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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem8lt10 7901 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 8  <  10
 
Theorem7lt10 7902 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 7  <  10
 
Theorem6lt10 7903 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 6  <  10
 
Theorem5lt10 7904 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 5  <  10
 
Theorem4lt10 7905 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 4  <  10
 
Theorem3lt10 7906 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 3  <  10
 
Theorem2lt10 7907 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 2  <  10
 
Theorem1lt10 7908 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 1  <  10
 
Theorem0ne2 7909 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
 0  =/=  2
 
Theorem1ne2 7910 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 1  =/=  2
 
Theorem1le2 7911 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  <_  2
 
Theorem2cnne0 7912 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 2  CC  2  =/=  0
 
Theorem2rene0 7913 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  RR  2  =/=  0
 
Theorem1le3 7914 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  <_  3
 
Theoremneg1mulneg1e1 7915  -u 1  x.  -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 -u 1  x.  -u 1  1
 
Theoremhalfre 7916 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  2  RR
 
Theoremhalfcn 7917 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 1  2  CC
 
Theoremhalfgt0 7918 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 0  <  1  2
 
Theoremhalflt1 7919 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 1  2  <  1
 
Theorem1mhlfehlf 7920 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 1  - 
 1  2  1  2
 
Theorem8th4div3 7921 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 1  8  x. 
 4  3  1  6
 
Theoremhalfpm6th 7922 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 7923 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 _i  x.  0  0
 
Theorem2mulicn 7924  2  x.  _i  CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  x.  _i  CC
 
Theoremiap0 7925 The imaginary unit  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 _i #  0
 
Theorem2muliap0 7926  2  x.  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 2  x.  _i #  0
 
Theorem2muline0 7927  2  x.  _i  =/=  0. See also 2muliap0 7926. (Contributed by David A. Wheeler, 8-Dec-2018.)
 2  x.  _i  =/=  0
 
3.4.5  Simple number properties
 
Theoremhalfcl 7928 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
 CC  2  CC
 
Theoremrehalfcl 7929 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 RR  2  RR
 
Theoremhalf0 7930 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
 CC  2  0  0
 
Theorem2halves 7931 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 CC  2  +  2
 
Theoremhalfpos2 7932 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 RR  0  <  0  <  2
 
Theoremhalfpos 7933 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  0  <  2  <
 
Theoremhalfnneg2 7934 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 RR  0  <_  0  <_  2
 
Theoremhalfaddsubcl 7935 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 CC  CC  +  2  CC  -  2  CC
 
Theoremhalfaddsub 7936 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 CC  CC  +  2  +  -  2  +  2  -  -  2
 
Theoremlt2halves 7937 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
 RR  RR  C  RR  <  C  2  <  C  2  +  <  C
 
Theoremaddltmul 7938 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
 RR  RR 
 2  <  2  <  +  <  x.
 
Theoremnominpos 7939* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
 RR  0  <  RR  0  <  <
 
Theoremavglt1 7940 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 RR  RR  <  <  +  2
 
Theoremavglt2 7941 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 RR  RR  <  +  2  <
 
Theoremavgle1 7942 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 RR  RR  <_  <_  +  2
 
Theoremavgle2 7943 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
 RR  RR  <_  +  2  <_
 
Theorem2timesd 7944 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 2  x.  +
 
Theoremtimes2d 7945 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  2  +
 
Theoremhalfcld 7946 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     2 
 CC
 
Theorem2halvesd 7947 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     2  +  2
 
Theoremrehalfcld 7948 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     2 
 RR
 
Theoremlt2halvesd 7949 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 7950 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 RR   =>     2  RR
 
Theoremadd1p1 7951 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 N  CC  N  +  1  +  1  N  +  2
 
Theoremsub1m1 7952 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 N  CC  N  -  1  -  1  N  -  2
 
Theoremcnm2m1cnm3 7953 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 7954* 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 7955* 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 7956* 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 7957 Extend class notation to include the class of nonnegative integers.
 NN0
 
Definitiondf-n0 7958 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

 NN0  NN  u.  { 0 }
 
Theoremelnn0 7959 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
 NN0  NN  0
 
Theoremnnssnn0 7960 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

 NN  C_  NN0
 
Theoremnn0ssre 7961 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

 NN0  C_  RR
 
Theoremnn0sscn 7962 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

 NN0  C_  CC
 
Theoremnn0ex 7963 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

 NN0  _V
 
Theoremnnnn0 7964 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 NN  NN0
 
Theoremnnnn0i 7965 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 N  NN   =>     N  NN0
 
Theoremnn0re 7966 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 NN0  RR
 
Theoremnn0cn 7967 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 NN0  CC
 
Theoremnn0rei 7968 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 NN0   =>     RR
 
Theoremnn0cni 7969 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 NN0   =>     CC
 
Theoremdfn2 7970 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 7971 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 7972 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 0  NN0
 
Theorem1nn0 7973 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 1  NN0
 
Theorem2nn0 7974 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 2  NN0
 
Theorem3nn0 7975 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 3  NN0
 
Theorem4nn0 7976 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 4  NN0
 
Theorem5nn0 7977 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 5  NN0
 
Theorem6nn0 7978 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  NN0
 
Theorem7nn0 7979 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  NN0
 
Theorem8nn0 7980 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  NN0
 
Theorem9nn0 7981 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 9  NN0
 
Theorem10nn0 7982 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

 10  NN0
 
Theoremnn0ge0 7983 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 7984 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 NN0  <  0
 
Theoremnn0ge0i 7985 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 N  NN0   =>     0  <_  N
 
Theoremnn0le0eq0 7986 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 7987 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 7988 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 7989 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 7990 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 7991 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 7992 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 7993 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 7994 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 7995 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 M  NN0   &     N  NN0   =>     M  +  N  NN0
 
Theoremnn0mulcli 7996 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 M  NN0   &     N  NN0   =>     M  x.  N  NN0
 
Theoremnn0p1nn 7997 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 7998 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 N  NN0  N  +  1  NN0
 
Theoremnnm1nn0 7999 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 8000 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
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