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Theorem List for Metamath Proof Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltmulgt12d 10301 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremgt0divd 10302 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0divd 10303 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgecld 10304 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge0d 10305 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1d 10306 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul2d 10307 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1d 10308 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2d 10309 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv1d 10310 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv1d 10311 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldivd 10312 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldiv2d 10313 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemuldivd 10314 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlemuldiv2d 10315 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdivmuld 10316 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdivmul2d 10317 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmuld 10318 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmul2d 10319 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1dd 10320 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltmul2dd 10321 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdiv1dd 10322 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv1dd 10323 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv12ad 10324 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv23d 10325 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv23d 10326 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2mul2divd 10327 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

5.5.2  Infinity and the extended real number system (cont.)

Syntaxcxne 10328 Extend class notation to include the negative of an extended real.

Syntaxcxmu 10330 Extend class notation to include multiplication of extended reals.

Definitiondf-xneg 10331 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)

Definitiondf-xadd 10332* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-xmul 10333* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfxr 10334 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theoremmnfxr 10335 Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltxr 10336 The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)

Theoremelxr 10337 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

Theorempnfnemnf 10338 Plus and minus infinity are distinguished elements of . (Contributed by NM, 14-Oct-2005.)

Theoremxrnemnf 10339 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrnepnf 10340 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrltnr 10341 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Theoremltpnf 10342 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnflt 10343 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltpnf 10344 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltxr 10345 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Theorempnfnlt 10346 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremnltmnf 10347 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

Theorempnfge 10348 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)

Theoremmnfle 10349 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)

Theoremxrltnsym 10350 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Theoremxrltnsym2 10351 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremxrlttri 10352 Ordering on the extended reals satisfies strict trichotomy. (Contributed by NM, 14-Oct-2005.)

Theoremxrlttr 10353 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

Theoremxrltso 10354 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Theoremxrlttri2 10355 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)

Theoremxrlttri3 10356 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrleloe 10357 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleltne 10358 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrltlen 10359 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdfle2 10360 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdflt2 10361 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremxrltle 10362 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleid 10363 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri 10364 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri3 10365 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

Theoremxrlelttr 10366 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrltletr 10367 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrletr 10368 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrlttrd 10369 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrlelttrd 10370 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltletrd 10371 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrletrd 10372 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltne 10373 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Theoremnltpnft 10374 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Theoremngtmnft 10375 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Theoremxrrebnd 10376 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Theoremxrre 10377 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Theoremxrre2 10378 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Theoremge0gtmnf 10379 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremge0nemnf 10380 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrrege0 10381 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrmax1 10382 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)

Theoremxrmax2 10383 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)

Theoremxrmin1 10384 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)

Theoremxrmin2 10385 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)

Theoremxrmaxeq 10386 The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremxrmineq 10387 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremxrmaxlt 10388 Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.)

Theoremxrltmin 10389 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)

Theoremxrmaxle 10390 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremxrlemin 10391 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremmax1 10392 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)

Theoremmax1ALT 10393 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)

Theoremmax2 10394 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)

Theoremmin1 10395 The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.)

Theoremmin2 10396 The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.)

Theoremmaxle 10397 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005.)

Theoremlemin 10398 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)

Theoremmaxlt 10399 Two ways of saying the maximum of two numbers is less than a third. (Contributed by NM, 3-Aug-2007.)

Theoremltmin 10400 Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.)

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