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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxneg11 10801 Extended real version of neg11 9352. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltnegi 10802 Forward direction of xltneg 10803. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltneg 10803 Extended real version of ltneg 9528. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleneg 10804 Extended real version of leneg 9531. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg1 10805 Extended real version of lt0neg1 9534. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg2 10806 Extended real version of lt0neg2 9535. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxle0neg1 10807 Extended real version of le0neg1 9536. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxle0neg2 10808 Extended real version of le0neg2 9537. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxaddval 10809 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddf 10810 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulval 10811 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf1 10812 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf2 10813 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf1 10814 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf2 10815 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfaddmnf 10816 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 10817 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 10818 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexsub 10819 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddnemnf 10820 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddnepnf 10821 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegid 10822 Extended real version of negid 9348. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcl 10823 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcom 10824 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)

Theoremxaddid1 10825 Extended real version of addid1 9246. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddid2 10826 Extended real version of addid2 9249. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegdi 10827 Extended real version of xnegdi 10827. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass 10828 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 10829, where is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass2 10829 Associativity of extended real addition. See xaddass 10828 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxpncan 10830 Extended real version of pncan 9311. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnpcan 10831 Extended real version of npcan 9314. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1a 10832 Extended real version of leadd1 9496; note that the converse implication is not true, unlike the real version (for example but ). (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd2a 10833 Commuted form of xleadd1a 10832. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1 10834 Weakened version of xleadd1a 10832 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltadd1 10835 Extended real version of ltadd1 9495. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxltadd2 10836 Extended real version of ltadd2 9177. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddge0 10837 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxle2add 10838 Extended real version of le2add 9510. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxlt2add 10839 Extended real version of lt2add 9513. Note that ltleadd 9511, which has weaker assumptions, is not true for the extended reals (since fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxsubge0 10840 Extended real version of subge0 9541. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxposdif 10841 Extended real version of posdif 9521. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxlesubadd 10842 Under certain conditions, the conclusion of lesubadd 9500 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxmullem 10843 Lemma for rexmul 10850. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmullem2 10844 Lemma for xmulneg1 10848. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulcom 10845 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul01 10846 Extended real version of mul01 9245. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul02 10847 Extended real version of mul02 9244. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg1 10848 Extended real version of mulneg1 9470. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg2 10849 Extended real version of mulneg2 9471. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexmul 10850 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulf 10851 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulcl 10852 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1 10853 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf2 10854 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf1 10855 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf2 10856 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1n 10857 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid1 10858 Extended real version of mulid1 9088. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid2 10859 Extended real version of mulid2 9089. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulm1 10860 Extended real version of mulm1 9475. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem2 10861 Lemma for xmulass 10866. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulgt0 10862 Extended real version of mulgt0 9153. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulge0 10863 Extended real version of mulge0 9545. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem 10864* Lemma for xmulass 10866. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem3 10865 Lemma for xmulass 10866. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulass 10866 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10828 which has to avoid the "undefined" combinations and . The equivalent "undefined" expression here would be , but since this is defined to equal any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul1a 10867 Extended real version of lemul1a 9864. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2a 10868 Extended real version of lemul2a 9865. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxlemul1 10869 Extended real version of lemul1 9862. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2 10870 Extended real version of lemul2 9863. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul1 10871 Extended real version of ltmul1 9860. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul2 10872 Extended real version of ltmul2 9861. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxadddi 10874 Distributive property for extended real addition and multiplication. Like xaddass 10828, this has an unusual domain of correctness due to counterexamples like . In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddir 10875 Commuted version of xadddi 10874. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2 10876 The assumption that the multiplier be real in xadddi 10874 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2r 10877 Commuted version of xadddi2 10876. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremx2times 10878 Extended real version of 2times 10099. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcld 10879 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxaddcld 10880 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxmulcld 10881 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxadd4d 10882 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 9289. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

5.5.3  Supremum on the extended reals

Theoremxrsupexmnf 10883* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)

Theoremxrinfmexpnf 10884* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupsslem 10885* Lemma for xrsupss 10887. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmsslem 10886* Lemma for xrinfmss 10888. (Contributed by NM, 19-Jan-2006.)

Theoremxrsupss 10887* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss 10888* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)

Theoremxrinfmss2 10889* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)

Theoremxrub 10890* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxr 10891* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)

Theoremsupxr2 10892* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)

Theoremsupxrcl 10893 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)

Theoremsupxrun 10894 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)

Theoreminfmxrcl 10895 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by Mario Carneiro, 16-Mar-2014.)

Theoremsupxrmnf 10896 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrpnf 10897 The supremum of a set of extended reals containing plus infnity is plus infinity. (Contributed by NM, 15-Oct-2005.)

Theoremsupxrunb1 10898* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrunb2 10899* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Theoremsupxrbnd1 10900* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)

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