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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3re 8801 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 8802 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem3ex 8803 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  _V
 
Theorem4re 8804 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 8805 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 8806 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem5cn 8807 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  5  e.  CC
 
Theorem6re 8808 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem6cn 8809 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  6  e.  CC
 
Theorem7re 8810 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem7cn 8811 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  7  e.  CC
 
Theorem8re 8812 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem8cn 8813 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  8  e.  CC
 
Theorem9re 8814 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem9cn 8815 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  9  e.  CC
 
Theorem0le0 8816 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem0le2 8817 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  0  <_  2
 
Theorem2pos 8818 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 8819 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem2ap0 8820 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  2 #  0
 
Theorem3pos 8821 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 8822 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem3ap0 8823 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  3 #  0
 
Theorem4pos 8824 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem4ne0 8825 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  4  =/=  0
 
Theorem4ap0 8826 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  4 #  0
 
Theorem5pos 8827 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 8828 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 8829 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 8830 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 8831 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
4.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

 
Theoremneg1cn 8832 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theoremneg1rr 8833 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  RR
 
Theoremneg1ne0 8834 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  =/=  0
 
Theoremneg1lt0 8835 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  <  0
 
Theoremneg1ap0 8836 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  -u 1 #  0
 
Theoremnegneg1e1 8837  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u -u 1  =  1
 
Theorem1pneg1e0 8838  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  -u 1
 )  =  0
 
Theorem0m0e0 8839 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0  -  0
 )  =  0
 
Theorem1m0e1 8840 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  -  0
 )  =  1
 
Theorem0p1e1 8841 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theoremfv0p1e1 8842 Function value at  N  +  1 with  N replaced by  0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
 |-  ( N  =  0 
 ->  ( F `  ( N  +  1 )
 )  =  ( F `
  1 ) )
 
Theorem1p0e1 8843 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  0 )  =  1
 
Theorem1p1e2 8844 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 8845 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8872. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem1e2m1 8846 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  =  ( 2  -  1 )
 
Theorem3m1e2 8847 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem4m1e3 8848 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
 |-  ( 4  -  1
 )  =  3
 
Theorem5m1e4 8849 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
 |-  ( 5  -  1
 )  =  4
 
Theorem6m1e5 8850 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
 |-  ( 6  -  1
 )  =  5
 
Theorem7m1e6 8851 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
 |-  ( 7  -  1
 )  =  6
 
Theorem8m1e7 8852 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
 |-  ( 8  -  1
 )  =  7
 
Theorem9m1e8 8853 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
 |-  ( 9  -  1
 )  =  8
 
Theorem2p2e4 8854 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
 |-  ( 2  +  2 )  =  4
 
Theorem2times 8855 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
 |-  ( A  e.  CC  ->  ( 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2 8856 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 8857 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 8858 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2div2e1 8859 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  /  2
 )  =  1
 
Theorem2p1e3 8860 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem1p2e3 8861 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  2 )  =  3
 
Theorem3p1e4 8862 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 8863 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 8864 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 8865 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 8866 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 8867 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem3p2e5 8868 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 8869 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 8870 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 8871 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 8872 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 8873 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 8874 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 8875 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem6p2e8 8876 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 8877 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
 
Theorem7p2e9 8878 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 7  +  2 )  =  9
 
Theorem1t1e1 8879 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  x.  1
 )  =  1
 
Theorem2t1e2 8880 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( 2  x.  1
 )  =  2
 
Theorem2t2e4 8881 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 |-  ( 2  x.  2
 )  =  4
 
Theorem3t1e3 8882 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 3  x.  1
 )  =  3
 
Theorem3t2e6 8883 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 |-  ( 3  x.  2
 )  =  6
 
Theorem3t3e9 8884 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 |-  ( 3  x.  3
 )  =  9
 
Theorem4t2e8 8885 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem2t0e0 8886 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  0
 )  =  0
 
Theorem4d2e2 8887 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 8888 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 8889 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 8890 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 8891 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 8892 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 8893 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 8894 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 8895 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem1lt2 8896 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 8897 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 8898 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 8899 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 8900 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
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