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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelioo3g 10901 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 10902 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 10903 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremubioo 10904 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremiooval2 10905* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooin 10906 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooss1 10907 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiooss2 10908 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiocval 10909* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremicoval 10910* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccval 10911* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1 10912 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo2 10913 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremelioc1 10914 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 10915 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 10916 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccid 10917 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremico0 10918 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremioc0 10919 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremicc0 10920 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremubioc1 10921 The upper bound belongs to an open-below, closed-above interval. See ubicc2 10970. (Contributed by FL, 29-May-2014.)

Theoremlbico1 10922 The lower bound belongs to a closed-below, open-above interval. See lbicc2 10969. (Contributed by FL, 29-May-2014.)

Theoremiccleub 10923 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeffrey Hankins, 14-Jul-2009.)

Theoremelioo5 10924 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooxr 10925 A non-empty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)

Theoremeliooord 10926 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 10927 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremioossre 10928 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)

Theoremelioc2 10929 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelico2 10930 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2 10931 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

Theoremelicc2i 10932 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremelicc4 10933 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)

Theoremiccss 10934 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 10935 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremicossico 10936 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)

Theoremiccss2 10937 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 10938 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremiccssioo2 10939 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremiccssico2 10940 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremioomax 10941 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)

Theoremiccmax 10942 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremioopos 10943 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)

Theoremioorp 10944 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)

Theoremiooshf 10945 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)

Theoremiocssre 10946 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)

Theoremicossre 10947 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremiccssre 10948 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)

Theoremiccssxr 10949 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)

Theoremiocssxr 10950 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 10951 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 10952 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)

Theoremiccsupr 10953* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 9924). (Contributed by Paul Chapman, 21-Jan-2008.)

Theoremelioopnf 10954 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremelioomnf 10955 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremelicopnf 10956 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrepos 10957 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)

Theoremioof 10958 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 10959 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 10960 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Theoremdfioo2 10961* 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 10962 Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremelrege0 10963 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremelxrge0 10964 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremge0addcl 10965 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0mulcl 10966 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0xaddcl 10967 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremge0xmulcl 10968 The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremlbicc2 10969 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 10970 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 10971 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theorem1elunit 10972 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremiooneg 10973 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremiccneg 10974 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremicoshft 10975 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremicoshftf1o 10976* 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 10977 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 10978 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremsnunioo 10979 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 10980 The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremprunioo 10981 The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremioodisj 10982 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 10983 Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremdifreicc 10984 The class difference of and a closed interval. (Contributed by FL, 18-Jun-2007.)

Theoremiccsplit 10985 Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftr 10986 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftri 10987 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftl 10988 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftli 10989 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdil 10990 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdili 10991 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntr 10992 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntri 10993 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremlincmb01cmp 10994 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 10995* Describe a bijection from to an arbitrary nontrivial closed interval . (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremiccen 10996 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 10997 A complex number is positive real iff is in . Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremunitssre 10998 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

5.5.5  Finite intervals of integers

Syntaxcfz 10999 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 11000* Define an operation that produces a finite set of sequential integers. Read " " as "the set of integers from to inclusive." See fzval 11001 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)

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