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Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn1suc 9001* If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <-> 
 th ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ch )   =>    |-  ( A  e.  NN  ->  th )
 
Theoremnnaddcl 9002 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  +  B )  e.  NN )
 
Theoremnnmulcl 9003 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 9004 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  x.  B )  e.  NN
 
Theoremnnge1 9005 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnnle1eq1 9006 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
 |-  ( A  e.  NN  ->  ( A  <_  1  <->  A  =  1 ) )
 
Theoremnngt0 9007 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 9008 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN  ->  -.  A  <  1
 )
 
Theorem0nnn 9009 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 9010 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnnap0 9011 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( A  e.  NN  ->  A #  0 )
 
Theoremnngt0i 9012 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnap0i 9013 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
 |-  A  e.  NN   =>    |-  A #  0
 
Theoremnnne0i 9014 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnn2ge 9015* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  E. x  e.  NN  ( A  <_  x  /\  B  <_  x ) )
 
Theoremnn1gt1 9016 A positive integer is either one or greater than one. This is for  NN; 0elnn 4651 is a similar theorem for  om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  1  <  A ) )
 
Theoremnngt1ne1 9017 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
 |-  ( A  e.  NN  ->  ( 1  <  A  <->  A  =/=  1 ) )
 
Theoremnndivre 9018 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( A  /  N )  e.  RR )
 
Theoremnnrecre 9019 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 9020 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 9021 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  <-> 
 ( B  -  A )  e.  NN )
 )
 
Theoremnnsubi 9022 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 9023* Two ways to express " A divides  B " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( E. x  e.  NN  ( A  x.  x )  =  B  <->  ( B  /  A )  e.  NN ) )
 
Theoremnndivtr 9024 Transitive property of divisibility: if  A divides  B and  B divides  C, then  A divides  C. Typically,  C would be an integer, although the theorem holds for complex  C. (Contributed by NM, 3-May-2005.)
 |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  /\  ( ( B  /  A )  e.  NN  /\  ( C  /  B )  e. 
 NN ) )  ->  ( C  /  A )  e.  NN )
 
Theoremnnge1d 9025 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  1  <_  A )
 
Theoremnngt0d 9026 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  0  <  A )
 
Theoremnnne0d 9027 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnap0d 9028 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A #  0 )
 
Theoremnnrecred 9029 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 9030 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 9031 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 9032 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 )
 
4.4.3  Decimal representation of numbers

The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7879 through df-9 9048), 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 7879 and df-1 7880).

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 9033 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 9034 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 9035 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 9036 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 9037 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 9038 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 9039 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 9040 Extend class notation to include the number 9.
 class 
 9
 
Definitiondf-2 9041 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 9042 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 9043 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 9044 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 9045 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 9046 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 9047 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 9048 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Theorem0ne1 9049  0  =/=  1 (common case). See aso 1ap0 8609. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  1
 
Theorem1ne0 9050  1  =/=  0. See aso 1ap0 8609. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  1  =/=  0
 
Theorem1m1e0 9051  ( 1  -  1 )  =  0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  -  1
 )  =  0
 
Theorem2re 9052 The number 2 is real. (Contributed by NM, 27-May-1999.)
 |-  2  e.  RR
 
Theorem2cn 9053 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 |-  2  e.  CC
 
Theorem2ex 9054 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  2  e.  _V
 
Theorem2cnd 9055 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  2  e.  CC )
 
Theorem3re 9056 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 9057 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem3ex 9058 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  _V
 
Theorem4re 9059 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 9060 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 9061 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem5cn 9062 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  5  e.  CC
 
Theorem6re 9063 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem6cn 9064 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  6  e.  CC
 
Theorem7re 9065 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem7cn 9066 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  7  e.  CC
 
Theorem8re 9067 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem8cn 9068 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  8  e.  CC
 
Theorem9re 9069 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem9cn 9070 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  9  e.  CC
 
Theorem0le0 9071 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem0le2 9072 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  0  <_  2
 
Theorem2pos 9073 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 9074 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem2ap0 9075 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  2 #  0
 
Theorem3pos 9076 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 9077 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem3ap0 9078 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  3 #  0
 
Theorem4pos 9079 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem4ne0 9080 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  4  =/=  0
 
Theorem4ap0 9081 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  4 #  0
 
Theorem5pos 9082 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 9083 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 9084 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 9085 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 9086 The number 9 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  9
 
4.4.4  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 9087 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theoremneg1rr 9088 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  RR
 
Theoremneg1ne0 9089 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  =/=  0
 
Theoremneg1lt0 9090 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  <  0
 
Theoremneg1ap0 9091 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  -u 1 #  0
 
Theoremnegneg1e1 9092  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u -u 1  =  1
 
Theorem1pneg1e0 9093  1  +  -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  -u 1
 )  =  0
 
Theorem0m0e0 9094 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 0  -  0
 )  =  0
 
Theorem1m0e1 9095 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  -  0
 )  =  1
 
Theorem0p1e1 9096 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 0  +  1 )  =  1
 
Theoremfv0p1e1 9097 Function value at  N  +  1 with  N replaced by  0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
 |-  ( N  =  0 
 ->  ( F `  ( N  +  1 )
 )  =  ( F `
  1 ) )
 
Theorem1p0e1 9098 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( 1  +  0 )  =  1
 
Theorem1p1e2 9099 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
 |-  ( 1  +  1 )  =  2
 
Theorem2m1e1 9100 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9127. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 2  -  1
 )  =  1
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