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Theorem List for Intuitionistic Logic Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnne0d 8401 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnap0d 8402 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A #  0 )
 
Theoremnnrecred 8403 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  (
 1  /  A )  e.  RR )
 
Theoremnnaddcld 8404 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcld 8405 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  x.  B )  e.  NN )
 
Theoremnndivred 8406 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  /  B )  e.  RR )
 
3.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7301 through df-9 8423), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7301 and df-1 7302).

Integers can also be exhibited as sums of powers of 10 (e.g. the number 103 can be expressed as  ( (; 1 0 ^ 2 )  +  3 )) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as 
( 7 ^ 7 )  -  2.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

 
Syntaxc2 8407 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 8408 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 8409 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 8410 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 8411 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 8412 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 8413 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 8414 Extend class notation to include the number 9.
 class 
 9
 
Syntaxc10 8415 Extend class notation to include the number 10.
 class  10
 
Definitiondf-2 8416 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 8417 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 8418 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 8419 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 8420 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 8421 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 8422 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 8423 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Theorem0ne1 8424  0  =/=  1 (common case). See aso 1ap0 8008. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  1
 
Theorem1ne0 8425  1  =/=  0. See aso 1ap0 8008. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  1  =/=  0
 
Theorem1m1e0 8426  ( 1  -  1 )  =  0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  -  1
 )  =  0
 
Theorem2re 8427 The number 2 is real. (Contributed by NM, 27-May-1999.)
 |-  2  e.  RR
 
Theorem2cn 8428 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 |-  2  e.  CC
 
Theorem2ex 8429 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  2  e.  _V
 
Theorem2cnd 8430 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  2  e.  CC )
 
Theorem3re 8431 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 8432 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem3ex 8433 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  _V
 
Theorem4re 8434 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 8435 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 8436 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem5cn 8437 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  5  e.  CC
 
Theorem6re 8438 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem6cn 8439 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  6  e.  CC
 
Theorem7re 8440 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem7cn 8441 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  7  e.  CC
 
Theorem8re 8442 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem8cn 8443 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  8  e.  CC
 
Theorem9re 8444 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem9cn 8445 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  9  e.  CC
 
Theorem0le0 8446 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem0le2 8447 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  0  <_  2
 
Theorem2pos 8448 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 8449 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem2ap0 8450 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  2 #  0
 
Theorem3pos 8451 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 8452 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem3ap0 8453 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  3 #  0
 
Theorem4pos 8454 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem4ne0 8455 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  4  =/=  0
 
Theorem4ap0 8456 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  4 #  0
 
Theorem5pos 8457 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 8458 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 8459 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 8460 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 8461 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
3.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

 
Theoremneg1cn 8462 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theoremneg1rr 8463 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  RR
 
Theoremneg1ne0 8464 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  =/=  0
 
Theoremneg1lt0 8465 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  <  0
 
Theoremneg1ap0 8466 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  -u 1 #  0
 
Theoremnegneg1e1 8467  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u -u 1  =  1
 
Theorem1pneg1e0 8468  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  -u 1
 )  =  0
 
Theorem0m0e0 8469 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0  -  0
 )  =  0
 
Theorem1m0e1 8470 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  -  0
 )  =  1
 
Theorem0p1e1 8471 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theorem1p0e1 8472 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  0 )  =  1
 
Theorem1p1e2 8473 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 8474 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8495. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
 
Theorem1e2m1 8475 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  1  =  ( 2  -  1 )
 
Theorem3m1e2 8476 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
 |-  ( 3  -  1
 )  =  2
 
Theorem2p2e4 8477 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
 |-  ( 2  +  2 )  =  4
 
Theorem2times 8478 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
 |-  ( A  e.  CC  ->  ( 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2 8479 A number times 2. (Contributed by NM, 16-Oct-2007.)
 |-  ( A  e.  CC  ->  ( A  x.  2
 )  =  ( A  +  A ) )
 
Theorem2timesi 8480 Two times a number. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( 2  x.  A )  =  ( A  +  A )
 
Theoremtimes2i 8481 A number times 2. (Contributed by NM, 11-May-2004.)
 |-  A  e.  CC   =>    |-  ( A  x.  2 )  =  ( A  +  A )
 
Theorem2div2e1 8482 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 2  /  2
 )  =  1
 
Theorem2p1e3 8483 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 2  +  1 )  =  3
 
Theorem1p2e3 8484 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  2 )  =  3
 
Theorem3p1e4 8485 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 3  +  1 )  =  4
 
Theorem4p1e5 8486 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 4  +  1 )  =  5
 
Theorem5p1e6 8487 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 5  +  1 )  =  6
 
Theorem6p1e7 8488 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 6  +  1 )  =  7
 
Theorem7p1e8 8489 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 7  +  1 )  =  8
 
Theorem8p1e9 8490 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
 |-  ( 8  +  1 )  =  9
 
Theorem3p2e5 8491 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  2 )  =  5
 
Theorem3p3e6 8492 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 3  +  3 )  =  6
 
Theorem4p2e6 8493 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  2 )  =  6
 
Theorem4p3e7 8494 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  3 )  =  7
 
Theorem4p4e8 8495 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 4  +  4 )  =  8
 
Theorem5p2e7 8496 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  2 )  =  7
 
Theorem5p3e8 8497 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  3 )  =  8
 
Theorem5p4e9 8498 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 5  +  4 )  =  9
 
Theorem6p2e8 8499 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  2 )  =  8
 
Theorem6p3e9 8500 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
 |-  ( 6  +  3 )  =  9
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