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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhalfre 8901 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  RR
 
Theoremhalfcn 8902 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  CC
 
Theoremhalfgt0 8903 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalfge0 8904 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2 )
 
Theoremhalflt1 8905 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 8906 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
 
Theorem8th4div3 8907 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 |-  ( ( 1  / 
 8 )  x.  (
 4  /  3 )
 )  =  ( 1 
 /  6 )
 
Theoremhalfpm6th 8908 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
 |-  ( ( ( 1 
 /  2 )  -  ( 1  /  6
 ) )  =  ( 1  /  3 ) 
 /\  ( ( 1 
 /  2 )  +  ( 1  /  6
 ) )  =  ( 2  /  3 ) )
 
Theoremit0e0 8909 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( _i  x.  0
 )  =  0
 
Theorem2mulicn 8910  ( 2  x.  _i )  e.  CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  e.  CC
 
Theoremiap0 8911 The imaginary unit  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  _i #  0
 
Theorem2muliap0 8912  2  x.  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  ( 2  x.  _i ) #  0
 
Theorem2muline0 8913  ( 2  x.  _i )  =/=  0. See also 2muliap0 8912. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  =/=  0
 
4.4.5  Simple number properties
 
Theoremhalfcl 8914 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 8915 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 8916 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 8917 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 8918 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 8919 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 8920 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 8921 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 8922 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 8923 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 8924 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 8925* 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 8926 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavglt2 8927 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( ( A  +  B )  /  2
 )  <  B )
 )
 
Theoremavgle1 8928 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A  <_  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavgle2 8929 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 8930 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2d 8931 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  2 )  =  ( A  +  A ) )
 
Theoremhalfcld 8932 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 8933 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  /  2
 )  +  ( A 
 /  2 ) )  =  A )
 
Theoremrehalfcld 8934 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 RR )
 
Theoremlt2halvesd 8935 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 8936 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  ( A  / 
 2 )  e.  RR
 
Theoremadd1p1 8937 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 |-  ( N  e.  CC  ->  ( ( N  +  1 )  +  1
 )  =  ( N  +  2 ) )
 
Theoremsub1m1 8938 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 |-  ( N  e.  CC  ->  ( ( N  -  1 )  -  1
 )  =  ( N  -  2 ) )
 
Theoremcnm2m1cnm3 8939 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 8940 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 8941 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 8942* 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 8943* 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 8944* 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 8945 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 8946 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
 
Theoremelnn0 8947 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 8948 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN  C_  NN0
 
Theoremnn0ssre 8949 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  C_  RR
 
Theoremnn0sscn 8950 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  CC
 
Theoremnn0ex 8951 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
 |- 
 NN0  e.  _V
 
Theoremnnnn0 8952 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN  ->  A  e.  NN0 )
 
Theoremnnnn0i 8953 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 |-  N  e.  NN   =>    |-  N  e.  NN0
 
Theoremnn0re 8954 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  RR )
 
Theoremnn0cn 8955 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  CC )
 
Theoremnn0rei 8956 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 8957 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 8958 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 8959 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 8960 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 8961 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 8962 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 8963 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 8964 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
 
Theorem5nn0 8965 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  5  e.  NN0
 
Theorem6nn0 8966 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  6  e.  NN0
 
Theorem7nn0 8967 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  7  e.  NN0
 
Theorem8nn0 8968 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  8  e.  NN0
 
Theorem9nn0 8969 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  9  e.  NN0
 
Theoremnn0ge0 8970 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 8971 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 8972 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  0  <_  N
 
Theoremnn0le0eq0 8973 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 8974 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 8975 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 8976 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 8977 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 8978 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 8979 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 8980 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 8981 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 8982 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 8983 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 8984 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 8985 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 8986 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 8987 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 8988 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 8989 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 8990 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 8991 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 8992 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 8993 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 8994 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 8995 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 8996 '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 8997 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 8998 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 8999 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 9000 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
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