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Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrletri3 10701 Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

Theoremxrlelttr 10702 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrltletr 10703 Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Theoremxrletr 10704 Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.)

Theoremxrlttrd 10705 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrlelttrd 10706 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltletrd 10707 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrletrd 10708 Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxrltne 10709 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

Theoremnltpnft 10710 An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Theoremngtmnft 10711 An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)

Theoremxrrebnd 10712 An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

Theoremxrre 10713 A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

Theoremxrre2 10714 An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

Theoremxrre3 10715 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

Theoremge0gtmnf 10716 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremge0nemnf 10717 A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrrege0 10718 A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxrmax1 10719 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)

Theoremxrmax2 10720 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.)

Theoremxrmin1 10721 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)

Theoremxrmin2 10722 The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.)

Theoremxrmaxeq 10723 The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremxrmineq 10724 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.)

Theoremxrmaxlt 10725 Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.)

Theoremxrltmin 10726 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.)

Theoremxrmaxle 10727 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremxrlemin 10728 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremmax1 10729 A number is less than or equal to the maximum of it and another. See also max1ALT 10730. (Contributed by NM, 3-Apr-2005.)

Theoremmax1ALT 10730 A number is less than or equal to the maximum of it and another. This version of max1 10729 omits the antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 10729 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.)

Theoremmax2 10731 A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.)

Theoremmin1 10732 The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.)

Theoremmin2 10733 The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.)

Theoremmaxle 10734 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005.)

Theoremlemin 10735 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.)

Theoremmaxlt 10736 Two ways of saying the maximum of two numbers is less than a third. (Contributed by NM, 3-Aug-2007.)

Theoremltmin 10737 Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.)

Theoremmax0sub 10738 Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremifle 10739 An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremz2ge 10740* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)

Theoremqbtwnre 10741* The rational numbers are dense in : any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)

Theoremqbtwnxr 10742* The rational numbers are dense in : any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremqsqueeze 10743* If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.)

Theoremqextltlem 10744* Lemma for qextlt 10745 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremqextlt 10745* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremqextle 10746* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremxralrple 10747* Show that is less than by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremalrple 10748* Show that is less than by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremxnegeq 10749 Equality of two extended numbers with in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxnegex 10750 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegpnf 10751 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Theoremxnegmnf 10752 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremrexneg 10753 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxneg0 10754 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcl 10755 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegneg 10756 Extended real version of negneg 9307. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxneg11 10757 Extended real version of neg11 9308. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltnegi 10758 Forward direction of xltneg 10759. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltneg 10759 Extended real version of ltneg 9484. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleneg 10760 Extended real version of leneg 9487. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg1 10761 Extended real version of lt0neg1 9490. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg2 10762 Extended real version of lt0neg2 9491. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxle0neg1 10763 Extended real version of le0neg1 9492. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxle0neg2 10764 Extended real version of le0neg2 9493. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxaddval 10765 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddf 10766 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulval 10767 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf1 10768 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf2 10769 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf1 10770 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf2 10771 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfaddmnf 10772 Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmnfaddpnf 10773 Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexadd 10774 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexsub 10775 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddnemnf 10776 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddnepnf 10777 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegid 10778 Extended real version of negid 9304. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcl 10779 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcom 10780 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)

Theoremxaddid1 10781 Extended real version of addid1 9202. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddid2 10782 Extended real version of addid2 9205. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegdi 10783 Extended real version of xnegdi 10783. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass 10784 Associativity of extended real addition. The correct condition here is "it is not the case that both and appear as one of , i.e. ", but this condition is difficult to work with, so we break the theorem into two parts: this one, where is not present in , and xaddass2 10785, where is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass2 10785 Associativity of extended real addition. See xaddass 10784 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxpncan 10786 Extended real version of pncan 9267. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnpcan 10787 Extended real version of npcan 9270. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1a 10788 Extended real version of leadd1 9452; note that the converse implication is not true, unlike the real version (for example but ). (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd2a 10789 Commuted form of xleadd1a 10788. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1 10790 Weakened version of xleadd1a 10788 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltadd1 10791 Extended real version of ltadd1 9451. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxltadd2 10792 Extended real version of ltadd2 9133. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddge0 10793 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxle2add 10794 Extended real version of le2add 9466. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxlt2add 10795 Extended real version of lt2add 9469. Note that ltleadd 9467, which has weaker assumptions, is not true for the extended reals (since fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxsubge0 10796 Extended real version of subge0 9497. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxposdif 10797 Extended real version of posdif 9477. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxlesubadd 10798 Under certain conditions, the conclusion of lesubadd 9456 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxmullem 10799 Lemma for rexmul 10806. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmullem2 10800 Lemma for xmulneg1 10804. (Contributed by Mario Carneiro, 20-Aug-2015.)

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