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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltp1d 8901 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <  ( A  +  1 ) )
 
Theoremlep1d 8902 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  <_  ( A  +  1 ) )
 
Theoremltm1d 8903 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <  A )
 
Theoremlem1d 8904 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  -  1 )  <_  A )
 
Theoremrecgt0d 8905 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  0  <  ( 1  /  A ) )
 
Theoremdivgt0d 8906 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  /  B ) )
 
Theoremmulgt1d 8907 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  1  <  B )   =>    |-  ( ph  ->  1  <  ( A  x.  B ) )
 
Theoremlemulge11d 8908 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( A  x.  B ) )
 
Theoremlemulge12d 8909 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  1  <_  B )   =>    |-  ( ph  ->  A  <_  ( B  x.  A ) )
 
Theoremlemul1ad 8910 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  C ) )
 
Theoremlemul2ad 8911 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( C  x.  A )  <_  ( C  x.  B ) )
 
Theoremltmul12ad 8912 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  0  <_  C )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  ->  ( A  x.  C )  < 
 ( B  x.  D ) )
 
Theoremlemul12ad 8913 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremlemul12bd 8914 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  0 
 <_  D )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  C 
 <_  D )   =>    |-  ( ph  ->  ( A  x.  C )  <_  ( B  x.  D ) )
 
Theoremmulle0r 8915 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  <_  0  /\  0  <_  B ) )  ->  ( A  x.  B )  <_  0 )
 
4.3.10  Suprema
 
Theoremlbreu 8916* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  E! x  e.  S  A. y  e.  S  x  <_  y
 )
 
Theoremlbcl 8917* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y
 )  e.  S )
 
Theoremlble 8918* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 ->  ( iota_ x  e.  S  A. y  e.  S  x  <_  y )  <_  A )
 
Theoremlbinf 8919* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  =  ( iota_ x  e.  S  A. y  e.  S  x  <_  y ) )
 
Theoremlbinfcl 8920* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y
 )  -> inf ( S ,  RR ,  <  )  e.  S )
 
Theoremlbinfle 8921* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( ( S  C_  RR  /\  E. x  e.  S  A. y  e.  S  x  <_  y  /\  A  e.  S ) 
 -> inf ( S ,  RR ,  <  )  <_  A )
 
Theoremsuprubex 8922* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  B  <_  sup ( A ,  RR ,  <  ) )
 
Theoremsuprlubex 8923* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( B  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  B  <  z ) )
 
Theoremsuprnubex 8924* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( -.  B  <  sup ( A ,  RR ,  <  )  <->  A. z  e.  A  -.  B  <  z ) )
 
Theoremsuprleubex 8925* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
 |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e. 
 RR  ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  <_  B  <->  A. z  e.  A  z 
 <_  B ) )
 
Theoremnegiso 8926 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  F  =  ( x  e.  RR  |->  -u x )   =>    |-  ( F  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' F  =  F )
 
Theoremdfinfre 8927* The infimum of a set of reals  A. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
 |-  ( A  C_  RR  -> inf ( A ,  RR ,  <  )  =  U. { x  e.  RR  |  ( A. y  e.  A  x  <_  y  /\  A. y  e.  RR  ( x  <  y  ->  E. z  e.  A  z  <  y
 ) ) } )
 
Theoremsup3exmid 8928* If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
 |-  ( ( u  C_  RR  /\  E. w  w  e.  u  /\  E. x  e.  RR  A. y  e.  u  y  <_  x )  ->  E. x  e.  RR  ( A. y  e.  u  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
 x  ->  E. z  e.  u  y  <  z ) ) )   =>    |- DECID  ph
 
4.3.11  Imaginary and complex number properties
 
Theoremcrap0 8929 The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A #  0  \/  B #  0
 ) 
 <->  ( A  +  ( _i  x.  B ) ) #  0 ) )
 
Theoremcreur 8930* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  E! x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcreui 8931* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  E! y  e.  RR  E. x  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremcju 8932* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  E! x  e.  CC  ( ( A  +  x )  e.  RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) )
 
4.4  Integer sets
 
4.4.1  Positive integers (as a subset of complex numbers)
 
Syntaxcn 8933 Extend class notation to include the class of positive integers.
 class  NN
 
Definitiondf-inn 8934* Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8935 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
 |- 
 NN  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremdfnn2 8935* Definition of the set of positive integers. Another name for df-inn 8934. (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 ) }
 
Theorempeano5nni 8936* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( ( 1  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  NN  C_  A )
 
Theoremnnssre 8937 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |- 
 NN  C_  RR
 
Theoremnnsscn 8938 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 NN  C_  CC
 
Theoremnnex 8939 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 NN  e.  _V
 
Theoremnnre 8940 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  RR )
 
Theoremnncn 8941 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  ( A  e.  NN  ->  A  e.  CC )
 
Theoremnnrei 8942 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  RR
 
Theoremnncni 8943 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  e.  CC
 
Theorem1nn 8944 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
 |-  1  e.  NN
 
Theorempeano2nn 8945 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
 
Theoremnnred 8946 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 8947 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 8948 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A  +  1 )  e.  NN )
 
4.4.2  Principle of mathematical induction
 
Theoremnnind 8949* 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 step. See nnaddcl 8953 for an example of its use. This is an alternative for Metamath 100 proof #74. (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 8950* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 8949 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.) (New usage is discouraged.) (Proof modification is discouraged.)

 |-  ( 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 8951 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1 )  e.  NN ) )
 
Theoremnn1suc 8952* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. 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 8953 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcl 8954 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 8955 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  x.  B )  e.  NN
 
Theoremnnge1 8956 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnnle1eq1 8957 A positive integer 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 8958 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 8959 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN  ->  -.  A  <  1
 )
 
Theorem0nnn 8960 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 8961 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnnap0 8962 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( A  e.  NN  ->  A #  0 )
 
Theoremnngt0i 8963 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnap0i 8964 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
 |-  A  e.  NN   =>    |-  A #  0
 
Theoremnnne0i 8965 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnn2ge 8966* There exists a positive integer 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 ) )
 
Theoremnn1gt1 8967 A positive integer is either one or greater than one. This is for  NN; 0elnn 4630 is a similar theorem for  om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
 
Theoremnngt1ne1 8968 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
 |-  ( A  e.  NN  ->  ( 1  <  A  <->  A  =/=  1 ) )
 
Theoremnndivre 8969 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( A  /  N )  e.  RR )
 
Theoremnnrecre 8970 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 8971 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 8972 Subtraction of positive integers. (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 8973 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 8974* Two ways to express " A divides  B " for positive integers. (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 8975 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 8976 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  1  <_  A )
 
Theoremnngt0d 8977 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  0  <  A )
 
Theoremnnne0d 8978 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnap0d 8979 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A #  0 )
 
Theoremnnrecred 8980 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremnnaddcld 8981 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcld 8982 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  x.  B )  e.  NN )
 
Theoremnndivred 8983 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR )
 
4.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7832 through df-9 8999), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7832 and df-1 7833).

Integers can also be exhibited as sums of powers of 10 (e.g., the number 103 can be expressed as  ( (; 1 0 ^ 2 )  +  3 )) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as 
( 7 ^ 7 )  -  2.

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 8984 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 8985 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 8986 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 8987 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 8988 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 8989 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 8990 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 8991 Extend class notation to include the number 9.
 class 
 9
 
Definitiondf-2 8992 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 8993 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 8994 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 8995 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 8996 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 8997 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 8998 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 8999 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Theorem0ne1 9000  0  =/=  1 (common case). See aso 1ap0 8561. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  1
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