HomeHome Metamath Proof Explorer
Theorem List (p. 101 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21491)
  Hilbert Space Explorer  Hilbert Space Explorer
(21492-23014)
  Users' Mathboxes  Users' Mathboxes
(23015-31422)
 

Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnm1nn0 10001 A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  ->  ( N  -  1
 )  e.  NN0 )
 
Theoremelnn0nn 10002 The nonnegative integer property expressed in terms of natural numbers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  <->  ( N  e.  CC  /\  ( N  +  1 )  e.  NN ) )
 
Theoremelnnnn0 10003 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  NN  <->  ( N  e.  CC  /\  ( N  -  1 )  e. 
 NN0 ) )
 
Theoremelnnnn0b 10004 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  0  <  N ) )
 
Theoremelnnnn0c 10005 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
 
Theoremnn0addge1 10006 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( A  +  N )
 )
 
Theoremnn0addge2 10007 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( N  +  A )
 )
 
Theoremnn0addge1i 10008 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( A  +  N )
 
Theoremnn0addge2i 10009 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( N  +  A )
 
Theoremnn0sub 10010 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )
 
Theoremnn0le2xi 10011 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  N  <_  ( 2  x.  N )
 
Theoremnn0lele2xi 10012 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( N  <_  M  ->  N  <_  ( 2  x.  M ) )
 
Theoremnn0supp 10013 Two ways to write the support of a function on  NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( F : I --> NN0  ->  ( `' F " ( _V  \  {
 0 } ) )  =  ( `' F " NN ) )
 
Theoremnnnn0d 10014 A natural number is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 10015 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 10016 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 10017 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  0  <_  A )
 
Theoremnn0addcld 10018 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN0 )
 
Theoremnn0mulcld 10019 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   &    |-  ( ph  ->  B  e.  NN0 )   =>    |-  ( ph  ->  ( A  x.  B )  e. 
 NN0 )
 
5.4.7  Integers (as a subset of complex numbers)
 
Syntaxcz 10020 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 10021 Define the set of integers, which are the positive and negative natural numbers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
 |- 
 ZZ  =  { n  e.  RR  |  ( n  =  0  \/  n  e.  NN  \/  -u n  e.  NN ) }
 
Theoremelz 10022 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )
 ) )
 
Theoremnnnegz 10023 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 10024 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 10025 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 10026 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 10027 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 10028 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 10029 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 10030 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 10031 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theoremelnn0z 10032 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 10033 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
 
Theoremelznn0 10034 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
 
Theoremelznn 10035 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 )
 ) )
 
Theoremelz2 10036* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the natural numbers. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  ZZ  <->  E. x  e.  NN  E. y  e.  NN  N  =  ( x  -  y ) )
 
Theoremdfz2 10037 Alternative definition of the integers, based on elz2 10036. (Contributed by Mario Carneiro, 16-May-2014.)
 |- 
 ZZ  =  (  -  " ( NN  X.  NN ) )
 
TheoremzexALT 10038 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.)
 |- 
 ZZ  e.  _V
 
Theoremnnssz 10039 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 10040 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 10041 A natural number is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 10042 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 10043 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 10044 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 10045 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  1  <_  N ) )
 
Theoremznnnlt1 10046 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)
 |-  ( N  e.  ZZ  ->  ( -.  N  e.  NN 
 <->  N  <  1 ) )
 
Theoremnnzrab 10047 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 10048 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 10049 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem2z 10050 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theoremznegcl 10051 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremznegclb 10052 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  ( A  e.  CC  ->  ( A  e.  ZZ  <->  -u A  e.  ZZ ) )
 
Theoremnn0negz 10053 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  -u N  e.  ZZ )
 
Theoremnn0negzi 10054 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  -u N  e.  ZZ
 
Theoremzaddcl 10055 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  +  N )  e.  ZZ )
 
Theorempeano2z 10056 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
 |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
 
Theoremzsubcl 10057 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
 
Theorempeano2zm 10058 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( N  e.  ZZ  ->  ( N  -  1
 )  e.  ZZ )
 
Theoremzrevaddcl 10059 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
 |-  ( N  e.  ZZ  ->  ( ( M  e.  CC  /\  ( M  +  N )  e.  ZZ ) 
 <->  M  e.  ZZ )
 )
 
Theoremznnsub 10060 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9780.) (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( N  -  M )  e.  NN )
 )
 
Theoremznn0sub 10061 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 10010.) (Contributed by NM, 14-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( N  -  M )  e.  NN0 ) )
 
Theoremzmulcl 10062 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
 
Theoremzltp1le 10063 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( M  +  1 )  <_  N )
 )
 
Theoremzleltp1 10064 Integer ordering relation. (Contributed by NM, 10-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremzlem1lt 10065 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremzltlem1 10066 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremnnleltp1 10067 Natural number ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <_  B  <->  A  <  ( B  +  1 ) ) )
 
Theoremnnltp1le 10068 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( A  +  1 )  <_  B )
 )
 
Theoremnnaddm1cl 10069 Closure of addition of natural numbers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  +  B )  -  1 )  e.  NN )
 
Theoremnn0ltp1le 10070 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  ( M  +  1 ) 
 <_  N ) )
 
Theoremnn0leltp1 10071 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremnn0ltlem1 10072 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  M 
 <_  ( N  -  1
 ) ) )
 
Theoremnn0sub2 10073 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( N  -  M )  e. 
 NN0 )
 
Theoremnn0lt10b 10074 A nonnegative integer less than  1 is  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  =  0
 ) )
 
Theoremnn0lem1lt 10075 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
 
Theoremnnlem1lt 10076 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremnnltlem1 10077 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremzdiv 10078* Two ways to express " M divides  N. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( M  e.  NN  /\  N  e.  ZZ )  ->  ( E. k  e.  ZZ  ( M  x.  k )  =  N  <->  ( N  /  M )  e.  ZZ ) )
 
Theoremzdivadd 10079 Property of divisibility: if  D divides  A and  B then it divides  A  +  B. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( A  /  D )  e.  ZZ  /\  ( B  /  D )  e. 
 ZZ ) )  ->  ( ( A  +  B )  /  D )  e.  ZZ )
 
Theoremzdivmul 10080 Property of divisibility: if  D divides  A then it divides  B  x.  A. (Contributed by NM, 3-Oct-2008.)
 |-  ( ( ( D  e.  NN  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( A 
 /  D )  e. 
 ZZ )  ->  (
 ( B  x.  A )  /  D )  e. 
 ZZ )
 
Theoremzextle 10081* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <_  M  <->  k  <_  N ) )  ->  M  =  N )
 
Theoremzextlt 10082* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\ 
 A. k  e.  ZZ  ( k  <  M  <->  k  <  N ) )  ->  M  =  N )
 
Theoremrecnz 10083 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  1  <  A )  ->  -.  ( 1  /  A )  e.  ZZ )
 
Theorembtwnnz 10084 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  ZZ  /\  A  <  B  /\  B  <  ( A  +  1 ) ) 
 ->  -.  B  e.  ZZ )
 
Theoremgtndiv 10085 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  NN  /\  B  <  A ) 
 ->  -.  ( B  /  A )  e.  ZZ )
 
Theoremhalfnz 10086 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 |- 
 -.  ( 1  / 
 2 )  e.  ZZ
 
Theoremsuprzcl 10087* The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( A ,  RR ,  <  )  e.  A )
 
Theoremprime 10088* Two ways to express " A is a prime number (or 1)." See also isprm 12755. (Contributed by NM, 4-May-2005.)
 |-  ( A  e.  NN  ->  ( A. x  e. 
 NN  ( ( A 
 /  x )  e. 
 NN  ->  ( x  =  1  \/  x  =  A ) )  <->  A. x  e.  NN  ( ( 1  < 
 x  /\  x  <_  A 
 /\  ( A  /  x )  e.  NN )  ->  x  =  A ) ) )
 
Theoremmsqznn 10089 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)
 |-  ( ( A  e.  ZZ  /\  A  =/=  0
 )  ->  ( A  x.  A )  e.  NN )
 
Theoremzneo 10090 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A )  =/=  (
 ( 2  x.  B )  +  1 )
 )
 
Theoremnneo 10091 A natural number is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 |-  ( N  e.  NN  ->  ( ( N  / 
 2 )  e.  NN  <->  -.  ( ( N  +  1 )  /  2
 )  e.  NN )
 )
 
Theoremnneoi 10092 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
 |-  N  e.  NN   =>    |-  ( ( N 
 /  2 )  e. 
 NN 
 <->  -.  ( ( N  +  1 )  / 
 2 )  e.  NN )
 
Theoremzeo 10093 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
 |-  ( N  e.  ZZ  ->  ( ( N  / 
 2 )  e.  ZZ  \/  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theoremzeo2 10094 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  ( N  e.  ZZ  ->  ( ( N  / 
 2 )  e.  ZZ  <->  -.  ( ( N  +  1 )  /  2
 )  e.  ZZ )
 )
 
Theorempeano2uz2 10095* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
 |-  ( ( A  e.  ZZ  /\  B  e.  { x  e.  ZZ  |  A  <_  x } )  ->  ( B  +  1
 )  e.  { x  e.  ZZ  |  A  <_  x } )
 
Theorempeano5uzi 10096* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
 |-  N  e.  ZZ   =>    |-  ( ( N  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  { k  e.  ZZ  |  N  <_  k }  C_  A )
 
Theorempeano5uzti 10097* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
 |-  ( N  e.  ZZ  ->  ( ( N  e.  A  /\  A. x  e.  A  ( x  +  1 )  e.  A )  ->  { k  e. 
 ZZ  |  N  <_  k }  C_  A )
 )
 
Theoremdfuzi 10098* An expression for the upper integers that start at  N that is analogous to df-nn 9743 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
 |-  N  e.  ZZ   =>    |-  { z  e. 
 ZZ  |  N  <_  z }  =  |^| { x  |  ( N  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }
 
Theoremuzind 10099* Induction on the upper integers that start at  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 5-Jul-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <_  k )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )  ->  ta )
 
Theoremuzind2 10100* Induction on the upper integers that start after an integer  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 25-Jul-2005.)
 |-  ( j  =  ( M  +  1 ) 
 ->  ( ph  <->  ps ) )   &    |-  (
 j  =  k  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  k  e.  ZZ  /\  M  <  k )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ta )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31422
  Copyright terms: Public domain < Previous  Next >