Home Intuitionistic Logic ExplorerTheorem List (p. 79 of 134) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrecnd 7801 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)

Theoremreaddcld 7802 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremremulcld 7803 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)

4.2.2  Infinity and the extended real number system

Syntaxcpnf 7804 Plus infinity.

Syntaxcmnf 7805 Minus infinity.

Syntaxcxr 7806 The set of extended reals (includes plus and minus infinity).

Syntaxclt 7807 'Less than' predicate (extended to include the extended reals).

Syntaxcle 7808 Extend wff notation to include the 'less than or equal to' relation.

Definitiondf-pnf 7809 Define plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (df-mnf 7810). We use to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 7814 and mnfnre 7815, and we'll also be able to prove .

A simpler possibility is to define as and as , but that approach requires the Axiom of Regularity to show that and are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-mnf 7810 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (see mnfnre 7815). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-xr 7811 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)

Definitiondf-ltxr 7812* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)

Definitiondf-le 7813 Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.)

Theorempnfnre 7814 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremmnfnre 7815 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremressxr 7816 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremrexpssxrxp 7817 The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremrexr 7818 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)

Theorem0xr 7819 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremrenepnf 7820 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrenemnf 7821 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrexrd 7822 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenepnfd 7823 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenemnfd 7824 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempnfxr 7825 Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.)

Theorempnfex 7826 Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorempnfnemnf 7827 Plus and minus infinity are different elements of . (Contributed by NM, 14-Oct-2005.)

Theoremmnfnepnf 7828 Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremmnfxr 7829 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.)

Theoremrexri 7830 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)

Theoremrenfdisj 7831 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltrelxr 7832 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremltrel 7833 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

Theoremlerelxr 7834 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremlerel 7835 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremxrlenlt 7836 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremltxrlt 7837 The standard less-than and the extended real less-than are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

4.2.3  Restate the ordering postulates with extended real "less than"

Theoremaxltirr 7838 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7739 with ordering on the extended reals. New proofs should use ltnr 7848 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)

Theoremaxltwlin 7839 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7740 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.)

Theoremaxlttrn 7840 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7741 with ordering on the extended reals. New proofs should use lttr 7845 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Theoremaxltadd 7841 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7743 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxapti 7842 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7742 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.)

Theoremaxmulgt0 7843 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7744 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxsuploc 7844* An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7748 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)

4.2.4  Ordering on reals

Theoremlttr 7845 Alias for axlttrn 7840, for naming consistency with lttri 7875. New proofs should generally use this instead of ax-pre-lttrn 7741. (Contributed by NM, 10-Mar-2008.)

Theoremmulgt0 7846 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)

Theoremlenlt 7847 'Less than or equal to' expressed in terms of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-May-1999.)

Theoremltnr 7848 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltso 7849 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

Theoremgtso 7850 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.)

Theoremlttri3 7851 Tightness of real apartness. (Contributed by NM, 5-May-1999.)

Theoremletri3 7852 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremltleletr 7853 Transitive law, weaker form of . (Contributed by AV, 14-Oct-2018.)

Theoremletr 7854 Transitive law. (Contributed by NM, 12-Nov-1999.)

Theoremleid 7855 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltne 7856 'Less than' implies not equal. See also ltap 8402 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)

Theoremltnsym 7857 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)

Theoremltle 7858 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)

Theoremlelttr 7859 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)

Theoremltletr 7860 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)

Theoremltnsym2 7861 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremeqle 7862 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)

Theoremltnri 7863 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremeqlei 7864 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)

Theoremeqlei2 7865 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)

Theoremgtneii 7866 'Less than' implies not equal. See also gtapii 8403 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.)

Theoremltneii 7867 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremlttri3i 7868 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremletri3i 7869 Tightness of real apartness. (Contributed by NM, 14-May-1999.)

Theoremltnsymi 7870 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)

Theoremlenlti 7871 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)

Theoremltlei 7872 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)

Theoremltleii 7873 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)

Theoremltnei 7874 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)

Theoremlttri 7875 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 7876 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 7877 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 7878 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 7879 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremmulgt0i 7880 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 7881 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 7882 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 7883 'Less than' implies not equal. See also gtapd 8406 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 7884 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 7885 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletri3d 7886 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 7887 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 7888 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 7889 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnltled 7890 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremlensymd 7891 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremmulgt0d 7892 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 7893 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 7894 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremlttrd 7895 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

Theorem0lt1 7896 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)

Theoremltntri 7897 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, . Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)

4.2.5  Initial properties of the complex numbers

Theoremmul12 7898 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 7899 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 7900 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13306
 Copyright terms: Public domain < Previous  Next >