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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem6lt7 8101 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 8102 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 8103 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 8104 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 8105 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 8106 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 8107 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 8108 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 8109 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
 
Theorem4lt8 8110 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  8
 
Theorem3lt8 8111 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  8
 
Theorem2lt8 8112 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  8
 
Theorem1lt8 8113 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  8
 
Theorem8lt9 8114 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  8  <  9
 
Theorem7lt9 8115 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  7  <  9
 
Theorem6lt9 8116 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  6  <  9
 
Theorem5lt9 8117 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  5  <  9
 
Theorem4lt9 8118 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  4  <  9
 
Theorem3lt9 8119 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  3  <  9
 
Theorem2lt9 8120 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  2  <  9
 
Theorem1lt9 8121 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  9
 
Theorem9lt10 8122 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  9  <  10
 
Theorem8lt10 8123 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  8  <  10
 
Theorem7lt10 8124 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  7  <  10
 
Theorem6lt10 8125 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  6  <  10
 
Theorem5lt10 8126 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  5  <  10
 
Theorem4lt10 8127 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  4  <  10
 
Theorem3lt10 8128 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  3  <  10
 
Theorem2lt10 8129 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  2  <  10
 
Theorem1lt10 8130 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  10
 
Theorem0ne2 8131 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  2
 
Theorem1ne2 8132 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 |-  1  =/=  2
 
Theorem1le2 8133 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  2
 
Theorem2cnne0 8134 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  ( 2  e.  CC  /\  2  =/=  0 )
 
Theorem2rene0 8135 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  e.  RR  /\  2  =/=  0 )
 
Theorem1le3 8136 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  3
 
Theoremneg1mulneg1e1 8137  -u 1  x.  -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1  x.  -u 1
 )  =  1
 
Theoremhalfre 8138 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  RR
 
Theoremhalfcn 8139 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  CC
 
Theoremhalfgt0 8140 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalfge0 8141 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2 )
 
Theoremhalflt1 8142 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 8143 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
 
Theorem8th4div3 8144 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 |-  ( ( 1  / 
 8 )  x.  (
 4  /  3 )
 )  =  ( 1 
 /  6 )
 
Theoremhalfpm6th 8145 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 8146 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( _i  x.  0
 )  =  0
 
Theorem2mulicn 8147  ( 2  x.  _i )  e.  CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  e.  CC
 
Theoremiap0 8148 The imaginary unit  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  _i #  0
 
Theorem2muliap0 8149  2  x.  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  ( 2  x.  _i ) #  0
 
Theorem2muline0 8150  ( 2  x.  _i )  =/=  0. See also 2muliap0 8149. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  =/=  0
 
3.4.5  Simple number properties
 
Theoremhalfcl 8151 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 8152 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 8153 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 8154 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 8155 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 8156 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 8157 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 8158 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 8159 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 8160 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 8161 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 8162* 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 8163 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavglt2 8164 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( ( A  +  B )  /  2
 )  <  B )
 )
 
Theoremavgle1 8165 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A  <_  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavgle2 8166 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 8167 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2d 8168 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  2 )  =  ( A  +  A ) )
 
Theoremhalfcld 8169 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 8170 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  /  2
 )  +  ( A 
 /  2 ) )  =  A )
 
Theoremrehalfcld 8171 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 RR )
 
Theoremlt2halvesd 8172 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 8173 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  ( A  / 
 2 )  e.  RR
 
Theoremadd1p1 8174 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 |-  ( N  e.  CC  ->  ( ( N  +  1 )  +  1
 )  =  ( N  +  2 ) )
 
Theoremsub1m1 8175 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 |-  ( N  e.  CC  ->  ( ( N  -  1 )  -  1
 )  =  ( N  -  2 ) )
 
Theoremcnm2m1cnm3 8176 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 ) )
 
Theoremdiv4p1lem1div2 8177 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 ) )
 
3.4.6  The Archimedean property
 
Theoremarch 8178* 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 8179* 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 8180* 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 )
 
3.4.7  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 8181 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 8182 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
 
Theoremelnn0 8183 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 8184 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN  C_  NN0
 
Theoremnn0ssre 8185 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  C_  RR
 
Theoremnn0sscn 8186 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  CC
 
Theoremnn0ex 8187 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
 |- 
 NN0  e.  _V
 
Theoremnnnn0 8188 A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN  ->  A  e.  NN0 )
 
Theoremnnnn0i 8189 A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 |-  N  e.  NN   =>    |-  N  e.  NN0
 
Theoremnn0re 8190 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  RR )
 
Theoremnn0cn 8191 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  CC )
 
Theoremnn0rei 8192 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 8193 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 8194 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 8195 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 8196 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 8197 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 8198 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 8199 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 8200 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
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