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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem5p3e8 10101 5 + 3 = 8. (Contributed by NM, 11-May-2004.)

Theorem5p4e9 10102 5 + 4 = 9. (Contributed by NM, 11-May-2004.)

Theorem5p5e10 10103 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem6p2e8 10104 6 + 2 = 8. (Contributed by NM, 11-May-2004.)

Theorem6p3e9 10105 6 + 3 = 9. (Contributed by NM, 11-May-2004.)

Theorem6p4e10 10106 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem7p2e9 10107 7 + 2 = 9. (Contributed by NM, 11-May-2004.)

Theorem7p3e10 10108 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem8p2e10 10109 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem1t1e1 10110 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2t2e4 10111 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)

Theorem3t2e6 10112 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)

Theorem3t3e9 10113 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)

Theorem4t2e8 10114 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)

Theorem5t2e10 10115 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)

Theorem4d2e2 10116 One half of four is two. (Contributed by NM, 3-Sep-1999.)

Theorem2nn 10117 2 is a natural number. (Contributed by NM, 20-Aug-2001.)

Theorem3nn 10118 3 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem4nn 10119 4 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem5nn 10120 5 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem6nn 10121 6 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem7nn 10122 7 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem8nn 10123 8 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem9nn 10124 9 is a natural number. (Contributed by NM, 21-Oct-2012.)

Theorem10nn 10125 10 is a natural number. (Contributed by NM, 8-Nov-2012.)

Theorem1lt2 10126 1 is less than 2. (Contributed by NM, 24-Feb-2005.)

Theorem2lt3 10127 2 is less than 3. (Contributed by NM, 26-Sep-2010.)

Theorem1lt3 10128 1 is less than 3. (Contributed by NM, 26-Sep-2010.)

Theorem3lt4 10129 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt4 10130 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt4 10131 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt5 10132 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt5 10133 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt5 10134 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt5 10135 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem5lt6 10136 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt6 10137 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt6 10138 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt6 10139 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt6 10140 1 is less than 6. (Contributed by NM, 19-Oct-2012.)

Theorem6lt7 10141 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem5lt7 10142 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt7 10143 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt7 10144 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt7 10145 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt7 10146 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem7lt8 10147 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem6lt8 10148 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem5lt8 10149 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem4lt8 10150 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem3lt8 10151 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem2lt8 10152 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem1lt8 10153 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem8lt9 10154 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theorem7lt9 10155 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem6lt9 10156 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem5lt9 10157 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem4lt9 10158 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem3lt9 10159 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem2lt9 10160 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theorem1lt9 10161 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)

Theorem9lt10 10162 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorem8lt10 10163 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorem7lt10 10164 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem6lt10 10165 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem5lt10 10166 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem4lt10 10167 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem3lt10 10168 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem2lt10 10169 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem1lt10 10170 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)

Theorem1ne2 10171 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)

Theoremhalfgt0 10172 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)

Theoremhalflt1 10173 One-half is less than one. (Contributed by NM, 24-Feb-2005.)

Theorem1mhlfehlf 10174 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)

Theorem8th4div3 10175 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)

Theoremhalfpm6th 10176 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremhalfcl 10177 Closure of half of a number (frequently used special case). (Contributed by NM, 1-Jan-2006.)

Theoremrehalfcl 10178 Real closure of half. (Contributed by NM, 1-Jan-2006.)

Theoremhalf0 10179 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)

Theorem2halves 10180 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)

Theoremhalfpos2 10181 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)

Theoremhalfpos 10182 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremhalfnneg2 10183 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)

Theoremhalfaddsubcl 10184 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremhalfaddsub 10185 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremlt2halves 10186 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)

Theoremaddltmul 10187 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)

Theoremnominpos 10188* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)

Theoremavglt1 10189 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavglt2 10190 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle1 10191 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle2 10192 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremavgle 10193 The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005.) (Revised by Mario Carneiro, 28-May-2014.)

Theorem2timesd 10194 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremtimes2d 10195 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremhalfcld 10196 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2halvesd 10197 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcld 10198 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2halvesd 10199 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcli 10200 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)

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