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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnaddcld 7701 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   &     NN   =>     + 
 NN
 
Theoremnnmulcld 7702 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   &     NN   =>     x. 
 NN
 
Theoremnndivred 7703 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     NN   =>    
 RR
 
3.4.3  Decimal representation of numbers

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6678 and df-1 6679).

Only the digits 0 through 9 (df-0 6678 through df-9 7720) and the number 10 (df-10 7721) are explicitly defined.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

 
Syntaxc2 7704 Extend class notation to include the number 2.
 2
 
Syntaxc3 7705 Extend class notation to include the number 3.
 3
 
Syntaxc4 7706 Extend class notation to include the number 4.
 4
 
Syntaxc5 7707 Extend class notation to include the number 5.
 5
 
Syntaxc6 7708 Extend class notation to include the number 6.
 6
 
Syntaxc7 7709 Extend class notation to include the number 7.
 7
 
Syntaxc8 7710 Extend class notation to include the number 8.
 8
 
Syntaxc9 7711 Extend class notation to include the number 9.
 9
 
Syntaxc10 7712 Extend class notation to include the number 10.

 10
 
Definitiondf-2 7713 Define the number 2. (Contributed by NM, 27-May-1999.)
 2  1  +  1
 
Definitiondf-3 7714 Define the number 3. (Contributed by NM, 27-May-1999.)
 3  2  +  1
 
Definitiondf-4 7715 Define the number 4. (Contributed by NM, 27-May-1999.)
 4  3  +  1
 
Definitiondf-5 7716 Define the number 5. (Contributed by NM, 27-May-1999.)
 5  4  +  1
 
Definitiondf-6 7717 Define the number 6. (Contributed by NM, 27-May-1999.)
 6  5  +  1
 
Definitiondf-7 7718 Define the number 7. (Contributed by NM, 27-May-1999.)
 7  6  +  1
 
Definitiondf-8 7719 Define the number 8. (Contributed by NM, 27-May-1999.)
 8  7  +  1
 
Definitiondf-9 7720 Define the number 9. (Contributed by NM, 27-May-1999.)
 9  8  +  1
 
Definitiondf-10 7721 Define the number 10. See remarks under df-2 7713. (Contributed by NM, 5-Feb-2007.)

 10  9  +  1
 
Theorem0ne1 7722  0  =/=  1 (common case). See aso 1ap0 7334. (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  =/=  1
 
Theorem1ne0 7723  1  =/=  0. See aso 1ap0 7334. (Contributed by Jim Kingdon, 9-Mar-2020.)
 1  =/=  0
 
Theorem1m1e0 7724  1  -  1  0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 1  -  1  0
 
Theorem2re 7725 The number 2 is real. (Contributed by NM, 27-May-1999.)
 2  RR
 
Theorem2cn 7726 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 2  CC
 
Theorem2ex 7727 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  _V
 
Theorem2cnd 7728 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  CC
 
Theorem3re 7729 The number 3 is real. (Contributed by NM, 27-May-1999.)
 3  RR
 
Theorem3cn 7730 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 3  CC
 
Theorem3ex 7731 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 3  _V
 
Theorem4re 7732 The number 4 is real. (Contributed by NM, 27-May-1999.)
 4  RR
 
Theorem4cn 7733 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 4  CC
 
Theorem5re 7734 The number 5 is real. (Contributed by NM, 27-May-1999.)
 5  RR
 
Theorem5cn 7735 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 5  CC
 
Theorem6re 7736 The number 6 is real. (Contributed by NM, 27-May-1999.)
 6  RR
 
Theorem6cn 7737 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 6  CC
 
Theorem7re 7738 The number 7 is real. (Contributed by NM, 27-May-1999.)
 7  RR
 
Theorem7cn 7739 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 7  CC
 
Theorem8re 7740 The number 8 is real. (Contributed by NM, 27-May-1999.)
 8  RR
 
Theorem8cn 7741 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 8  CC
 
Theorem9re 7742 The number 9 is real. (Contributed by NM, 27-May-1999.)
 9  RR
 
Theorem9cn 7743 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 9  CC
 
Theorem10re 7744 The number 10 is real. (Contributed by NM, 5-Feb-2007.)

 10  RR
 
Theorem0le0 7745 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 0  <_  0
 
Theorem0le2 7746 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
 0  <_  2
 
Theorem2pos 7747 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 0  <  2
 
Theorem2ne0 7748 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 2  =/=  0
 
Theorem2ap0 7749 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 2 #  0
 
Theorem3pos 7750 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 0  <  3
 
Theorem3ne0 7751 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 3  =/=  0
 
Theorem4pos 7752 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 0  <  4
 
Theorem4ne0 7753 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 4  =/=  0
 
Theorem5pos 7754 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 0  <  5
 
Theorem6pos 7755 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 0  <  6
 
Theorem7pos 7756 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 0  <  7
 
Theorem8pos 7757 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 0  <  8
 
Theorem9pos 7758 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 0  <  9
 
Theorem10pos 7759 The number 10 is positive. (Contributed by NM, 5-Feb-2007.)
 0  <  10
 
3.4.4  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 7760 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 -u 1  CC
 
Theoremneg1rr 7761 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 -u 1  RR
 
Theoremneg1ne0 7762 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u 1  =/=  0
 
Theoremneg1lt0 7763 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u 1  <  0
 
Theoremneg1ap0 7764 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 -u 1 #  0
 
Theoremnegneg1e1 7765  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u -u 1  1
 
Theorem1pneg1e0 7766  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  -u 1  0
 
Theorem0m0e0 7767 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  -  0  0
 
Theorem1m0e1 7768 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  -  0  1
 
Theorem0p1e1 7769 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 0  +  1  1
 
Theorem1p0e1 7770 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  0  1
 
Theorem1p1e2 7771 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 1  +  1  2
 
Theorem2m1e1 7772 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 7794. (Contributed by David A. Wheeler, 4-Jan-2017.)
 2  -  1  1
 
Theorem1e2m1 7773 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  2  -  1
 
Theorem3m1e2 7774 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 3  -  1  2
 
Theorem2p2e4 7775 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 7776 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
 CC  2  x.  +
 
Theoremtimes2 7777 A number times 2. (Contributed by NM, 16-Oct-2007.)
 CC  x.  2  +
 
Theorem2timesi 7778 Two times a number. (Contributed by NM, 1-Aug-1999.)
 CC   =>     2  x.  +
 
Theoremtimes2i 7779 A number times 2. (Contributed by NM, 11-May-2004.)
 CC   =>     x.  2  +
 
Theorem2div2e1 7780 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  2  1
 
Theorem2p1e3 7781 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 2  +  1  3
 
Theorem1p2e3 7782 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  +  2  3
 
Theorem3p1e4 7783 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 3  +  1  4
 
Theorem4p1e5 7784 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 4  +  1  5
 
Theorem5p1e6 7785 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 5  +  1  6
 
Theorem6p1e7 7786 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 6  +  1  7
 
Theorem7p1e8 7787 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 7  +  1  8
 
Theorem8p1e9 7788 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 8  +  1  9
 
Theorem9p1e10 7789 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
 9  +  1  10
 
Theorem3p2e5 7790 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 3  +  2  5
 
Theorem3p3e6 7791 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 3  +  3  6
 
Theorem4p2e6 7792 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 4  +  2  6
 
Theorem4p3e7 7793 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 4  +  3  7
 
Theorem4p4e8 7794 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 4  +  4  8
 
Theorem5p2e7 7795 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 5  +  2  7
 
Theorem5p3e8 7796 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 5  +  3  8
 
Theorem5p4e9 7797 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 5  +  4  9
 
Theorem5p5e10 7798 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
 5  +  5  10
 
Theorem6p2e8 7799 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 6  +  2  8
 
Theorem6p3e9 7800 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 6  +  3  9
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