Theorem List for Intuitionistic Logic Explorer - 9001-9100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | ztri3or 9001 |
Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|
       |
|
Theorem | zletric 9002 |
Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|
       |
|
Theorem | zlelttric 9003 |
Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|
       |
|
Theorem | zltnle 9004 |
'Less than' expressed in terms of 'less than or equal to'. (Contributed
by Jim Kingdon, 14-Mar-2020.)
|
   
   |
|
Theorem | zleloe 9005 |
Integer 'Less than or equal to' expressed in terms of 'less than' or
'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
|
         |
|
Theorem | znnnlt1 9006 |
An integer is not a positive integer iff it is less than one.
(Contributed by NM, 13-Jul-2005.)
|
     |
|
Theorem | zletr 9007 |
Transitive law of ordering for integers. (Contributed by Alexander van
der Vekens, 3-Apr-2018.)
|
         |
|
Theorem | zrevaddcl 9008 |
Reverse closure law for addition of integers. (Contributed by NM,
11-May-2004.)
|
    
    |
|
Theorem | znnsub 9009 |
The positive difference of unequal integers is a positive integer.
(Generalization of nnsub 8669.) (Contributed by NM, 11-May-2004.)
|
     
   |
|
Theorem | nzadd 9010 |
The sum of a real number not being an integer and an integer is not an
integer. Note that "not being an integer" in this case means
"the
negation of is an integer" rather than "is apart from any
integer" (given
excluded middle, those two would be equivalent). (Contributed by AV,
19-Jul-2021.)
|
  
   
    |
|
Theorem | zmulcl 9011 |
Closure of multiplication of integers. (Contributed by NM,
30-Jul-2004.)
|
    
  |
|
Theorem | zltp1le 9012 |
Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof
shortened by Mario Carneiro, 16-May-2014.)
|
     
   |
|
Theorem | zleltp1 9013 |
Integer ordering relation. (Contributed by NM, 10-May-2004.)
|
         |
|
Theorem | zlem1lt 9014 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
     
   |
|
Theorem | zltlem1 9015 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
         |
|
Theorem | zgt0ge1 9016 |
An integer greater than
is greater than or equal to .
(Contributed by AV, 14-Oct-2018.)
|
     |
|
Theorem | nnleltp1 9017 |
Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | nnltp1le 9018 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|
     
   |
|
Theorem | nnaddm1cl 9019 |
Closure of addition of positive integers minus one. (Contributed by NM,
6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|
      
  |
|
Theorem | nn0ltp1le 9020 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | nn0leltp1 9021 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
|
         |
|
Theorem | nn0ltlem1 9022 |
Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | znn0sub 9023 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 9024.) (Contributed by NM, 14-Jul-2005.)
|
     
   |
|
Theorem | nn0sub 9024 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|
         |
|
Theorem | nn0n0n1ge2 9025 |
A nonnegative integer which is neither 0 nor 1 is greater than or equal to
2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
|
  
  |
|
Theorem | elz2 9026* |
Membership in the set of integers. Commonly used in constructions of
the integers as equivalence classes under subtraction of the positive
integers. (Contributed by Mario Carneiro, 16-May-2014.)
|
 
     |
|
Theorem | dfz2 9027 |
Alternate definition of the integers, based on elz2 9026.
(Contributed by
Mario Carneiro, 16-May-2014.)
|
     |
|
Theorem | nn0sub2 9028 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|
       |
|
Theorem | zapne 9029 |
Apartness is equivalent to not equal for integers. (Contributed by Jim
Kingdon, 14-Mar-2020.)
|
    #    |
|
Theorem | zdceq 9030 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
|
   DECID
  |
|
Theorem | zdcle 9031 |
Integer is
decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
|
   DECID   |
|
Theorem | zdclt 9032 |
Integer is
decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
|
   DECID   |
|
Theorem | zltlen 9033 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8311 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 14-Mar-2020.)
|
         |
|
Theorem | nn0n0n1ge2b 9034 |
A nonnegative integer is neither 0 nor 1 if and only if it is greater than
or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
|
   
   |
|
Theorem | nn0lt10b 9035 |
A nonnegative integer less than is .
(Contributed by Paul
Chapman, 22-Jun-2011.)
|
 
   |
|
Theorem | nn0lt2 9036 |
A nonnegative integer less than 2 must be 0 or 1. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
|
       |
|
Theorem | nn0le2is012 9037 |
A nonnegative integer which is less than or equal to 2 is either 0 or 1 or
2. (Contributed by AV, 16-Mar-2019.)
|
 
     |
|
Theorem | nn0lem1lt 9038 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
|
         |
|
Theorem | nnlem1lt 9039 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
     
   |
|
Theorem | nnltlem1 9040 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
         |
|
Theorem | nnm1ge0 9041 |
A positive integer decreased by 1 is greater than or equal to 0.
(Contributed by AV, 30-Oct-2018.)
|

    |
|
Theorem | nn0ge0div 9042 |
Division of a nonnegative integer by a positive number is not negative.
(Contributed by Alexander van der Vekens, 14-Apr-2018.)
|
  
    |
|
Theorem | zdiv 9043* |
Two ways to express " divides .
(Contributed by NM,
3-Oct-2008.)
|
      
     |
|
Theorem | zdivadd 9044 |
Property of divisibility: if divides
and then it divides
. (Contributed by NM, 3-Oct-2008.)
|
   
         
   |
|
Theorem | zdivmul 9045 |
Property of divisibility: if divides
then it divides
. (Contributed by NM, 3-Oct-2008.)
|
   
         |
|
Theorem | zextle 9046* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
   
 
  |
|
Theorem | zextlt 9047* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
   
 
  |
|
Theorem | recnz 9048 |
The reciprocal of a number greater than 1 is not an integer. (Contributed
by NM, 3-May-2005.)
|
  
    |
|
Theorem | btwnnz 9049 |
A number between an integer and its successor is not an integer.
(Contributed by NM, 3-May-2005.)
|
       |
|
Theorem | gtndiv 9050 |
A larger number does not divide a smaller positive integer. (Contributed
by NM, 3-May-2005.)
|
    
  |
|
Theorem | halfnz 9051 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|
   |
|
Theorem | 3halfnz 9052 |
Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|
   |
|
Theorem | suprzclex 9053* |
The supremum of a set of integers is an element of the set.
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
   
 
         
   |
|
Theorem | prime 9054* |
Two ways to express " is a prime number (or 1)." (Contributed by
NM, 4-May-2005.)
|
           
 
     |
|
Theorem | msqznn 9055 |
The square of a nonzero integer is a positive integer. (Contributed by
NM, 2-Aug-2004.)
|
  
    |
|
Theorem | zneo 9056 |
No even integer equals an odd integer (i.e. no integer can be both even
and odd). Exercise 10(a) of [Apostol] p.
28. (Contributed by NM,
31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
|
       
   |
|
Theorem | nneoor 9057 |
A positive integer is even or odd. (Contributed by Jim Kingdon,
15-Mar-2020.)
|
      
    |
|
Theorem | nneo 9058 |
A positive integer is even or odd but not both. (Contributed by NM,
1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
|
      
    |
|
Theorem | nneoi 9059 |
A positive integer is even or odd but not both. (Contributed by NM,
20-Aug-2001.)
|
         |
|
Theorem | zeo 9060 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|
      
    |
|
Theorem | zeo2 9061 |
An integer is even or odd but not both. (Contributed by Mario Carneiro,
12-Sep-2015.)
|
      
    |
|
Theorem | peano2uz2 9062* |
Second Peano postulate for upper integers. (Contributed by NM,
3-Oct-2004.)
|
  
  
     |
|
Theorem | peano5uzti 9063* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
|
     
  
   |
|
Theorem | peano5uzi 9064* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
|
  

   
  |
|
Theorem | dfuzi 9065* |
An expression for the upper integers that start at that is
analogous to dfnn2 8632 for positive integers. (Contributed by NM,
6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
|
      
    |
|
Theorem | uzind 9066* |
Induction on the upper integers that start at . The first four
hypotheses give us the substitution instances we need; the last two are
the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
|
                  
   
     
  |
|
Theorem | uzind2 9067* |
Induction on the upper integers that start after an integer .
The first four hypotheses give us the substitution instances we need;
the last two are the basis and the induction step. (Contributed by NM,
25-Jul-2005.)
|
                    
   
     
  |
|
Theorem | uzind3 9068* |
Induction on the upper integers that start at an integer . The
first four hypotheses give us the substitution instances we need, and
the last two are the basis and the induction step. (Contributed by NM,
26-Jul-2005.)
|
                  
   
 
     
 
  |
|
Theorem | nn0ind 9069* |
Principle of Mathematical Induction (inference schema) on nonnegative
integers. The first four hypotheses give us the substitution instances
we need; the last two are the basis and the induction step.
(Contributed by NM, 13-May-2004.)
|
    
   
     
    
     |
|
Theorem | fzind 9070* |
Induction on the integers from to
inclusive . The first
four hypotheses give us the substitution instances we need; the last two
are the basis and the induction step. (Contributed by Paul Chapman,
31-Mar-2011.)
|
    
   
     
         
 
         
    |
|
Theorem | fnn0ind 9071* |
Induction on the integers from to
inclusive . The first
four hypotheses give us the substitution instances we need; the last two
are the basis and the induction step. (Contributed by Paul Chapman,
31-Mar-2011.)
|
    
   
     
   
        
   |
|
Theorem | nn0ind-raph 9072* |
Principle of Mathematical Induction (inference schema) on nonnegative
integers. The first four hypotheses give us the substitution instances
we need; the last two are the basis and the induction step. Raph Levien
remarks: "This seems a bit painful. I wonder if an explicit
substitution version would be easier." (Contributed by Raph
Levien,
10-Apr-2004.)
|
    
   
     
    
     |
|
Theorem | zindd 9073* |
Principle of Mathematical Induction on all integers, deduction version.
The first five hypotheses give the substitutions; the last three are the
basis, the induction, and the extension to negative numbers.
(Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario
Carneiro, 4-Jan-2017.)
|
    
   
     
    
       
     
    
   |
|
Theorem | btwnz 9074* |
Any real number can be sandwiched between two integers. Exercise 2 of
[Apostol] p. 28. (Contributed by NM,
10-Nov-2004.)
|
       |
|
Theorem | nn0zd 9075 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | nnzd 9076 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | zred 9077 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | zcnd 9078 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | znegcld 9079 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
      |
|
Theorem | peano2zd 9080 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
|
   
   |
|
Theorem | zaddcld 9081 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zsubcld 9082 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zmulcld 9083 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zadd2cl 9084 |
Increasing an integer by 2 results in an integer. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
|
     |
|
Theorem | btwnapz 9085 |
A number between an integer and its successor is apart from any integer.
(Contributed by Jim Kingdon, 6-Jan-2023.)
|
             #   |
|
4.4.10 Decimal arithmetic
|
|
Syntax | cdc 9086 |
Constant used for decimal constructor.
|
;  |
|
Definition | df-dec 9087 |
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base  . For example,
;;;   ;;;    ;;;   1kp2ke3k 12629.
(Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV,
1-Aug-2021.)
|
;        |
|
Theorem | 9p1e10 9088 |
9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by
Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
|
  ;  |
|
Theorem | dfdec10 9089 |
Version of the definition of the "decimal constructor" using ;
instead of the symbol 10. Of course, this statement cannot be used as
definition, because it uses the "decimal constructor".
(Contributed by
AV, 1-Aug-2021.)
|
;  ; 
  |
|
Theorem | deceq1 9090 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
 ;
;   |
|
Theorem | deceq2 9091 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
 ;
;   |
|
Theorem | deceq1i 9092 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | deceq2i 9093 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | deceq12i 9094 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | numnncl 9095 |
Closure for a numeral (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
   
 |
|
Theorem | num0u 9096 |
Add a zero in the units place. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
 
     |
|
Theorem | num0h 9097 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
     |
|
Theorem | numcl 9098 |
Closure for a decimal integer (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
  

 |
|
Theorem | numsuc 9099 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
    
        |
|
Theorem | deccl 9100 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
|
;  |