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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1pneg1e0 8101  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  -u 1
 )  =  0
 
Theorem0m0e0 8102 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0  -  0
 )  =  0
 
Theorem1m0e1 8103 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  -  0
 )  =  1
 
Theorem0p1e1 8104 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theorem1p0e1 8105 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  0 )  =  1
 
Theorem1p1e2 8106 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 8107 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8128. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem1e2m1 8108 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  =  ( 2  -  1 )
 
Theorem3m1e2 8109 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem2p2e4 8110 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
 |-  ( 2  +  2 )  =  4
 
Theorem2times 8111 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 8112 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 8113 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 8114 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2div2e1 8115 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  /  2
 )  =  1
 
Theorem2p1e3 8116 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem1p2e3 8117 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  2 )  =  3
 
Theorem3p1e4 8118 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 8119 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 8120 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 8121 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 8122 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 8123 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem3p2e5 8124 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 8125 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 8126 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 8127 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 8128 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 8129 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 8130 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 8131 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem6p2e8 8132 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 8133 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
 
Theorem7p2e9 8134 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 7  +  2 )  =  9
 
Theorem1t1e1 8135 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  x.  1
 )  =  1
 
Theorem2t1e2 8136 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( 2  x.  1
 )  =  2
 
Theorem2t2e4 8137 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 |-  ( 2  x.  2
 )  =  4
 
Theorem3t1e3 8138 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 3  x.  1
 )  =  3
 
Theorem3t2e6 8139 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 |-  ( 3  x.  2
 )  =  6
 
Theorem3t3e9 8140 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 |-  ( 3  x.  3
 )  =  9
 
Theorem4t2e8 8141 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem2t0e0 8142 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  0
 )  =  0
 
Theorem4d2e2 8143 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 8144 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 8145 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 8146 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 8147 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 8148 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 8149 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 8150 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 8151 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem1lt2 8152 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 8153 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 8154 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 8155 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 8156 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
 
Theorem1lt4 8157 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  4
 
Theorem4lt5 8158 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  5
 
Theorem3lt5 8159 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  5
 
Theorem2lt5 8160 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  5
 
Theorem1lt5 8161 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  5
 
Theorem5lt6 8162 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  6
 
Theorem4lt6 8163 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  6
 
Theorem3lt6 8164 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  6
 
Theorem2lt6 8165 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  6
 
Theorem1lt6 8166 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 |-  1  <  6
 
Theorem6lt7 8167 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 8168 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 8169 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 8170 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 8171 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 8172 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 8173 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 8174 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 8175 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
 
Theorem4lt8 8176 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  8
 
Theorem3lt8 8177 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  8
 
Theorem2lt8 8178 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  8
 
Theorem1lt8 8179 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  8
 
Theorem8lt9 8180 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  8  <  9
 
Theorem7lt9 8181 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  7  <  9
 
Theorem6lt9 8182 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  6  <  9
 
Theorem5lt9 8183 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  5  <  9
 
Theorem4lt9 8184 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  4  <  9
 
Theorem3lt9 8185 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  3  <  9
 
Theorem2lt9 8186 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  2  <  9
 
Theorem1lt9 8187 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  9
 
Theorem0ne2 8188 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  2
 
Theorem1ne2 8189 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 |-  1  =/=  2
 
Theorem1le2 8190 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  2
 
Theorem2cnne0 8191 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  ( 2  e.  CC  /\  2  =/=  0 )
 
Theorem2rene0 8192 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  e.  RR  /\  2  =/=  0 )
 
Theorem1le3 8193 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  3
 
Theoremneg1mulneg1e1 8194  -u 1  x.  -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1  x.  -u 1
 )  =  1
 
Theoremhalfre 8195 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  RR
 
Theoremhalfcn 8196 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  CC
 
Theoremhalfgt0 8197 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalfge0 8198 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2 )
 
Theoremhalflt1 8199 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 8200 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
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