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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem3p1e4 9801 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 9802 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 9803 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 9804 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 9805 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 9806 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem9p1e10 9807 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 9  +  1 )  =  10
 
Theorem3p2e5 9808 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 9809 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 9810 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 9811 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 9812 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 9813 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 9814 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 9815 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem5p5e10 9816 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 5  +  5 )  =  10
 
Theorem6p2e8 9817 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 9818 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
 
Theorem6p4e10 9819 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 6  +  4 )  =  10
 
Theorem7p2e9 9820 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 7  +  2 )  =  9
 
Theorem7p3e10 9821 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 7  +  3 )  =  10
 
Theorem8p2e10 9822 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 8  +  2 )  =  10
 
Theorem1t1e1 9823 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  x.  1
 )  =  1
 
Theorem2t2e4 9824 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 |-  ( 2  x.  2
 )  =  4
 
Theorem3t2e6 9825 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 |-  ( 3  x.  2
 )  =  6
 
Theorem3t3e9 9826 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 |-  ( 3  x.  3
 )  =  9
 
Theorem4t2e8 9827 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem5t2e10 9828 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 5  x.  2
 )  =  10
 
Theorem4d2e2 9829 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 9830 2 is a natural number. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 9831 3 is a natural number. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 9832 4 is a natural number. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 9833 5 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 9834 6 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 9835 7 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 9836 8 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 9837 9 is a natural number. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem10nn 9838 10 is a natural number. (Contributed by NM, 8-Nov-2012.)
 |- 
 10  e.  NN
 
Theorem1lt2 9839 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 9840 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 9841 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 9842 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 9843 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
 
Theorem1lt4 9844 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  4
 
Theorem4lt5 9845 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  5
 
Theorem3lt5 9846 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  5
 
Theorem2lt5 9847 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  5
 
Theorem1lt5 9848 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  5
 
Theorem5lt6 9849 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  6
 
Theorem4lt6 9850 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  6
 
Theorem3lt6 9851 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  6
 
Theorem2lt6 9852 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  6
 
Theorem1lt6 9853 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 |-  1  <  6
 
Theorem6lt7 9854 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 9855 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 9856 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 9857 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 9858 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 9859 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 9860 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 9861 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 9862 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
 
Theorem4lt8 9863 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  8
 
Theorem3lt8 9864 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  8
 
Theorem2lt8 9865 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  8
 
Theorem1lt8 9866 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  8
 
Theorem8lt9 9867 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  8  <  9
 
Theorem7lt9 9868 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  7  <  9
 
Theorem6lt9 9869 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  6  <  9
 
Theorem5lt9 9870 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  5  <  9
 
Theorem4lt9 9871 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  4  <  9
 
Theorem3lt9 9872 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  3  <  9
 
Theorem2lt9 9873 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  2  <  9
 
Theorem1lt9 9874 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  9
 
Theorem9lt10 9875 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  9  <  10
 
Theorem8lt10 9876 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  8  <  10
 
Theorem7lt10 9877 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  7  <  10
 
Theorem6lt10 9878 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  6  <  10
 
Theorem5lt10 9879 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  5  <  10
 
Theorem4lt10 9880 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  4  <  10
 
Theorem3lt10 9881 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  3  <  10
 
Theorem2lt10 9882 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  2  <  10
 
Theorem1lt10 9883 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  10
 
Theorem1ne2 9884 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 |-  1  =/=  2
 
Theoremhalfgt0 9885 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalflt1 9886 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 9887 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
 
Theorem8th4div3 9888 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 |-  ( ( 1  / 
 8 )  x.  (
 4  /  3 )
 )  =  ( 1 
 /  6 )
 
Theoremhalfpm6th 9889 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
 |-  ( ( ( 1 
 /  2 )  -  ( 1  /  6
 ) )  =  ( 1  /  3 ) 
 /\  ( ( 1 
 /  2 )  +  ( 1  /  6
 ) )  =  ( 2  /  3 ) )
 
Theoremhalfcl 9890 Closure of half of a number (frequently used special case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 9891 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 9892 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  =  0  <->  A  =  0 )
 )
 
Theorem2halves 9893 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 9894 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 9895 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  ( A  /  2 )  <  A ) )
 
Theoremhalfnneg2 9896 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 9897 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B ) 
 /  2 )  e. 
 CC  /\  ( ( A  -  B )  / 
 2 )  e.  CC ) )
 
Theoremhalfaddsub 9898 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  B )  /  2 )  +  ( ( A  -  B )  /  2
 ) )  =  A  /\  ( ( ( A  +  B )  / 
 2 )  -  (
 ( A  -  B )  /  2 ) )  =  B ) )
 
Theoremlt2halves 9899 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  ( C  /  2 ) 
 /\  B  <  ( C  /  2 ) ) 
 ->  ( A  +  B )  <  C ) )
 
Theoremaddltmul 9900 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 2  <  A  /\  2  <  B ) ) 
 ->  ( A  +  B )  <  ( A  x.  B ) )
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