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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | recnd 7801 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
Theorem | readdcld 7802 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | remulcld 7803 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Syntax | cpnf 7804 | Plus infinity. |
Syntax | cmnf 7805 | Minus infinity. |
Syntax | cxr 7806 | The set of extended reals (includes plus and minus infinity). |
Syntax | clt 7807 | 'Less than' predicate (extended to include the extended reals). |
Syntax | cle 7808 | Extend wff notation to include the 'less than or equal to' relation. |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-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.) |
Definition | df-le 7813 | Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.) |
Theorem | pnfnre 7814 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Theorem | mnfnre 7815 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Theorem | ressxr 7816 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
Theorem | rexpssxrxp 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.) |
Theorem | rexr 7818 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
Theorem | 0xr 7819 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
Theorem | renepnf 7820 | No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | renemnf 7821 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | rexrd 7822 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | renepnfd 7823 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | renemnfd 7824 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | pnfxr 7825 | Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
Theorem | pnfex 7826 | Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | pnfnemnf 7827 | Plus and minus infinity are different elements of . (Contributed by NM, 14-Oct-2005.) |
Theorem | mnfnepnf 7828 | Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | mnfxr 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.) |
Theorem | rexri 7830 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
Theorem | renfdisj 7831 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | ltrelxr 7832 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | ltrel 7833 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
Theorem | lerelxr 7834 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | lerel 7835 | 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | xrlenlt 7836 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
Theorem | ltxrlt 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.) |
Theorem | axltirr 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.) |
Theorem | axltwlin 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.) |
Theorem | axlttrn 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.) |
Theorem | axltadd 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.) |
Theorem | axapti 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.) |
Theorem | axmulgt0 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.) |
Theorem | axsuploc 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.) |
Theorem | lttr 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.) |
Theorem | mulgt0 7846 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
Theorem | lenlt 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.) |
Theorem | ltnr 7848 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
Theorem | ltso 7849 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
Theorem | gtso 7850 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
Theorem | lttri3 7851 | Tightness of real apartness. (Contributed by NM, 5-May-1999.) |
Theorem | letri3 7852 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
Theorem | ltleletr 7853 | Transitive law, weaker form of . (Contributed by AV, 14-Oct-2018.) |
Theorem | letr 7854 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
Theorem | leid 7855 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
Theorem | ltne 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.) |
Theorem | ltnsym 7857 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
Theorem | ltle 7858 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
Theorem | lelttr 7859 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.) |
Theorem | ltletr 7860 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.) |
Theorem | ltnsym2 7861 | 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | eqle 7862 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
Theorem | ltnri 7863 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
Theorem | eqlei 7864 | Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) |
Theorem | eqlei2 7865 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
Theorem | gtneii 7866 | 'Less than' implies not equal. See also gtapii 8403 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.) |
Theorem | ltneii 7867 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Theorem | lttri3i 7868 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
Theorem | letri3i 7869 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
Theorem | ltnsymi 7870 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
Theorem | lenlti 7871 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
Theorem | ltlei 7872 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
Theorem | ltleii 7873 | 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
Theorem | ltnei 7874 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
Theorem | lttri 7875 | 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Theorem | lelttri 7876 | 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.) |
Theorem | ltletri 7877 | 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.) |
Theorem | letri 7878 | 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.) |
Theorem | le2tri3i 7879 | Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) |
Theorem | mulgt0i 7880 | The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
Theorem | mulgt0ii 7881 | The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
Theorem | ltnrd 7882 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | gtned 7883 | 'Less than' implies not equal. See also gtapd 8406 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltned 7884 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lttri3d 7885 | Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | letri3d 7886 | Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lenltd 7887 | 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltled 7888 | 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltnsymd 7889 | 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nltled 7890 | 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | lensymd 7891 | 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | mulgt0d 7892 | The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | letrd 7893 | Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
Theorem | lelttrd 7894 | Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
Theorem | lttrd 7895 | Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.) |
Theorem | 0lt1 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.) |
Theorem | ltntri 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.) |
Theorem | mul12 7898 | Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.) |
Theorem | mul32 7899 | Commutative/associative law. (Contributed by NM, 8-Oct-1999.) |
Theorem | mul31 7900 | Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
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