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Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnrei 10001 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 10002 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 10003 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  1  e.  NN
 
Theorempeano2nn 10004 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 10005* Alternate definition of the set of natural numbers. This was our original definition, before the current df-nn 9993 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 10006* 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 10007 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 10008 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 10009 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 10010* 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 10014 for an example of its use. See nn0ind 10358 for induction on nonnegative integers and uzind 10353, uzind4 10526 for induction on an arbitrary set of upper integers. See indstr 10537 for strong induction. See also nnindALT 10011. (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 10011* 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 10010 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 10012 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 10013* 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 10014 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 10015 Closure of multiplication of natural numbers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 10016 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 10017* 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 10018 A natural number is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnngt1ne1 10019 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 10020 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 10021 A natural number is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 10022 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 10023 Zero is not a natural number. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 10024 A natural number is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnngt0i 10025 A natural number is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnne0i 10026 A natural number is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnndivre 10027 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 10028 The reciprocal of a natural number is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 10029 The reciprocal of a natural number is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 10030 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 10031 Subtraction of natural numbers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 10032* 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 10033 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 10034 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  1  <_  A )
 
Theoremnngt0d 10035 A natural number is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  0  <  A )
 
Theoremnnne0d 10036 A natural number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnrecred 10037 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 10038 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 10039 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 10040 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 8989 and df-1 8990).

Only the digits 0 through 9 (df-0 8989 through df-9 10057) and the number 10 (df-10 10058) are explicitly defined.

We will later define the decimal constructor df-dec 10375, which will allow us to easily express larger integers in base 10. See deccl 10388 and the theorems that follow it. See also 4001prm 13456 (4001 is prime) and the proof of bpos 21069. 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 12781.

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 10041 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 10042 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 10043 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 10044 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 10045 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 10046 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 10047 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 10048 Extend class notation to include the number 9.
 class 
 9
 
Syntaxc10 10049 Extend class notation to include the number 10.
 class  10
 
Definitiondf-2 10050 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 10051 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 10052 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 10053 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 10054 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 10055 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 10056 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 10057 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Definitiondf-10 10058 Define the number 10. See remarks under df-2 10050. (Contributed by NM, 5-Feb-2007.)
 |- 
 10  =  ( 9  +  1 )
 
Theoremneg1cn 10059 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theorem1m1e0 10060  ( 1  -  1 )  =  0. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  -  1
 )  =  0
 
Theorem2re 10061 The number 2 is real. (Contributed by NM, 27-May-1999.)
 |-  2  e.  RR
 
Theorem2cn 10062 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 |-  2  e.  CC
 
Theorem3re 10063 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 10064 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem4re 10065 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 10066 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 10067 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem6re 10068 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem7re 10069 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem8re 10070 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem9re 10071 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem10re 10072 The number 10 is real. (Contributed by NM, 5-Feb-2007.)
 |- 
 10  e.  RR
 
Theorem0le0 10073 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem2pos 10074 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 10075 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem3pos 10076 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 10077 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem4pos 10078 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem5pos 10079 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 10080 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 10081 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 10082 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 10083 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
Theorem10pos 10084 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 10085 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theorem1p1e2 10086 One plus one equals two. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 10087 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 10107. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem3m1e2 10088 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem3m1e2OLD 10089  2 equals  3 minus  1. (Contributed by FL, 17-Oct-2010.) Obsolete version of 3m1e2 10088 as of 10-Dec-2017. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  2  =  ( 3  -  1 )
 
Theorem2p2e4 10090 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 10091 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 10092 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 10093 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 10094 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2p1e3 10095 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem3p1e4 10096 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 10097 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 10098 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 10099 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 10100 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
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