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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1nn 8001 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
 |-  1  e.  NN
 
Theorempeano2nn 8002 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
 
Theoremnnred 8003 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnncnd 8004 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  CC )
 
Theorempeano2nnd 8005 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  ( A  +  1 )  e.  NN )
 
3.4.2  Principle of mathematical induction
 
Theoremnnind 8006* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 8010 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
 |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  NN  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  NN  ->  ta )
 
TheoremnnindALT 8007* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis.

This ALT version of nnind 8006 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

 |-  ( y  e.  NN  ->  ( ch  ->  th )
 )   &    |- 
 ps   &    |-  ( x  =  1 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  NN  ->  ta )
 
Theoremnn1m1nn 8008 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
 |-  ( A  e.  NN  ->  ( A  =  1  \/  ( A  -  1 )  e.  NN ) )
 
Theoremnn1suc 8009* 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 8010 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 8011 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
 |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  x.  B )  e.  NN )
 
Theoremnnmulcli 8012 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 8013 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  1  <_  A )
 
Theoremnnle1eq1 8014 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 8015 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
 |-  ( A  e.  NN  ->  0  <  A )
 
Theoremnnnlt1 8016 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 8017 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
 |- 
 -.  0  e.  NN
 
Theoremnnne0 8018 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  e.  NN  ->  A  =/=  0 )
 
Theoremnnap0 8019 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
 |-  ( A  e.  NN  ->  A #  0 )
 
Theoremnngt0i 8020 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
 |-  A  e.  NN   =>    |-  0  <  A
 
Theoremnnne0i 8021 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  NN   =>    |-  A  =/=  0
 
Theoremnn2ge 8022* 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 8023 A positive integer is either one or greater than one. This is for  NN; 0elnn 4368 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 8024 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 8025 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 8026 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
 |-  ( N  e.  NN  ->  ( 1  /  N )  e.  RR )
 
Theoremnnrecgt0 8027 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
 |-  ( A  e.  NN  ->  0  <  ( 1 
 /  A ) )
 
Theoremnnsub 8028 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 8029 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
 |-  A  e.  NN   &    |-  B  e.  NN   =>    |-  ( A  <  B  <->  ( B  -  A )  e.  NN )
 
Theoremnndiv 8030* 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 8031 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 8032 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  1  <_  A )
 
Theoremnngt0d 8033 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  0  <  A )
 
Theoremnnne0d 8034 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  =/=  0 )
 
Theoremnnap0d 8035 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A #  0 )
 
Theoremnnrecred 8036 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 8037 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 8038 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 8039 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 6954 through df-9 8056), 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 6954 and df-1 6955).

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 8040 Extend class notation to include the number 2.
 class 
 2
 
Syntaxc3 8041 Extend class notation to include the number 3.
 class 
 3
 
Syntaxc4 8042 Extend class notation to include the number 4.
 class 
 4
 
Syntaxc5 8043 Extend class notation to include the number 5.
 class 
 5
 
Syntaxc6 8044 Extend class notation to include the number 6.
 class 
 6
 
Syntaxc7 8045 Extend class notation to include the number 7.
 class 
 7
 
Syntaxc8 8046 Extend class notation to include the number 8.
 class 
 8
 
Syntaxc9 8047 Extend class notation to include the number 9.
 class 
 9
 
Syntaxc10 8048 Extend class notation to include the number 10.
 class  10
 
Definitiondf-2 8049 Define the number 2. (Contributed by NM, 27-May-1999.)
 |-  2  =  ( 1  +  1 )
 
Definitiondf-3 8050 Define the number 3. (Contributed by NM, 27-May-1999.)
 |-  3  =  ( 2  +  1 )
 
Definitiondf-4 8051 Define the number 4. (Contributed by NM, 27-May-1999.)
 |-  4  =  ( 3  +  1 )
 
Definitiondf-5 8052 Define the number 5. (Contributed by NM, 27-May-1999.)
 |-  5  =  ( 4  +  1 )
 
Definitiondf-6 8053 Define the number 6. (Contributed by NM, 27-May-1999.)
 |-  6  =  ( 5  +  1 )
 
Definitiondf-7 8054 Define the number 7. (Contributed by NM, 27-May-1999.)
 |-  7  =  ( 6  +  1 )
 
Definitiondf-8 8055 Define the number 8. (Contributed by NM, 27-May-1999.)
 |-  8  =  ( 7  +  1 )
 
Definitiondf-9 8056 Define the number 9. (Contributed by NM, 27-May-1999.)
 |-  9  =  ( 8  +  1 )
 
Theorem0ne1 8057  0  =/=  1 (common case). See aso 1ap0 7655. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  0  =/=  1
 
Theorem1ne0 8058  1  =/=  0. See aso 1ap0 7655. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  1  =/=  0
 
Theorem1m1e0 8059  ( 1  -  1 )  =  0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  ( 1  -  1
 )  =  0
 
Theorem2re 8060 The number 2 is real. (Contributed by NM, 27-May-1999.)
 |-  2  e.  RR
 
Theorem2cn 8061 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
 |-  2  e.  CC
 
Theorem2ex 8062 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  2  e.  _V
 
Theorem2cnd 8063 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( ph  ->  2  e.  CC )
 
Theorem3re 8064 The number 3 is real. (Contributed by NM, 27-May-1999.)
 |-  3  e.  RR
 
Theorem3cn 8065 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
 |-  3  e.  CC
 
Theorem3ex 8066 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  3  e.  _V
 
Theorem4re 8067 The number 4 is real. (Contributed by NM, 27-May-1999.)
 |-  4  e.  RR
 
Theorem4cn 8068 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  4  e.  CC
 
Theorem5re 8069 The number 5 is real. (Contributed by NM, 27-May-1999.)
 |-  5  e.  RR
 
Theorem5cn 8070 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  5  e.  CC
 
Theorem6re 8071 The number 6 is real. (Contributed by NM, 27-May-1999.)
 |-  6  e.  RR
 
Theorem6cn 8072 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  6  e.  CC
 
Theorem7re 8073 The number 7 is real. (Contributed by NM, 27-May-1999.)
 |-  7  e.  RR
 
Theorem7cn 8074 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  7  e.  CC
 
Theorem8re 8075 The number 8 is real. (Contributed by NM, 27-May-1999.)
 |-  8  e.  RR
 
Theorem8cn 8076 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  8  e.  CC
 
Theorem9re 8077 The number 9 is real. (Contributed by NM, 27-May-1999.)
 |-  9  e.  RR
 
Theorem9cn 8078 The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  9  e.  CC
 
Theorem0le0 8079 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  <_  0
 
Theorem0le2 8080 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  0  <_  2
 
Theorem2pos 8081 The number 2 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  2
 
Theorem2ne0 8082 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
 |-  2  =/=  0
 
Theorem2ap0 8083 The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
 |-  2 #  0
 
Theorem3pos 8084 The number 3 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  3
 
Theorem3ne0 8085 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
 |-  3  =/=  0
 
Theorem3ap0 8086 The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  3 #  0
 
Theorem4pos 8087 The number 4 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  4
 
Theorem4ne0 8088 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  4  =/=  0
 
Theorem4ap0 8089 The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  4 #  0
 
Theorem5pos 8090 The number 5 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  5
 
Theorem6pos 8091 The number 6 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  6
 
Theorem7pos 8092 The number 7 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  7
 
Theorem8pos 8093 The number 8 is positive. (Contributed by NM, 27-May-1999.)
 |-  0  <  8
 
Theorem9pos 8094 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 8095 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  -u 1  e.  CC
 
Theoremneg1rr 8096 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
 |-  -u 1  e.  RR
 
Theoremneg1ne0 8097 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  =/=  0
 
Theoremneg1lt0 8098 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u 1  <  0
 
Theoremneg1ap0 8099 -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
 |-  -u 1 #  0
 
Theoremnegneg1e1 8100  -u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u -u 1  =  1
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