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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2ap0 9001 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  2 #  0
 
Theorem3pos 9002 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 9003 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem3ap0 9004 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  3 #  0
 
Theorem4pos 9005 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem4ne0 9006 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  4  =/=  0
 
Theorem4ap0 9007 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  4 #  0
 
Theorem5pos 9008 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 9009 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 9010 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 9011 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 9012 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 9013 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theoremneg1rr 9014 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  RR
 
Theoremneg1ne0 9015 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  =/=  0
 
Theoremneg1lt0 9016 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  <  0
 
Theoremneg1ap0 9017 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  -u 1 #  0
 
Theoremnegneg1e1 9018  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u -u 1  =  1
 
Theorem1pneg1e0 9019  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  -u 1
 )  =  0
 
Theorem0m0e0 9020 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0  -  0
 )  =  0
 
Theorem1m0e1 9021 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  -  0
 )  =  1
 
Theorem0p1e1 9022 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theoremfv0p1e1 9023 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 9024 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  0 )  =  1
 
Theorem1p1e2 9025 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 9026 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9053. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem1e2m1 9027 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  =  ( 2  -  1 )
 
Theorem3m1e2 9028 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem4m1e3 9029 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
 |-  ( 4  -  1
 )  =  3
 
Theorem5m1e4 9030 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
 |-  ( 5  -  1
 )  =  4
 
Theorem6m1e5 9031 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
 |-  ( 6  -  1
 )  =  5
 
Theorem7m1e6 9032 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
 |-  ( 7  -  1
 )  =  6
 
Theorem8m1e7 9033 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
 |-  ( 8  -  1
 )  =  7
 
Theorem9m1e8 9034 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
 |-  ( 9  -  1
 )  =  8
 
Theorem2p2e4 9035 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 9036 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 9037 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 9038 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 9039 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2div2e1 9040 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  /  2
 )  =  1
 
Theorem2p1e3 9041 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem1p2e3 9042 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  2 )  =  3
 
Theorem3p1e4 9043 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 9044 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 9045 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 9046 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 9047 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 9048 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem3p2e5 9049 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 9050 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 9051 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 9052 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 9053 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 9054 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 9055 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 9056 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem6p2e8 9057 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 9058 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
 
Theorem7p2e9 9059 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 7  +  2 )  =  9
 
Theorem1t1e1 9060 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  x.  1
 )  =  1
 
Theorem2t1e2 9061 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( 2  x.  1
 )  =  2
 
Theorem2t2e4 9062 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 |-  ( 2  x.  2
 )  =  4
 
Theorem3t1e3 9063 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 3  x.  1
 )  =  3
 
Theorem3t2e6 9064 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 |-  ( 3  x.  2
 )  =  6
 
Theorem3t3e9 9065 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 |-  ( 3  x.  3
 )  =  9
 
Theorem4t2e8 9066 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem2t0e0 9067 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  0
 )  =  0
 
Theorem4d2e2 9068 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 9069 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 9070 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 9071 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 9072 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 9073 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 9074 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 9075 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 9076 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem1lt2 9077 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 9078 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 9079 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 9080 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 9081 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
 
Theorem1lt4 9082 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  4
 
Theorem4lt5 9083 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  5
 
Theorem3lt5 9084 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  5
 
Theorem2lt5 9085 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  5
 
Theorem1lt5 9086 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  5
 
Theorem5lt6 9087 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  6
 
Theorem4lt6 9088 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  6
 
Theorem3lt6 9089 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  6
 
Theorem2lt6 9090 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  6
 
Theorem1lt6 9091 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 |-  1  <  6
 
Theorem6lt7 9092 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 9093 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 9094 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 9095 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 9096 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 9097 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 9098 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 9099 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 9100 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
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