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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremavglt2 7901 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 RR  RR  <  +  2  <
 
Theoremavgle1 7902 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 RR  RR  <_  <_  +  2
 
Theoremavgle2 7903 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
 RR  RR  <_  +  2  <_
 
Theorem2timesd 7904 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 2  x.  +
 
Theoremtimes2d 7905 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  2  +
 
Theoremhalfcld 7906 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     2 
 CC
 
Theorem2halvesd 7907 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     2  +  2
 
Theoremrehalfcld 7908 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     2 
 RR
 
Theoremlt2halvesd 7909 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     <  C  2   &     <  C  2   =>     +  <  C
 
Theoremrehalfcli 7910 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 RR   =>     2  RR
 
Theoremadd1p1 7911 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 N  CC  N  +  1  +  1  N  +  2
 
Theoremsub1m1 7912 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 N  CC  N  -  1  -  1  N  -  2
 
Theoremcnm2m1cnm3 7913 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.)
 CC  -  2  -  1  -  3
 
3.4.6  The Archimedean property
 
Theoremarch 7914* 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.)
 RR  n  NN  <  n
 
Theoremnnrecl 7915* 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.)
 RR  0  <  n 
 NN  1  n  <
 
Theorembndndx 7916* A bounded real sequence  k is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
 RR  k  NN  RR  <_  k  NN  <_  k
 
3.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 7917 Extend class notation to include the class of nonnegative integers.
 NN0
 
Definitiondf-n0 7918 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

 NN0  NN  u.  { 0 }
 
Theoremelnn0 7919 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
 NN0  NN  0
 
Theoremnnssnn0 7920 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

 NN  C_  NN0
 
Theoremnn0ssre 7921 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

 NN0  C_  RR
 
Theoremnn0sscn 7922 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

 NN0  C_  CC
 
Theoremnn0ex 7923 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

 NN0  _V
 
Theoremnnnn0 7924 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 NN  NN0
 
Theoremnnnn0i 7925 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 N  NN   =>     N  NN0
 
Theoremnn0re 7926 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 NN0  RR
 
Theoremnn0cn 7927 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 NN0  CC
 
Theoremnn0rei 7928 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 NN0   =>     RR
 
Theoremnn0cni 7929 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 NN0   =>     CC
 
Theoremdfn2 7930 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 7931 The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 N  NN  N  NN0  N  =/=  0
 
Theorem0nn0 7932 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 0  NN0
 
Theorem1nn0 7933 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 1  NN0
 
Theorem2nn0 7934 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 2  NN0
 
Theorem3nn0 7935 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 3  NN0
 
Theorem4nn0 7936 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 4  NN0
 
Theorem5nn0 7937 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  NN0
 
Theorem6nn0 7938 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  NN0
 
Theorem7nn0 7939 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  NN0
 
Theorem8nn0 7940 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  NN0
 
Theorem9nn0 7941 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  NN0
 
Theorem10nn0 7942 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

 10  NN0
 
Theoremnn0ge0 7943 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 N  NN0  0  <_  N
 
Theoremnn0nlt0 7944 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 NN0  <  0
 
Theoremnn0ge0i 7945 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 N  NN0   =>     0  <_  N
 
Theoremnn0le0eq0 7946 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
 N  NN0  N  <_  0  N  0
 
Theoremnn0p1gt0 7947 A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 N  NN0  0  <  N  +  1
 
Theoremnnnn0addcl 7948 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  NN  N  NN0  M  +  N  NN
 
Theoremnn0nnaddcl 7949 A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
 M  NN0  N  NN  M  +  N  NN
 
Theorem0mnnnnn0 7950 The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
 N  NN  0  -  N  e/  NN0
 
Theoremun0addcl 7951 If  S is closed under addition, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 S  C_ 
 CC   &     T  S  u.  { 0 }   &     M  S  N  S  M  +  N  S   =>     M  T  N  T  M  +  N  T
 
Theoremun0mulcl 7952 If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 S  C_ 
 CC   &     T  S  u.  { 0 }   &     M  S  N  S  M  x.  N  S   =>     M  T  N  T  M  x.  N  T
 
Theoremnn0addcl 7953 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 M  NN0  N  NN0  M  +  N  NN0
 
Theoremnn0mulcl 7954 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 M  NN0  N  NN0  M  x.  N  NN0
 
Theoremnn0addcli 7955 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 M  NN0   &     N  NN0   =>     M  +  N  NN0
 
Theoremnn0mulcli 7956 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 M  NN0   &     N  NN0   =>     M  x.  N  NN0
 
Theoremnn0p1nn 7957 A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
 N  NN0  N  +  1  NN
 
Theorempeano2nn0 7958 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 N  NN0  N  +  1  NN0
 
Theoremnnm1nn0 7959 A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
 N  NN  N  -  1  NN0
 
Theoremelnn0nn 7960 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  NN0  N  CC  N  +  1  NN
 
Theoremelnnnn0 7961 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 N  NN  N  CC  N  -  1  NN0
 
Theoremelnnnn0b 7962 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 N  NN  N  NN0  0  <  N
 
Theoremelnnnn0c 7963 The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 N  NN  N  NN0  1  <_  N
 
Theoremnn0addge1 7964 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 RR  N  NN0  <_  +  N
 
Theoremnn0addge2 7965 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 RR  N  NN0  <_  N  +
 
Theoremnn0addge1i 7966 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 RR   &     N  NN0   =>     <_  +  N
 
Theoremnn0addge2i 7967 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 RR   &     N  NN0   =>     <_  N  +
 
Theoremnn0le2xi 7968 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
 N  NN0   =>     N  <_  2  x.  N
 
Theoremnn0lele2xi 7969 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 M  NN0   &     N  NN0   =>     N  <_  M  N  <_  2  x.  M
 
Theoremnn0supp 7970 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 7971 A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 NN   =>     NN0
 
Theoremnn0red 7972 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 NN0   =>     RR
 
Theoremnn0cnd 7973 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 NN0   =>     CC
 
Theoremnn0ge0d 7974 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 NN0   =>     0  <_
 
Theoremnn0addcld 7975 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 NN0   &     NN0   =>     + 
 NN0
 
Theoremnn0mulcld 7976 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
 NN0   &     NN0   =>     x. 
 NN0
 
Theoremnn0readdcl 7977 Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 NN0  NN0  +  RR
 
Theoremnn0ge2m1nn 7978 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  NN0  2  <_  N  N  -  1  NN
 
Theoremnn0ge2m1nn0 7979 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  NN0  2  <_  N  N  -  1  NN0
 
Theoremnn0nndivcl 7980 Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 K  NN0  L  NN  K  L  RR
 
3.4.8  Integers (as a subset of complex numbers)
 
Syntaxcz 7981 Extend class notation to include the class of integers.

 ZZ
 
Definitiondf-z 7982 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  RR  |  n  0  n  NN  -u n  NN }
 
Theoremelz 7983 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
 N  ZZ  N  RR  N  0  N  NN  -u N  NN
 
Theoremnnnegz 7984 The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
 N  NN  -u N  ZZ
 
Theoremzre 7985 An integer is a real. (Contributed by NM, 8-Jan-2002.)
 N  ZZ  N  RR
 
Theoremzcn 7986 An integer is a complex number. (Contributed by NM, 9-May-2004.)
 N  ZZ  N  CC
 
Theoremzrei 7987 An integer is a real number. (Contributed by NM, 14-Jul-2005.)
 ZZ   =>     RR
 
Theoremzssre 7988 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

 ZZ  C_  RR
 
Theoremzsscn 7989 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

 ZZ  C_  CC
 
Theoremzex 7990 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

 ZZ  _V
 
Theoremelnnz 7991 Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
 N  NN  N  ZZ  0  <  N
 
Theorem0z 7992 Zero is an integer. (Contributed by NM, 12-Jan-2002.)
 0  ZZ
 
Theorem0zd 7993 Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  ZZ
 
Theoremelnn0z 7994 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
 N  NN0  N  ZZ  0  <_  N
 
Theoremelznn0nn 7995 Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
 N  ZZ  N  NN0  N  RR  -u N  NN
 
Theoremelznn0 7996 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
 N  ZZ  N  RR  N  NN0  -u N  NN0
 
Theoremelznn 7997 Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
 N  ZZ  N  RR  N  NN  -u N  NN0
 
Theoremnnssz 7998 Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

 NN  C_  ZZ
 
Theoremnn0ssz 7999 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

 NN0  C_  ZZ
 
Theoremnnz 8000 A positive integer is an integer. (Contributed by NM, 9-May-2004.)
 N  NN  N  ZZ
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