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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnncni 9801 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 9802 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  1  e.  NN
 
Theorempeano2nn 9803 Peano postulate: a successor of a natural number is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
 
Theoremdfnn2 9804* Alternate definition of the set of natural numbers. This was our original definition, before the current df-nn 9792 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremdfnn3 9805* Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)
 |- 
 NN  =  |^| { x  |  ( x  C_  RR  /\  1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
 
Theoremnnred 9806 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 9807 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 9808 Peano postulate: a successor of a natural number is a natural number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A  +  1 )  e.  NN )
 
5.4.2  Principle of mathematical induction
 
Theoremnnind 9809* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 9813 for an example of its use. See nn0ind 10155 for induction on nonnegative integers and uzind 10150, uzind4 10323 for induction on an arbitrary set of upper integers. See indstr 10334 for strong induction. See also nnindALT 9810. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN  ->  ta )
 
TheoremnnindALT 9810* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis.

This ALT version of nnind 9809 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.)

 |-  ( y  e.  NN  ->  ( ch  ->  th )
 )   &    |- 
 ps   &    |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  NN  ->  ta )
 
Theoremnn1m1nn 9811 Every natural number is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1 )  e.  NN ) )
 
Theoremnn1suc 9812* If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <-> 
 th ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ch )   =>    |-  ( A  e.  NN  ->  th )
 
Theoremnnaddcl 9813 Closure of addition of natural numbers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcl 9814 Closure of multiplication of natural numbers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 9815 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  x.  B )  e.  NN
 
Theoremnn2ge 9816* There exists a natural number greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  E. x  e.  NN  ( A  <_  x  /\  B  <_  x ) )
 
Theoremnnge1 9817 A natural number is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnngt1ne1 9818 A natural number is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
 |-  ( A  e.  NN  ->  ( 1  <  A  <->  A  =/=  1 ) )
 
Theoremnnle1eq1 9819 A natural number is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
 |-  ( A  e.  NN  ->  ( A  <_  1  <->  A  =  1 ) )
 
Theoremnngt0 9820 A natural number is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 9821 A natural number is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN  ->  -.  A  <  1
 )
 
Theorem0nnn 9822 Zero is not a natural number. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 9823 A natural number is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnngt0i 9824 A natural number is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnne0i 9825 A natural number is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnndivre 9826 The quotient of a real and a natural number is real. (Contributed by NM, 28-Nov-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( A  /  N )  e.  RR )
 
Theoremnnrecre 9827 The reciprocal of a natural number is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 9828 The reciprocal of a natural number is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 9829 Subtraction of natural numbers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  NN )
 )
 
Theoremnnsubi 9830 Subtraction of natural numbers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 9831* Two ways to express " A divides  B " for natural numbers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. x  e.  NN  ( A  x.  x )  =  B  <->  ( B  /  A )  e.  NN ) )
 
Theoremnndivtr 9832 Transitive property of divisibility: if  A divides  B and  B divides  C, then  A divides  C. Typically,  C would be an integer, although the theorem holds for complex  C. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  /\  ( ( B  /  A )  e.  NN  /\  ( C  /  B )  e. 
 NN ) )  ->  ( C  /  A )  e.  NN )
 
Theoremnnge1d 9833 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  1  <_  A )
 
Theoremnngt0d 9834 A natural number is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  0  <  A )
 
Theoremnnne0d 9835 A natural number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnrecred 9836 The reciprocal of a natural number is real. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremnnaddcld 9837 Closure of addition of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcld 9838 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  x.  B )  e.  NN )
 
Theoremnndivred 9839 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR )
 
5.4.3  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 8789 and df-1 8790).

Only the digits 0 through 9 (df-0 8789 through df-9 9856) and the number 10 (df-10 9857) are explicitly defined.

We will later define the decimal constructor df-dec 10172, which will allow us to easily express larger integers in base 10. See deccl 10185 and the theorems that follow it. See also 4001prm 13190 (4001 is prime) and the proof of bpos 20585. Note that the decimal constructor builds on the definitions in this section.

Integers can also be exhibited as sums of powers of 10 or as some other expression built from operations on the numbers 0 through 10. For example, the prime number 823541 can be expressed as  ( 7 ^ 7 )  -  2. Decimals can be expressed as ratios of integers, as in cos2bnd 12515.

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 9840 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 9841 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 9842 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 9843 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 9844 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 9845 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 9846 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 9847 Extend class notation to include the number 9.
 class 
 9
 
Syntaxc10 9848 Extend class notation to include the number 10.
 class  10
 
Definitiondf-2 9849 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 9850 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 9851 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 9852 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 9853 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 9854 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 9855 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 9856 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Definitiondf-10 9857 Define the number 10. See remarks under df-2 9849. (Contributed by NM, 5-Feb-2007.)
 |- 
 10  =  ( 9  +  1 )
 
Theoremneg1cn 9858 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theorem1m1e0 9859  ( 1  -  1 )  =  0. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  -  1
 )  =  0
 
Theorem2re 9860 The number 2 is real. (Contributed by NM, 27-May-1999.)
 |-  2  e.  RR
 
Theorem2cn 9861 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 |-  2  e.  CC
 
Theorem3re 9862 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 9863 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem4re 9864 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 9865 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 9866 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem6re 9867 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem7re 9868 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem8re 9869 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem9re 9870 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem10re 9871 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
 |- 
 10  e.  RR
 
Theorem0le0 9872 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem2pos 9873 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 9874 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem3pos 9875 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 9876 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem4pos 9877 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem5pos 9878 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 9879 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 9880 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 9881 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 9882 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
Theorem10pos 9883 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 9884 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theorem1p1e2 9885 One plus one equals two. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 9886 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9906. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem3m1e2 9887 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem3m1e2OLD 9888  2 equals  3 minus  1. (Contributed by FL, 17-Oct-2010.) Obsolete version of 3m1e2 9887 as of 10-Dec-2017. (New usage is discouraged.)
 |-  2  =  ( 3  -  1 )
 
Theorem2p2e4 9889 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 9890 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 9891 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 9892 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 9893 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2p1e3 9894 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem3p1e4 9895 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 9896 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 9897 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 9898 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 9899 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 9900 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
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