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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4t2e8 8501 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
 |-  ( 4  x.  2
 )  =  8
 
Theorem2t0e0 8502 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  0
 )  =  0
 
Theorem4d2e2 8503 One half of four is two. (Contributed by NM, 3-Sep-1999.)
 |-  ( 4  /  2
 )  =  2
 
Theorem2nn 8504 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
 |-  2  e.  NN
 
Theorem3nn 8505 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  3  e.  NN
 
Theorem4nn 8506 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
 |-  4  e.  NN
 
Theorem5nn 8507 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  e.  NN
 
Theorem6nn 8508 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  e.  NN
 
Theorem7nn 8509 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  e.  NN
 
Theorem8nn 8510 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  8  e.  NN
 
Theorem9nn 8511 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
 |-  9  e.  NN
 
Theorem1lt2 8512 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
 |-  1  <  2
 
Theorem2lt3 8513 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  2  <  3
 
Theorem1lt3 8514 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
 |-  1  <  3
 
Theorem3lt4 8515 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  4
 
Theorem2lt4 8516 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  4
 
Theorem1lt4 8517 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  4
 
Theorem4lt5 8518 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  5
 
Theorem3lt5 8519 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  5
 
Theorem2lt5 8520 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  5
 
Theorem1lt5 8521 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  5
 
Theorem5lt6 8522 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  6
 
Theorem4lt6 8523 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  6
 
Theorem3lt6 8524 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  6
 
Theorem2lt6 8525 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  6
 
Theorem1lt6 8526 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
 |-  1  <  6
 
Theorem6lt7 8527 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  7
 
Theorem5lt7 8528 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  7
 
Theorem4lt7 8529 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  7
 
Theorem3lt7 8530 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  7
 
Theorem2lt7 8531 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  7
 
Theorem1lt7 8532 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  7
 
Theorem7lt8 8533 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  7  <  8
 
Theorem6lt8 8534 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  6  <  8
 
Theorem5lt8 8535 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  5  <  8
 
Theorem4lt8 8536 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  4  <  8
 
Theorem3lt8 8537 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  3  <  8
 
Theorem2lt8 8538 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  2  <  8
 
Theorem1lt8 8539 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
 |-  1  <  8
 
Theorem8lt9 8540 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
 |-  8  <  9
 
Theorem7lt9 8541 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  7  <  9
 
Theorem6lt9 8542 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  6  <  9
 
Theorem5lt9 8543 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  5  <  9
 
Theorem4lt9 8544 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  4  <  9
 
Theorem3lt9 8545 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  3  <  9
 
Theorem2lt9 8546 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
 |-  2  <  9
 
Theorem1lt9 8547 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  9
 
Theorem0ne2 8548 0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  2
 
Theorem1ne2 8549 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 |-  1  =/=  2
 
Theorem1le2 8550 1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  2
 
Theorem2cnne0 8551 2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  ( 2  e.  CC  /\  2  =/=  0 )
 
Theorem2rene0 8552 2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  e.  RR  /\  2  =/=  0 )
 
Theorem1le3 8553 1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  <_  3
 
Theoremneg1mulneg1e1 8554  -u 1  x.  -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( -u 1  x.  -u 1
 )  =  1
 
Theoremhalfre 8555 One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  RR
 
Theoremhalfcn 8556 One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  /  2
 )  e.  CC
 
Theoremhalfgt0 8557 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalfge0 8558 One-half is not negative. (Contributed by AV, 7-Jun-2020.)
 |-  0  <_  ( 1  /  2 )
 
Theoremhalflt1 8559 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 8560 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
 
Theorem8th4div3 8561 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 |-  ( ( 1  / 
 8 )  x.  (
 4  /  3 )
 )  =  ( 1 
 /  6 )
 
Theoremhalfpm6th 8562 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 8563 i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( _i  x.  0
 )  =  0
 
Theorem2mulicn 8564  ( 2  x.  _i )  e.  CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  e.  CC
 
Theoremiap0 8565 The imaginary unit  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  _i #  0
 
Theorem2muliap0 8566  2  x.  _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  ( 2  x.  _i ) #  0
 
Theorem2muline0 8567  ( 2  x.  _i )  =/=  0. See also 2muliap0 8566. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  x.  _i )  =/=  0
 
3.4.5  Simple number properties
 
Theoremhalfcl 8568 Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 8569 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 8570 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 8571 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 8572 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 8573 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 8574 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 8575 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 8576 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 8577 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 8578 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 8579* 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 8580 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavglt2 8581 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( ( A  +  B )  /  2
 )  <  B )
 )
 
Theoremavgle1 8582 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A  <_  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavgle2 8583 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 8584 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2d 8585 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  2 )  =  ( A  +  A ) )
 
Theoremhalfcld 8586 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 8587 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  /  2
 )  +  ( A 
 /  2 ) )  =  A )
 
Theoremrehalfcld 8588 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 RR )
 
Theoremlt2halvesd 8589 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 8590 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  ( A  / 
 2 )  e.  RR
 
Theoremadd1p1 8591 Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
 |-  ( N  e.  CC  ->  ( ( N  +  1 )  +  1
 )  =  ( N  +  2 ) )
 
Theoremsub1m1 8592 Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
 |-  ( N  e.  CC  ->  ( ( N  -  1 )  -  1
 )  =  ( N  -  2 ) )
 
Theoremcnm2m1cnm3 8593 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 8594 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 8595 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 8596* 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 8597* 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 8598* 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 8599 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 8600 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
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