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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxmulcom 10801 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul01 10802 Extended real version of mul01 9201. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul02 10803 Extended real version of mul02 9200. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg1 10804 Extended real version of mulneg1 9426. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg2 10805 Extended real version of mulneg2 9427. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexmul 10806 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulf 10807 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulcl 10808 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1 10809 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf2 10810 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf1 10811 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf2 10812 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1n 10813 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid1 10814 Extended real version of mulid1 9044. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid2 10815 Extended real version of mulid2 9045. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulm1 10816 Extended real version of mulm1 9431. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem2 10817 Lemma for xmulass 10822. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulgt0 10818 Extended real version of mulgt0 9109. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulge0 10819 Extended real version of mulge0 9501. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem 10820* Lemma for xmulass 10822. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem3 10821 Lemma for xmulass 10822. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulass 10822 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10784 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 10823 Extended real version of lemul1a 9820. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2a 10824 Extended real version of lemul2a 9821. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxlemul1 10825 Extended real version of lemul1 9818. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2 10826 Extended real version of lemul2 9819. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul1 10827 Extended real version of ltmul1 9816. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul2 10828 Extended real version of ltmul2 9817. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxadddi 10830 Distributive property for extended real addition and multiplication. Like xaddass 10784, 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 10831 Commuted version of xadddi 10830. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2 10832 The assumption that the multiplier be real in xadddi 10830 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2r 10833 Commuted version of xadddi2 10832. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremx2times 10834 Extended real version of 2times 10055. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcld 10835 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxaddcld 10836 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxmulcld 10837 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxadd4d 10838 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 9245. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

5.5.3  Supremum on the extended reals

Theoremxrsupexmnf 10839* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Theoremxrinfmexpnf 10840* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupsslem 10841* Lemma for xrsupss 10843. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmsslem 10842* Lemma for xrinfmss 10844. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupss 10843* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss 10844* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss2 10845* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremxrub 10846* 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 10847* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)

Theoremsupxr2 10848* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxrcl 10849 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)

Theoremsupxrun 10850 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

Theoreminfmxrcl 10851 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 10852 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrpnf 10853 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrunb1 10854* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrunb2 10855* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrbnd1 10856* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremsupxrbnd2 10857* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremxrsup0 10858 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrub 10859 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 10860* 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 10861* 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 10862* 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 10863 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrgtmnf 10864 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

Theoremsupxrre1 10865 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrre2 10866 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrss 10867 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoreminfmxrlb 10868 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 10869* 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 10870* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.)

Theoremxrinfm0 10871 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 10872 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 10873 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 10874 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 10875 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 10876* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 10877* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 10878* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 10879* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 10880* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 10881* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 10882* 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 10883* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 10884* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 10885* 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 10886* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 10887* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxun 10888* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxin 10889* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxss1 10890* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 10891* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 10892* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxub 10893* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremixxlb 10894* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

Theoremiooex 10895 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremiooval 10896* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremioo0 10897 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremioon0 10898 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 10899 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 10900 An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

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