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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrprecred 10401 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalfcld 10402 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreclt1d 10403 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt1d 10404 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpaddcld 10405 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpmulcld 10406 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpdivcld 10407 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrecd 10408 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerecd 10409 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrec1d 10410 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerec2d 10411 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2ad 10412 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv2d 10413 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2d 10414 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivdivd 10415 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0p1rpd 10416 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrerpdivcld 10417 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltsubrpd 10418 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrpd 10419 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrp2d 10420 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt11d 10421 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt12d 10422 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremgt0divd 10423 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0divd 10424 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgecld 10425 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge0d 10426 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1d 10427 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul2d 10428 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 10429 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2d 10430 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv1d 10431 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv1d 10432 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldivd 10433 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldiv2d 10434 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemuldivd 10435 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlemuldiv2d 10436 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdivmuld 10437 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdivmul2d 10438 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmuld 10439 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmul2d 10440 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1dd 10441 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltmul2dd 10442 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 10443 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv1dd 10444 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv12ad 10445 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv23d 10446 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv23d 10447 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2mul2divd 10448 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 10449 Extend class notation to include the negative of an extended real.

Syntaxcxmu 10451 Extend class notation to include multiplication of extended reals.

Definitiondf-xneg 10452 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)

Definitiondf-xadd 10453* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-xmul 10454* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfxr 10455 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theoremmnfxr 10456 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 10457 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 10458 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

Theorempnfnemnf 10459 Plus and minus infinity are distinguished elements of . (Contributed by NM, 14-Oct-2005.)

Theoremxrnemnf 10460 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrnepnf 10461 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrltnr 10462 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Theoremltpnf 10463 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnflt 10464 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltpnf 10465 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltxr 10466 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Theorempnfnlt 10467 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremnltmnf 10468 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

Theorempnfge 10469 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)

Theoremmnfle 10470 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)

Theoremxrltnsym 10471 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Theoremxrltnsym2 10472 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremxrlttri 10473 Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 8811 or axlttri 8894. (Contributed by NM, 14-Oct-2005.)

Theoremxrlttr 10474 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

Theoremxrltso 10475 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Theoremxrlttri2 10476 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)

Theoremxrlttri3 10477 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrleloe 10478 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleltne 10479 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrltlen 10480 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdfle2 10481 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdflt2 10482 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremxrltle 10483 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleid 10484 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri 10485 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri3 10486 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

Theoremxrlelttr 10487 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrltletr 10488 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrletr 10489 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrlttrd 10490 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrlelttrd 10491 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltletrd 10492 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrletrd 10493 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltne 10494 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Theoremnltpnft 10495 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Theoremngtmnft 10496 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Theoremxrrebnd 10497 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Theoremxrre 10498 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Theoremxrre2 10499 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Theoremxrre3 10500 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

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