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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem5re 9701 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem6re 9702 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem7re 9703 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem8re 9704 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem9re 9705 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem10re 9706 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
 |- 
 10  e.  RR
 
Theorem0le0 9707 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem2pos 9708 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 9709 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem3pos 9710 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 9711 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem4pos 9712 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem5pos 9713 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 9714 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 9715 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 9716 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 9717 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
Theorem10pos 9718 The number 10 is positive. (Contributed by NM, 5-Feb-2007.)
 |-  0  <  10
 
5.4.4  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.

 
Theorem0p1e1 9719 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theorem1p1e2 9720 One plus one equals two. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2p2e4 9721 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpegif/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
 |-  ( 2  +  2 )  =  4
 
Theorem2times 9722 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2 9723 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 9724 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 9725 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2p1e3 9726 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem3p1e4 9727 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 9728 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 9729 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 9730 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 9731 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 9732 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem9p1e10 9733 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 9  +  1 )  =  10
 
Theorem3p2e5 9734 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 9735 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 9736 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 9737 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 9738 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 9739 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 9740 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 9741 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem5p5e10 9742 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 5  +  5 )  =  10
 
Theorem6p2e8 9743 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 9744 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
 
Theorem6p4e10 9745 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 6  +  4 )  =  10
 
Theorem7p2e9 9746 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 7  +  2 )  =  9
 
Theorem7p3e10 9747 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 7  +  3 )  =  10
 
Theorem8p2e10 9748 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 8  +  2 )  =  10
 
Theorem1t1e1 9749 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  x.  1
 )  =  1
 
Theorem2t2e4 9750 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
 |-  ( 2  x.  2
 )  =  4
 
Theorem3t2e6 9751 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
 |-  ( 3  x.  2
 )  =  6
 
Theorem3t3e9 9752 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
 |-  ( 3  x.  3
 )  =  9
 
Theorem4t2e8 9753 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem5t2e10 9754 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
 |-  ( 5  x.  2
 )  =  10
 
Theorem4d2e2 9755 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 9756 2 is a natural number. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 9757 3 is a natural number. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 9758 4 is a natural number. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 9759 5 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 9760 6 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 9761 7 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 9762 8 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 9763 9 is a natural number. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem10nn 9764 10 is a natural number. (Contributed by NM, 8-Nov-2012.)
 |- 
 10  e.  NN
 
Theorem1lt2 9765 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 9766 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 9767 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 9768 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 9769 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
 
Theorem1lt4 9770 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  4
 
Theorem4lt5 9771 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  5
 
Theorem3lt5 9772 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  5
 
Theorem2lt5 9773 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  5
 
Theorem1lt5 9774 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  5
 
Theorem5lt6 9775 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  6
 
Theorem4lt6 9776 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  6
 
Theorem3lt6 9777 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  6
 
Theorem2lt6 9778 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  6
 
Theorem1lt6 9779 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 |-  1  <  6
 
Theorem6lt7 9780 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 9781 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 9782 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 9783 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 9784 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 9785 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 9786 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 9787 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 9788 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
 
Theorem4lt8 9789 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  8
 
Theorem3lt8 9790 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  8
 
Theorem2lt8 9791 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  8
 
Theorem1lt8 9792 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  8
 
Theorem8lt9 9793 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  8  <  9
 
Theorem7lt9 9794 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  7  <  9
 
Theorem6lt9 9795 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  6  <  9
 
Theorem5lt9 9796 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  5  <  9
 
Theorem4lt9 9797 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  4  <  9
 
Theorem3lt9 9798 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  3  <  9
 
Theorem2lt9 9799 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  2  <  9
 
Theorem1lt9 9800 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  9
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