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Theorem List for Metamath Proof Explorer - 10601-10700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdifrp 10601 Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)

Theoremelrpd 10602 Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnrpd 10603 A natural number is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpred 10604 A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpxrd 10605 A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnd 10606 A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgt0d 10607 A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpge0d 10608 A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpne0d 10609 A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpregt0d 10610 A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprege0d 10611 A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprene0d 10612 A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpcnne0d 10613 A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpreccld 10614 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrprecred 10615 Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrphalfcld 10616 Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremreclt1d 10617 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt1d 10618 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpaddcld 10619 Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpmulcld 10620 Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpdivcld 10621 Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrecd 10622 The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerecd 10623 The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltrec1d 10624 Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlerec2d 10625 Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2ad 10626 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv2d 10627 Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv2d 10628 Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivdivd 10629 Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0p1rpd 10630 A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrerpdivcld 10631 Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltsubrpd 10632 Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrpd 10633 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltaddrp2d 10634 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt11d 10635 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmulgt12d 10636 Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremgt0divd 10637 Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremge0divd 10638 Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrpgecld 10639 A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivge0d 10640 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1d 10641 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul2d 10642 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 10643 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2d 10644 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv1d 10645 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv1d 10646 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldivd 10647 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmuldiv2d 10648 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemuldivd 10649 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlemuldiv2d 10650 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltdivmuld 10651 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdivmul2d 10652 'Less than' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmuld 10653 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremledivmul2d 10654 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul1dd 10655 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltmul2dd 10656 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 10657 Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv1dd 10658 Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlediv12ad 10659 Comparison of ratio of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltdiv23d 10660 Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlediv23d 10661 Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlt2mul2divd 10662 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 10663 Extend class notation to include the negative of an extended real.

Syntaxcxmu 10665 Extend class notation to include multiplication of extended reals.

Definitiondf-xneg 10666 Define the negative of an extended real number. (Contributed by FL, 26-Dec-2011.)

Definitiondf-xadd 10667* Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Definitiondf-xmul 10668* Define multiplication over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfxr 10669 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theoremmnfxr 10670 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 10671 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 10672 Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)

Theorempnfnemnf 10673 Plus and minus infinity are distinguished elements of . (Contributed by NM, 14-Oct-2005.)

Theoremxrnemnf 10674 An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrnepnf 10675 An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrltnr 10676 The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Theoremltpnf 10677 Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnflt 10678 Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltpnf 10679 Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Theoremmnfltxr 10680 Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Theorempnfnlt 10681 No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremnltmnf 10682 No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.)

Theorempnfge 10683 Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.)

Theoremnn0pnfge0 10684 If a number is a nonnegative integer or positive infinity, it is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

Theoremmnfle 10685 Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.)

Theoremxrltnsym 10686 Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.)

Theoremxrltnsym2 10687 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremxrlttri 10688 Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 9020 or axlttri 9103. (Contributed by NM, 14-Oct-2005.)

Theoremxrlttr 10689 Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

Theoremxrltso 10690 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.)

Theoremxrlttri2 10691 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.)

Theoremxrlttri3 10692 Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrleloe 10693 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleltne 10694 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrltlen 10695 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdfle2 10696 Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremdflt2 10697 Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.)

Theoremxrltle 10698 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrleid 10699 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.)

Theoremxrletri 10700 Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)

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