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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrexmul 11501 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulf 11502 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulcl 11503 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1 11504 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf2 11505 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf1 11506 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf2 11507 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1n 11508 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid1 11509 Extended real version of mulid1 9584. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid2 11510 Extended real version of mulid2 9585. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulm1 11511 Extended real version of mulm1 10004. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem2 11512 Lemma for xmulass 11517. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulgt0 11513 Extended real version of mulgt0 9655. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulge0 11514 Extended real version of mulge0 10076. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem 11515* Lemma for xmulass 11517. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem3 11516 Lemma for xmulass 11517. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulass 11517 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 11479 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 11518 Extended real version of lemul1a 10403. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2a 11519 Extended real version of lemul2a 10404. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxlemul1 11520 Extended real version of lemul1 10401. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2 11521 Extended real version of lemul2 10402. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul1 11522 Extended real version of ltmul1 10399. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul2 11523 Extended real version of ltmul2 10400. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxadddi 11525 Distributive property for extended real addition and multiplication. Like xaddass 11479, 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 11526 Commuted version of xadddi 11525. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2 11527 The assumption that the multiplier be real in xadddi 11525 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2r 11528 Commuted version of xadddi2 11527. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremx2times 11529 Extended real version of 2times 10672. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcld 11530 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxaddcld 11531 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxmulcld 11532 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxadd4d 11533 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 9802. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

5.5.3  Supremum and infimum on the extended reals

Theoremxrsupexmnf 11534* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Theoremxrinfmexpnf 11535* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupsslem 11536* Lemma for xrsupss 11538. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmsslem 11537* Lemma for xrinfmss 11539. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupss 11538* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss 11539* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss2 11540* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremxrub 11541* 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 11542* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)

Theoremsupxr2 11543* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxrcl 11544 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)

Theoremsupxrun 11545 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

TheoreminfmxrclOLD 11546 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.) Obsolete version of infxrcl 11563 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsupxrmnf 11547 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrpnf 11548 The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrunb1 11549* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrunb2 11550* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrbnd1 11551* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremsupxrbnd2 11552* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

Theoremxrsup0 11553 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrub 11554 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 11555* 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 11556* 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 11557* 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 11558 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrgtmnf 11559 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

Theoremsupxrre1 11560 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrre2 11561 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)

Theoremsupxrss 11562 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoreminfxrcl 11563 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.)
inf

Theoreminfxrlb 11564 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.) (Revised by AV, 5-Sep-2020.)
inf

Theoreminfxrgelb 11565* 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 AV, 5-Sep-2020.)
inf

Theoreminfxrre 11566* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 5-Sep-2020.)
inf inf

Theoremxrinf0 11567 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
inf

TheoreminfmxrlbOLD 11568 A member of a set of extended reals is greater than or equal to the set's infimum. Note that we did not introduce a notation for the infimum, so we represent it as the supremum for the opposite order relation. (Contributed by Mario Carneiro, 16-Mar-2014.) Obsolete version of infxrlb 11564 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoreminfmxrgelbOLD 11569* 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.) Obsolete version of infxrgelb 11565 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

TheoreminfmxrreOLD 11570* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) Obsolete version of infxrre 11566 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremxrinfm0OLD 11571 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) Obsolete version of xrinf0 11567 as of 5-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoreminfxrss 11572 Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
inf inf

TheoreminfmxrssOLD 11573 Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of infxrss 11572 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremreltre 11574* For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.)

Theoremrpltrp 11575* For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.)

Theoremreltxrnmnf 11576* For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.)

Theoreminfmremnf 11577 The infimum of the reals is minus infinity. (Contributed by AV, 5-Sep-2020.)
inf

Theoreminfmrp1 11578 The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)
inf

5.5.4  Real number intervals

Syntaxcioo 11579 Extend class notation with the set of open intervals of extended reals.

Syntaxcioc 11580 Extend class notation with the set of open-below, closed-above intervals of extended reals.

Syntaxcico 11581 Extend class notation with the set of closed-below, open-above intervals of extended reals.

Syntaxcicc 11582 Extend class notation with the set of closed intervals of extended reals.

Definitiondf-ioo 11583* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ioc 11584* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-ico 11585* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Definitiondf-icc 11586* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)

Theoremixxval 11587* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremelixx1 11588* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxf 11589* 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 11590* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremixxssxr 11591* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)

Theoremelixx3g 11592* 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 11593* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremixxdisj 11594* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxun 11595* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremixxin 11596* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremixxss1 11597* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss2 11598* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxss12 11599* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremixxub 11600* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

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