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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiap0 9101 The imaginary unit  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  _i #  0
 
Theorem2muliap0 9102  2  x.  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  ( 2  x.  _i ) #  0
 
Theorem2muline0 9103  ( 2  x.  _i )  =/=  0. See also 2muliap0 9102. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  =/=  0
 
4.4.5  Simple number properties
 
Theoremhalfcl 9104 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 9105 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 9106 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  =  0  <->  A  =  0 )
 )
 
Theorem2halves 9107 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 9108 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 9109 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  ( A  /  2 )  <  A ) )
 
Theoremhalfnneg2 9110 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 9111 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B ) 
 /  2 )  e. 
 CC  /\  ( ( A  -  B )  / 
 2 )  e.  CC ) )
 
Theoremhalfaddsub 9112 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  B )  /  2 )  +  ( ( A  -  B )  /  2
 ) )  =  A  /\  ( ( ( A  +  B )  / 
 2 )  -  (
 ( A  -  B )  /  2 ) )  =  B ) )
 
Theoremlt2halves 9113 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  ( C  /  2 ) 
 /\  B  <  ( C  /  2 ) ) 
 ->  ( A  +  B )  <  C ) )
 
Theoremaddltmul 9114 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 2  <  A  /\  2  <  B ) ) 
 ->  ( A  +  B )  <  ( A  x.  B ) )
 
Theoremnominpos 9115* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
 |- 
 -.  E. x  e.  RR  ( 0  <  x  /\  -.  E. y  e. 
 RR  ( 0  < 
 y  /\  y  <  x ) )
 
Theoremavglt1 9116 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavglt2 9117 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( ( A  +  B )  /  2
 )  <  B )
 )
 
Theoremavgle1 9118 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A  <_  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavgle2 9119 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( ( A  +  B )  /  2
 )  <_  B )
 )
 
Theorem2timesd 9120 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2d 9121 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  2 )  =  ( A  +  A ) )
 
Theoremhalfcld 9122 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 CC )
 
Theorem2halvesd 9123 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  /  2
 )  +  ( A 
 /  2 ) )  =  A )
 
Theoremrehalfcld 9124 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 RR )
 
Theoremlt2halvesd 9125 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( C  /  2
 ) )   &    |-  ( ph  ->  B  <  ( C  / 
 2 ) )   =>    |-  ( ph  ->  ( A  +  B )  <  C )
 
Theoremrehalfcli 9126 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  ( A  / 
 2 )  e.  RR
 
Theoremadd1p1 9127 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 |-  ( N  e.  CC  ->  ( ( N  +  1 )  +  1
 )  =  ( N  +  2 ) )
 
Theoremsub1m1 9128 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 |-  ( N  e.  CC  ->  ( ( N  -  1 )  -  1
 )  =  ( N  -  2 ) )
 
Theoremcnm2m1cnm3 9129 Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 |-  ( A  e.  CC  ->  ( ( A  -  2 )  -  1
 )  =  ( A  -  3 ) )
 
Theoremxp1d2m1eqxm1d2 9130 A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
 |-  ( X  e.  CC  ->  ( ( ( X  +  1 )  / 
 2 )  -  1
 )  =  ( ( X  -  1 ) 
 /  2 ) )
 
Theoremdiv4p1lem1div2 9131 An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( N  e.  RR  /\  6  <_  N )  ->  ( ( N 
 /  4 )  +  1 )  <_  ( ( N  -  1 ) 
 /  2 ) )
 
4.4.6  The Archimedean property
 
Theoremarch 9132* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
 |-  ( A  e.  RR  ->  E. n  e.  NN  A  <  n )
 
Theoremnnrecl 9133* There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. n  e.  NN  ( 1  /  n )  <  A )
 
Theorembndndx 9134* A bounded real sequence  A ( k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
 |-  ( E. x  e. 
 RR  A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k )
 
4.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 9135 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 9136 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
 
Theoremelnn0 9137 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 )
 )
 
Theoremnnssnn0 9138 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN  C_  NN0
 
Theoremnn0ssre 9139 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  C_  RR
 
Theoremnn0sscn 9140 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  CC
 
Theoremnn0ex 9141 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
 |- 
 NN0  e.  _V
 
Theoremnnnn0 9142 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN  ->  A  e.  NN0 )
 
Theoremnnnn0i 9143 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 |-  N  e.  NN   =>    |-  N  e.  NN0
 
Theoremnn0re 9144 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  RR )
 
Theoremnn0cn 9145 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  CC )
 
Theoremnn0rei 9146 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 9147 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 9148 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 9149 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 9150 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 9151 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 9152 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 9153 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 9154 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
 
Theorem5nn0 9155 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  5  e.  NN0
 
Theorem6nn0 9156 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  6  e.  NN0
 
Theorem7nn0 9157 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  7  e.  NN0
 
Theorem8nn0 9158 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  8  e.  NN0
 
Theorem9nn0 9159 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  9  e.  NN0
 
Theoremnn0ge0 9160 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 9161 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 9162 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  0  <_  N
 
Theoremnn0le0eq0 9163 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 9164 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 9165 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 9166 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 9167 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 9168 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 9169 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 9170 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 9171 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 9172 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 9173 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 9174 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 9175 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 9176 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 9177 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 9178 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 9179 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 9180 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 9181 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 9182 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 9183 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 9184 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 9185 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 9186 '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 9187 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 9188 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 9189 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 9190 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 9191 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 9192 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 9193 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 9194 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 9195 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 9196 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 9197 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 7958.

 
Syntaxcxnn0 9198 The set of extended nonnegative integers.
 class NN0*
 
Definitiondf-xnn0 9199 Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers  RR*, see df-xr 7958. If we assumed excluded middle, this would be essentially the same as ℕ as defined at df-nninf 7097 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
 |- NN0*  =  ( NN0  u.  { +oo } )
 
Theoremelxnn0 9200 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 ) )
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