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Type | Label | Description |
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Statement | ||
Theorem | adddiri 7801 | Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.) |
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Theorem | recni 7802 | A real number is a complex number. (Contributed by NM, 1-Mar-1995.) |
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Theorem | readdcli 7803 | Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.) |
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Theorem | remulcli 7804 | Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.) |
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Theorem | 1red 7805 | 1 is an real number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
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Theorem | 1cnd 7806 | 1 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.) |
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Theorem | mulid1d 7807 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulid2d 7808 | Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | addcld 7809 | Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulcld 7810 | Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulcomd 7811 | Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | addassd 7812 | Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulassd 7813 | Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | adddid 7814 | Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | adddird 7815 | Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | adddirp1d 7816 | Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | joinlmuladdmuld 7817 | Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
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Theorem | recnd 7818 | Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.) |
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Theorem | readdcld 7819 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | remulcld 7820 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
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Syntax | cpnf 7821 | Plus infinity. |
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Syntax | cmnf 7822 | Minus infinity. |
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Syntax | cxr 7823 | The set of extended reals (includes plus and minus infinity). |
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Syntax | clt 7824 | 'Less than' predicate (extended to include the extended reals). |
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Syntax | cle 7825 | Extend wff notation to include the 'less than or equal to' relation. |
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Definition | df-pnf 7826 |
Define plus infinity. Note that the definition is arbitrary, requiring
only that ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
A simpler possibility is to define |
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Definition | df-mnf 7827 |
Define minus infinity as the power set of plus infinity. Note that the
definition is arbitrary, requiring only that ![]() ![]() ![]() |
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Definition | df-xr 7828 | 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.) |
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Definition | df-ltxr 7829* |
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,
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Definition | df-le 7830 | Define 'less than or equal to' on the extended real subset of complex numbers. (Contributed by NM, 13-Oct-2005.) |
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Theorem | pnfnre 7831 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
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Theorem | mnfnre 7832 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
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Theorem | ressxr 7833 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
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Theorem | rexpssxrxp 7834 | 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.) |
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Theorem | rexr 7835 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
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Theorem | 0xr 7836 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
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Theorem | renepnf 7837 | No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | renemnf 7838 | No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | rexrd 7839 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | renepnfd 7840 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | renemnfd 7841 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | pnfxr 7842 | Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
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Theorem | pnfex 7843 | Plus infinity exists (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | pnfnemnf 7844 |
Plus and minus infinity are different elements of ![]() |
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Theorem | mnfnepnf 7845 | Minus and plus infinity are different (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | mnfxr 7846 | 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.) |
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Theorem | rexri 7847 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | renfdisj 7848 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | ltrelxr 7849 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
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Theorem | ltrel 7850 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
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Theorem | lerelxr 7851 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
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Theorem | lerel 7852 | 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
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Theorem | xrlenlt 7853 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.) |
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Theorem | ltxrlt 7854 |
The standard less-than ![]() ![]() ![]() |
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Theorem | axltirr 7855 | Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7756 with ordering on the extended reals. New proofs should use ltnr 7865 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.) |
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Theorem | axltwlin 7856 | Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7757 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.) |
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Theorem | axlttrn 7857 | Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-lttrn 7758 with ordering on the extended reals. New proofs should use lttr 7862 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axltadd 7858 | Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7760 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axapti 7859 | Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7759 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | axmulgt0 7860 | The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7761 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axsuploc 7861* | 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 7765 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
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Theorem | lttr 7862 | Alias for axlttrn 7857, for naming consistency with lttri 7892. New proofs should generally use this instead of ax-pre-lttrn 7758. (Contributed by NM, 10-Mar-2008.) |
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Theorem | mulgt0 7863 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
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Theorem | lenlt 7864 | '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.) |
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Theorem | ltnr 7865 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | ltso 7866 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
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Theorem | gtso 7867 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
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Theorem | lttri3 7868 | Tightness of real apartness. (Contributed by NM, 5-May-1999.) |
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Theorem | letri3 7869 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
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Theorem | ltleletr 7870 |
Transitive law, weaker form of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | letr 7871 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
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Theorem | leid 7872 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | ltne 7873 | 'Less than' implies not equal. See also ltap 8419 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
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Theorem | ltnsym 7874 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
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Theorem | ltle 7875 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
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Theorem | lelttr 7876 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.) |
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Theorem | ltletr 7877 | Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.) |
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Theorem | ltnsym2 7878 | 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | eqle 7879 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
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Theorem | ltnri 7880 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | eqlei 7881 | Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) |
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Theorem | eqlei2 7882 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
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Theorem | gtneii 7883 | 'Less than' implies not equal. See also gtapii 8420 which is the same for apartness. (Contributed by Mario Carneiro, 30-Sep-2013.) |
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Theorem | ltneii 7884 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | lttri3i 7885 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
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Theorem | letri3i 7886 | Tightness of real apartness. (Contributed by NM, 14-May-1999.) |
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Theorem | ltnsymi 7887 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
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Theorem | lenlti 7888 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
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Theorem | ltlei 7889 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
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Theorem | ltleii 7890 | 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
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Theorem | ltnei 7891 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
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Theorem | lttri 7892 | 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
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Theorem | lelttri 7893 | 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.) |
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Theorem | ltletri 7894 | 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.) |
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Theorem | letri 7895 | 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.) |
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Theorem | le2tri3i 7896 | Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) |
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Theorem | mulgt0i 7897 | The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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Theorem | mulgt0ii 7898 | The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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Theorem | ltnrd 7899 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | gtned 7900 | 'Less than' implies not equal. See also gtapd 8423 which is the same but for apartness. (Contributed by Mario Carneiro, 27-May-2016.) |
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