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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0rei 9101 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 9102 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 9103 The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |- 
 NN  =  ( NN0  \  { 0 } )
 
Theoremelnnne0 9104 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
 
Theorem0nn0 9105 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 9106 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 9107 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 9108 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 9109 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
 
Theorem5nn0 9110 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  5  e.  NN0
 
Theorem6nn0 9111 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  6  e.  NN0
 
Theorem7nn0 9112 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  7  e.  NN0
 
Theorem8nn0 9113 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  8  e.  NN0
 
Theorem9nn0 9114 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  9  e.  NN0
 
Theoremnn0ge0 9115 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  -> 
 0  <_  N )
 
Theoremnn0nlt0 9116 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN0  ->  -.  A  <  0 )
 
Theoremnn0ge0i 9117 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  0  <_  N
 
Theoremnn0le0eq0 9118 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( N  <_  0  <->  N  =  0
 ) )
 
Theoremnn0p1gt0 9119 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( N  e.  NN0  -> 
 0  <  ( N  +  1 ) )
 
Theoremnnnn0addcl 9120 A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN )
 
Theoremnn0nnaddcl 9121 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  +  N )  e.  NN )
 
Theorem0mnnnnn0 9122 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
 |-  ( N  e.  NN  ->  ( 0  -  N )  e/  NN0 )
 
Theoremun0addcl 9123 If  S is closed under addition, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  +  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  +  N )  e.  T )
 
Theoremun0mulcl 9124 If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  x.  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  x.  N )  e.  T )
 
Theoremnn0addcl 9125 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN0 )
 
Theoremnn0mulcl 9126 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 )
 
Theoremnn0addcli 9127 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  +  N )  e.  NN0
 
Theoremnn0mulcli 9128 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  x.  N )  e.  NN0
 
Theoremnn0p1nn 9129 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN )
 
Theorempeano2nn0 9130 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 9131 A positive integer 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 9132 The nonnegative integer property expressed in terms of positive integers. (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 9133 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  NN  <->  ( N  e.  CC  /\  ( N  -  1 )  e. 
 NN0 ) )
 
Theoremelnnnn0b 9134 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  0  <  N ) )
 
Theoremelnnnn0c 9135 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
 
Theoremnn0addge1 9136 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 9137 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 9138 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 9139 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 )
 
Theoremnn0le2xi 9140 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 9141 '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 9142 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 9143 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 9144 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 9145 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 9146 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 9147 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 9148 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 )
 
Theoremnn0readdcl 9149 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  +  B )  e.  RR )
 
Theoremnn0ge2m1nn 9150 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN )
 
Theoremnn0ge2m1nn0 9151 If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  NN0 )
 
Theoremnn0nndivcl 9152 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( K  /  L )  e.  RR )
 
4.4.8  Extended nonnegative integers

The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers  RR*, see df-xr 7916.

 
Syntaxcxnn0 9153 The set of extended nonnegative integers.
 class NN0*
 
Definitiondf-xnn0 9154 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers  RR*, see df-xr 7916. If we assumed excluded middle, this would be essentially the same as ℕ as defined at df-nninf 7064 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
 |- NN0*  =  ( NN0  u.  { +oo } )
 
Theoremelxnn0 9155 An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  <->  ( A  e.  NN0 
 \/  A  = +oo ) )
 
Theoremnn0ssxnn0 9156 The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
 |- 
 NN0  C_ NN0*
 
Theoremnn0xnn0 9157 A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  e. NN0* )
 
Theoremxnn0xr 9158 An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  e.  RR* )
 
Theorem0xnn0 9159 Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |-  0  e. NN0*
 
Theorempnf0xnn0 9160 Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
 |- +oo  e. NN0*
 
Theoremnn0nepnf 9161 No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e.  NN0  ->  A  =/= +oo )
 
Theoremnn0xnn0d 9162 A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e. NN0* )
 
Theoremnn0nepnfd 9163 No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  =/= +oo )
 
Theoremxnn0nemnf 9164 No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  A  =/= -oo )
 
Theoremxnn0xrnemnf 9165 The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( A  e. NN0*  ->  ( A  e.  RR*  /\  A  =/= -oo ) )
 
Theoremxnn0nnn0pnf 9166 An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
 |-  ( ( N  e. NN0*  /\ 
 -.  N  e.  NN0 )  ->  N  = +oo )
 
4.4.9  Integers (as a subset of complex numbers)
 
Syntaxcz 9167 Extend class notation to include the class of integers.
 class  ZZ
 
Definitiondf-z 9168 Define the set of integers, which are the positive and negative integers 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 9169 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 9170 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  ( N  e.  NN  -> 
 -u N  e.  ZZ )
 
Theoremzre 9171 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  ZZ  ->  N  e.  RR )
 
Theoremzcn 9172 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  ->  N  e.  CC )
 
Theoremzrei 9173 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 |-  A  e.  ZZ   =>    |-  A  e.  RR
 
Theoremzssre 9174 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  RR
 
Theoremzsscn 9175 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
 |- 
 ZZ  C_  CC
 
Theoremzex 9176 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |- 
 ZZ  e.  _V
 
Theoremelnnz 9177 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  <  N ) )
 
Theorem0z 9178 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 |-  0  e.  ZZ
 
Theorem0zd 9179 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  0  e.  ZZ )
 
Theoremelnn0z 9180 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  <->  ( N  e.  ZZ  /\  0  <_  N ) )
 
Theoremelznn0nn 9181 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
 
Theoremelznn0 9182 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 9183 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
 |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 )
 ) )
 
Theoremnnssz 9184 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
 |- 
 NN  C_  ZZ
 
Theoremnn0ssz 9185 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  ZZ
 
Theoremnnz 9186 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN  ->  N  e.  ZZ )
 
Theoremnn0z 9187 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  N  e.  ZZ )
 
Theoremnnzi 9188 A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN   =>    |-  N  e.  ZZ
 
Theoremnn0zi 9189 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  N  e.  NN0   =>    |-  N  e.  ZZ
 
Theoremelnnz1 9190 Positive integer 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 ) )
 
Theoremnnzrab 9191 Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN  =  { x  e.  ZZ  |  1  <_  x }
 
Theoremnn0zrab 9192 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
 |- 
 NN0  =  { x  e.  ZZ  |  0  <_  x }
 
Theorem1z 9193 One is an integer. (Contributed by NM, 10-May-2004.)
 |-  1  e.  ZZ
 
Theorem1zzd 9194 1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  ( ph  ->  1  e.  ZZ )
 
Theorem2z 9195 Two is an integer. (Contributed by NM, 10-May-2004.)
 |-  2  e.  ZZ
 
Theorem3z 9196 3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  ZZ
 
Theorem4z 9197 4 is an integer. (Contributed by BJ, 26-Mar-2020.)
 |-  4  e.  ZZ
 
Theoremznegcl 9198 Closure law for negative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  ZZ  -> 
 -u N  e.  ZZ )
 
Theoremneg1z 9199 -1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  ZZ
 
Theoremznegclb 9200 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 ) )
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