10762 The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremsnunico 10763 The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremprunioo 10764 The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremioodisj 10765 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremioojoin 10766 Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremdifreicc 10767 The class difference of and a closed interval. (Contributed by FL, 18-Jun-2007.)

Theoremiccsplit 10768 Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftr 10769 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftri 10770 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftl 10771 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftli 10772 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdil 10773 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdili 10774 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntr 10775 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntri 10776 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremlincmb01cmp 10777 A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)

Theoremiccf1o 10778* Describe a bijection from to an arbitrary nontrivial closed interval . (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremiccen 10779 Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremxov1plusxeqvd 10780 A complex number is positive real iff is in . Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremunitssre 10781 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

5.5.5  Finite intervals of integers

Syntaxcfz 10782 Extend class notation to include the notation for a contiguous finite set of integers. Read " " as "the set of integers from to inclusive."

Definitiondf-fz 10783* Define an operation that produces a finite set of sequential integers. Read " " as "the set of integers from to inclusive." See fzval 10784 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)

Theoremfzval 10784* The value of a finite set of sequential integers. E.g., means the set . A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where _k means our ; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremfzval2 10785 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremfzf 10786 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremelfz1 10787 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)

Theoremelfz 10788 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)

Theoremelfz2 10789 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfz5 10790 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)

Theoremelfz4 10791 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuzb 10792 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz 10793 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz 10794 A member of a finite set of sequential integers belongs to a set of upper integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz3 10795 Membership in a finite set of sequential integers implies membership in a set of upper integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzel2 10796 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzel1 10797 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzelz 10798 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle1 10799 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle2 10800 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

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