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Type | Label | Description |
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Statement | ||
Theorem | readdcld 9701 | Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | remulcld 9702 | Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
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Syntax | cpnf 9703 | Plus infinity. |
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Syntax | cmnf 9704 | Minus infinity. |
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Syntax | cxr 9705 | The set of extended reals (includes plus and minus infinity). |
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Syntax | clt 9706 | 'Less than' predicate (extended to include the extended reals). |
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Syntax | cle 9707 | Extend wff notation to include the 'less than or equal to' relation. |
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Definition | df-pnf 9708 |
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 9709 |
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 9710 | 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 9711* |
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 9712 | Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 9751 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.) |
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Theorem | pnfnre 9713 | Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
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Theorem | mnfnre 9714 | Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
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Theorem | ressxr 9715 | The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) |
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Theorem | rexpssxrxp 9716 | 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 9717 | A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
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Theorem | 0xr 9718 | Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
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Theorem | renepnf 9719 | 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 9720 | 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 9721 | A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | renepnfd 9722 | No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | renemnfd 9723 | No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | rexri 9724 | A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.) |
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Theorem | renfdisj 9725 | 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 9726 | 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
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Theorem | ltrel 9727 | 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.) |
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Theorem | lerelxr 9728 | 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
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Theorem | lerel 9729 | '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 9730 | '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 | xrlenltd 9731 | 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | xrltnle 9732 | 'Less than' expressed in terms of 'less than or equal to', for extended reals. (Contributed by NM, 6-Feb-2007.) |
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Theorem | xrnltled 9733 | 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ssxr 9734 | The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | ltxrlt 9735 |
The standard less-than ![]() ![]() ![]() |
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Theorem | axlttri 9736 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 9644 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axlttrn 9737 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 9645 with ordering on the extended reals. New proofs should use lttr 9741 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axltadd 9738 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 9646 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axmulgt0 9739 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 9647 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | axsup 9740* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 9648 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
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Theorem | lttr 9741 | Alias for axlttrn 9737, for naming consistency with lttri 9791. New proofs should generally use this instead of ax-pre-lttrn 9645. (Contributed by NM, 10-Mar-2008.) |
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Theorem | mulgt0 9742 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
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Theorem | lenlt 9743 | 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.) |
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Theorem | ltnle 9744 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
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Theorem | ltso 9745 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
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Theorem | gtso 9746 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
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Theorem | lttri2 9747 | Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.) |
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Theorem | lttri3 9748 | Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.) |
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Theorem | lttri4 9749 | Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | letri3 9750 | Trichotomy law. (Contributed by NM, 14-May-1999.) |
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Theorem | leloe 9751 | 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.) |
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Theorem | eqlelt 9752 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.) |
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Theorem | ltle 9753 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
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Theorem | leltne 9754 | 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.) |
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Theorem | lelttr 9755 | Transitive law. (Contributed by NM, 23-May-1999.) |
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Theorem | ltletr 9756 | Transitive law. (Contributed by NM, 25-Aug-1999.) |
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Theorem | ltleletr 9757 | Transitive law, weaker form of ltletr 9756. (Contributed by AV, 14-Oct-2018.) |
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Theorem | letr 9758 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
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Theorem | ltnr 9759 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | leid 9760 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | ltne 9761 | 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
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Theorem | ltneOLD 9762 | 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | ltnsym 9763 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
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Theorem | ltnsym2 9764 | '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 | letric 9765 | Trichotomy law. (Contributed by NM, 18-Aug-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | ltlen 9766 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.) |
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Theorem | eqle 9767 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
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Theorem | eqled 9768 | Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltadd2 9769 | Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | ne0gt0 9770 | A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.) |
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Theorem | lecasei 9771 | Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.) |
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Theorem | lelttric 9772 | Trichotomy law. (Contributed by NM, 4-Apr-2005.) |
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Theorem | ltlecasei 9773 | Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
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Theorem | ltnri 9774 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
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Theorem | eqlei 9775 | 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 9776 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
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Theorem | gtneii 9777 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.) |
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Theorem | ltneii 9778 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
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Theorem | lttri2i 9779 | Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.) |
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Theorem | lttri3i 9780 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
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Theorem | letri3i 9781 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
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Theorem | leloei 9782 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999.) |
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Theorem | ltleni 9783 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.) |
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Theorem | ltnsymi 9784 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
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Theorem | lenlti 9785 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
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Theorem | ltnlei 9786 | 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
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Theorem | ltlei 9787 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
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Theorem | ltleii 9788 | 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
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Theorem | ltnei 9789 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
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Theorem | letrii 9790 | Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.) |
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Theorem | lttri 9791 | 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
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Theorem | lelttri 9792 | 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.) |
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Theorem | ltletri 9793 | 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.) |
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Theorem | letri 9794 | 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.) |
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Theorem | le2tri3i 9795 | Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) |
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Theorem | ltadd2i 9796 | Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by OpenAI, 25-Mar-2020.) |
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Theorem | ltadd2iOLD 9797 | Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.) Obsolete version of ltadd2i 9796 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | mulgt0i 9798 | The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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Theorem | mulgt0ii 9799 | The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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Theorem | ltnrd 9800 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
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