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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0addcld 9901 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 9902 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 9903 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 9904 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 9905 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 9906 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 9907 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 9908 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 9909 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 9910 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 9911 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 9912 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 9913 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 9914 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theoremelnn0z 9915 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 9916 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 9917 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 9918 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 9919* 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 9920 Alternative definition of the integers, based on elz2 9919. (Contributed by Mario Carneiro, 16-May-2014.)
 |- 
 ZZ  =  (  -  " ( NN  X.  NN ) )
 
TheoremzexALT 9921 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 9922 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 9923 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 9924 A natural number is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 9925 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 9926 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 9927 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 9928 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 9929 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 9930 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 9931 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 9932 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem2z 9933 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theoremznegcl 9934 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremznegclb 9935 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 9936 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  -u N  e.  ZZ )
 
Theoremnn0negzi 9937 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  -u N  e.  ZZ
 
Theoremzaddcl 9938 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 9939 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
 |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
 
Theoremzsubcl 9940 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N )  e.  ZZ )
 
Theorempeano2zm 9941 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
 |-  ( N  e.  ZZ  ->  ( N  -  1
 )  e.  ZZ )
 
Theoremzrevaddcl 9942 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 9943 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9664.) (Contributed by NM, 11-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <-> 
 ( N  -  M )  e.  NN )
 )
 
Theoremznn0sub 9944 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9893.) (Contributed by NM, 14-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( N  -  M )  e.  NN0 ) )
 
Theoremzmulcl 9945 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  x.  N )  e.  ZZ )
 
Theoremzltp1le 9946 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 9947 Integer ordering relation. (Contributed by NM, 10-May-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremzlem1lt 9948 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremzltlem1 9949 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremnnleltp1 9950 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 9951 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( A  +  1 )  <_  B )
 )
 
Theoremnnaddm1cl 9952 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 9953 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 9954 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  M  <  ( N  +  1 ) ) )
 
Theoremnn0ltlem1 9955 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 9956 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0  /\  M  <_  N )  ->  ( N  -  M )  e. 
 NN0 )
 
Theoremnn0lt10b 9957 A nonnegative integer less than  1 is  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( N  <  1  <->  N  =  0
 ) )
 
Theoremnn0lem1lt 9958 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( M  -  1 )  <  N ) )
 
Theoremnnlem1lt 9959 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <_  N  <-> 
 ( M  -  1
 )  <  N )
 )
 
Theoremnnltlem1 9960 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)
 |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  M  <_  ( N  -  1 ) ) )
 
Theoremzdiv 9961* 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 9962 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 9963 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 9964* 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 9965* 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 9966 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 9967 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 9968 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 9969 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 |- 
 -.  ( 1  / 
 2 )  e.  ZZ
 
Theoremsuprzcl 9970* 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 9971* Two ways to express " A is a prime number (or 1)." See also isprm 12634. (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 9972 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 9973 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 9974 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 9975 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 9976 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 9977 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 9978* 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 9979* 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 9980* 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 9981* An expression for the upper integers that start at  N that is analogous to df-n 9627 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 9982* 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 9983* 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 )
 
Theoremuzind3 9984* Induction on the upper integers that start at an integer  M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 26-Jul-2005.)
 |-  ( j  =  M  ->  ( ph  <->  ps ) )   &    |-  (
 j  =  m  ->  ( ph  <->  ch ) )   &    |-  (
 j  =  ( m  +  1 )  ->  ( ph  <->  th ) )   &    |-  (
 j  =  N  ->  (
 ph 
 <->  ta ) )   &    |-  ( M  e.  ZZ  ->  ps )   &    |-  ( ( M  e.  ZZ  /\  m  e.  { k  e.  ZZ  |  M  <_  k }
 )  ->  ( ch  ->  th ) )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  { k  e.  ZZ  |  M  <_  k }
 )  ->  ta )
 
TheoremuzindOLD 9985* Induction on the upper integers that start at an integer  B. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list. Anyone is welcome to try. (Contributed by NM, 11-May-2004.) (New usage is discouraged.)

 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( ( y  e. 
 ZZ  /\  B  e.  ZZ )  /\  B  <_  y )  ->  ( ch  ->  th ) )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  B  <_  A )  ->  ta )
 
Theoremuzind3OLD 9986* Induction on the set of upper integers that starts at  B. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 9-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( B  e.  ZZ  /\  y  e.  { z  e.  ZZ  |  B  <_  z } )  ->  ( ch  ->  th ) )   =>    |-  ( ( B  e.  ZZ  /\  A  e.  { z  e.  ZZ  |  B  <_  z }
 )  ->  ta )
 
Theoremnn0ind 9987* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 13-May-2004.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN0  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN0  ->  ta )
 
Theoremfzind 9988* Induction on the integers from  M to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( x  =  M  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N )  ->  ps )   &    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( y  e.  ZZ  /\  M  <_  y  /\  y  <  N ) ) 
 ->  ( ch  ->  th )
 )   =>    |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) )  ->  ta )
 
Theoremfnn0ind 9989* Induction on the integers from  0 to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  ( N  e.  NN0  ->  ps )   &    |-  (
 ( N  e.  NN0  /\  y  e.  NN0  /\  y  <  N )  ->  ( ch  ->  th ) )   =>    |-  ( ( N  e.  NN0  /\  K  e.  NN0  /\  K  <_  N )  ->  ta )
 
Theoremnn0indALT 9990* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by NM, 28-Nov-2005.)
 |-  ( y  e.  NN0  ->  ( ch  ->  th )
 )   &    |- 
 ps   &    |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  NN0 
 ->  ta )
 
Theoremnn0ind-raph 9991* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN0  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN0  ->  ta )
 
Theoremzindd 9992* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
 |-  ( x  =  0 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ta ) )   &    |-  ( x  =  -u y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  et ) )   &    |-  ( ze  ->  ps )   &    |-  ( ze  ->  ( y  e.  NN0  ->  ( ch  ->  ta )
 ) )   &    |-  ( ze  ->  ( y  e.  NN  ->  ( ch  ->  th )
 ) )   =>    |-  ( ze  ->  ( A  e.  ZZ  ->  et ) )
 
Theorembtwnz 9993* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
 |-  ( A  e.  RR  ->  ( E. x  e. 
 ZZ  x  <  A  /\  E. y  e.  ZZ  A  <  y ) )
 
Theoremnn0zd 9994 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremnnzd 9995 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  ZZ )
 
Theoremzred 9996 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremzcnd 9997 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremznegcld 9998 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  -u A  e.  ZZ )
 
Theorempeano2zd 9999 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   =>    |-  ( ph  ->  ( A  +  1 )  e.  ZZ )
 
Theoremzaddcld 10000 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   =>    |-  ( ph  ->  ( A  +  B )  e.  ZZ )
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