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

Theoremxmulge0 10901 Extended real version of mulge0 9583. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem 10902* Lemma for xmulass 10904. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem3 10903 Lemma for xmulass 10904. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulass 10904 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10866 which has to avoid the "undefined" combinations and . The equivalent "undefined" expression here would be , but since this is defined to equal any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul1a 10905 Extended real version of lemul1a 9902. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2a 10906 Extended real version of lemul2a 9903. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxlemul1 10907 Extended real version of lemul1 9900. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2 10908 Extended real version of lemul2 9901. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul1 10909 Extended real version of ltmul1 9898. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul2 10910 Extended real version of ltmul2 9899. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxadddilem 10911 Lemma for xadddi 10912. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi 10912 Distributive property for extended real addition and multiplication. Like xaddass 10866, this has an unusual domain of correctness due to counterexamples like . In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddir 10913 Commuted version of xadddi 10912. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2 10914 The assumption that the multiplier be real in xadddi 10912 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2r 10915 Commuted version of xadddi2 10914. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremx2times 10916 Extended real version of 2times 10137. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcld 10917 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxaddcld 10918 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxmulcld 10919 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxadd4d 10920 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 9327. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

5.5.3  Supremum on the extended reals

Theoremxrsupexmnf 10921* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Theoremxrinfmexpnf 10922* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupsslem 10923* Lemma for xrsupss 10925. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmsslem 10924* Lemma for xrinfmss 10926. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupss 10925* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss 10926* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss2 10927* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremxrub 10928* 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 10929* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)

Theoremsupxr2 10930* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxrcl 10931 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)

Theoremsupxrun 10932 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

Theoreminfmxrcl 10933 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 10934 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrpnf 10935 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrunb1 10936* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrunb2 10937* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrbnd1 10938* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremsupxrbnd2 10939* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremxrsup0 10940 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrub 10941 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 10942* 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 10943* 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 10944* 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 10945 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrgtmnf 10946 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

Theoremsupxrre1 10947 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrre2 10948 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrss 10949 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoreminfmxrlb 10950 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 10951* 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 10952* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.)

Theoremxrinfm0 10953 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 10954 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 10955 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 10956 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 10957 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 10958* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 10959* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 10960* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 10961* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 10962* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 10963* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 10964* 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 10965* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 10966* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 10967* 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 10968* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 10969* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxun 10970* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxin 10971* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxss1 10972* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 10973* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 10974* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxub 10975* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremixxlb 10976* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremiooex 10977 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooval 10978* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremioo0 10979 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremioon0 10980 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 10981 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 10982 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 10983 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 10984 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 10985 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremubioo 10986 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremiooval2 10987* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooin 10988 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooss1 10989 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremiooss2 10990 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiocval 10991* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremicoval 10992* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccval 10993* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo1 10994 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremelioo2 10995 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremelioc1 10996 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 10997 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 10998 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiccid 10999 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremico0 11000 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

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