Type | Label | Description |
Statement |
|
Theorem | zletr 9301 |
Transitive law of ordering for integers. (Contributed by Alexander van
der Vekens, 3-Apr-2018.)
|
         |
|
Theorem | zrevaddcl 9302 |
Reverse closure law for addition of integers. (Contributed by NM,
11-May-2004.)
|
    
    |
|
Theorem | znnsub 9303 |
The positive difference of unequal integers is a positive integer.
(Generalization of nnsub 8957.) (Contributed by NM, 11-May-2004.)
|
     
   |
|
Theorem | nzadd 9304 |
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 9305 |
Closure of multiplication of integers. (Contributed by NM,
30-Jul-2004.)
|
    
  |
|
Theorem | zltp1le 9306 |
Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof
shortened by Mario Carneiro, 16-May-2014.)
|
     
   |
|
Theorem | zleltp1 9307 |
Integer ordering relation. (Contributed by NM, 10-May-2004.)
|
         |
|
Theorem | zlem1lt 9308 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
     
   |
|
Theorem | zltlem1 9309 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|
         |
|
Theorem | zgt0ge1 9310 |
An integer greater than
is greater than or equal to .
(Contributed by AV, 14-Oct-2018.)
|
     |
|
Theorem | nnleltp1 9311 |
Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | nnltp1le 9312 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|
     
   |
|
Theorem | nnaddm1cl 9313 |
Closure of addition of positive integers minus one. (Contributed by NM,
6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|
      
  |
|
Theorem | nn0ltp1le 9314 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | nn0leltp1 9315 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
|
         |
|
Theorem | nn0ltlem1 9316 |
Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
|
         |
|
Theorem | znn0sub 9317 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 9318.) (Contributed by NM, 14-Jul-2005.)
|
     
   |
|
Theorem | nn0sub 9318 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|
         |
|
Theorem | ltsubnn0 9319 |
Subtracting a nonnegative integer from a nonnegative integer which is
greater than the first one results in a nonnegative integer. (Contributed
by Alexander van der Vekens, 6-Apr-2018.)
|
     
   |
|
Theorem | nn0negleid 9320 |
A nonnegative integer is greater than or equal to its negative.
(Contributed by AV, 13-Aug-2021.)
|
 
  |
|
Theorem | difgtsumgt 9321 |
If the difference of a real number and a nonnegative integer is greater
than another real number, the sum of the real number and the nonnegative
integer is also greater than the other real number. (Contributed by AV,
13-Aug-2021.)
|
           |
|
Theorem | nn0n0n1ge2 9322 |
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 9323* |
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 9324 |
Alternate definition of the integers, based on elz2 9323.
(Contributed by
Mario Carneiro, 16-May-2014.)
|
     |
|
Theorem | nn0sub2 9325 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|
       |
|
Theorem | zapne 9326 |
Apartness is equivalent to not equal for integers. (Contributed by Jim
Kingdon, 14-Mar-2020.)
|
    #    |
|
Theorem | zdceq 9327 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
|
   DECID
  |
|
Theorem | zdcle 9328 |
Integer is
decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
|
   DECID   |
|
Theorem | zdclt 9329 |
Integer is
decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
|
   DECID   |
|
Theorem | zltlen 9330 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8588 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 14-Mar-2020.)
|
         |
|
Theorem | nn0n0n1ge2b 9331 |
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 9332 |
A nonnegative integer less than is .
(Contributed by Paul
Chapman, 22-Jun-2011.)
|
 
   |
|
Theorem | nn0lt2 9333 |
A nonnegative integer less than 2 must be 0 or 1. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
|
       |
|
Theorem | nn0le2is012 9334 |
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 9335 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
|
         |
|
Theorem | nnlem1lt 9336 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
     
   |
|
Theorem | nnltlem1 9337 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
|
         |
|
Theorem | nnm1ge0 9338 |
A positive integer decreased by 1 is greater than or equal to 0.
(Contributed by AV, 30-Oct-2018.)
|

    |
|
Theorem | nn0ge0div 9339 |
Division of a nonnegative integer by a positive number is not negative.
(Contributed by Alexander van der Vekens, 14-Apr-2018.)
|
  
    |
|
Theorem | zdiv 9340* |
Two ways to express " divides .
(Contributed by NM,
3-Oct-2008.)
|
      
     |
|
Theorem | zdivadd 9341 |
Property of divisibility: if divides
and then it divides
. (Contributed by NM, 3-Oct-2008.)
|
   
         
   |
|
Theorem | zdivmul 9342 |
Property of divisibility: if divides
then it divides
. (Contributed by NM, 3-Oct-2008.)
|
   
         |
|
Theorem | zextle 9343* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
   
 
  |
|
Theorem | zextlt 9344* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
|
   
 
  |
|
Theorem | recnz 9345 |
The reciprocal of a number greater than 1 is not an integer. (Contributed
by NM, 3-May-2005.)
|
  
    |
|
Theorem | btwnnz 9346 |
A number between an integer and its successor is not an integer.
(Contributed by NM, 3-May-2005.)
|
       |
|
Theorem | gtndiv 9347 |
A larger number does not divide a smaller positive integer. (Contributed
by NM, 3-May-2005.)
|
    
  |
|
Theorem | halfnz 9348 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
|
   |
|
Theorem | 3halfnz 9349 |
Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)
|
   |
|
Theorem | suprzclex 9350* |
The supremum of a set of integers is an element of the set.
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
   
 
         
   |
|
Theorem | prime 9351* |
Two ways to express " is a prime number (or 1)". (Contributed by
NM, 4-May-2005.)
|
           
 
     |
|
Theorem | msqznn 9352 |
The square of a nonzero integer is a positive integer. (Contributed by
NM, 2-Aug-2004.)
|
  
    |
|
Theorem | zneo 9353 |
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 9354 |
A positive integer is even or odd. (Contributed by Jim Kingdon,
15-Mar-2020.)
|
      
    |
|
Theorem | nneo 9355 |
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 9356 |
A positive integer is even or odd but not both. (Contributed by NM,
20-Aug-2001.)
|
         |
|
Theorem | zeo 9357 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
|
      
    |
|
Theorem | zeo2 9358 |
An integer is even or odd but not both. (Contributed by Mario Carneiro,
12-Sep-2015.)
|
      
    |
|
Theorem | peano2uz2 9359* |
Second Peano postulate for upper integers. (Contributed by NM,
3-Oct-2004.)
|
  
  
     |
|
Theorem | peano5uzti 9360* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
|
     
  
   |
|
Theorem | peano5uzi 9361* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
|
  

   
  |
|
Theorem | dfuzi 9362* |
An expression for the upper integers that start at that is
analogous to dfnn2 8920 for positive integers. (Contributed by NM,
6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
|
      
    |
|
Theorem | uzind 9363* |
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 9364* |
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 9365* |
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 9366* |
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 9367* |
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 9368* |
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 9369* |
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 9370* |
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 9371* |
Any real number can be sandwiched between two integers. Exercise 2 of
[Apostol] p. 28. (Contributed by NM,
10-Nov-2004.)
|
       |
|
Theorem | nn0zd 9372 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | nnzd 9373 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | zred 9374 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | zcnd 9375 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
|
     |
|
Theorem | znegcld 9376 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
      |
|
Theorem | peano2zd 9377 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
|
   
   |
|
Theorem | zaddcld 9378 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zsubcld 9379 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zmulcld 9380 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
|
         |
|
Theorem | zadd2cl 9381 |
Increasing an integer by 2 results in an integer. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
|
     |
|
Theorem | btwnapz 9382 |
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 9383 |
Constant used for decimal constructor.
|
;  |
|
Definition | df-dec 9384 |
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base  . For example,
;;;   ;;;    ;;;   1kp2ke3k 14446.
(Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV,
1-Aug-2021.)
|
;        |
|
Theorem | 9p1e10 9385 |
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 9386 |
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 9387 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
 ;
;   |
|
Theorem | deceq2 9388 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
 ;
;   |
|
Theorem | deceq1i 9389 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | deceq2i 9390 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | deceq12i 9391 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
|
; ;  |
|
Theorem | numnncl 9392 |
Closure for a numeral (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
   
 |
|
Theorem | num0u 9393 |
Add a zero in the units place. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
 
     |
|
Theorem | num0h 9394 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
|
     |
|
Theorem | numcl 9395 |
Closure for a decimal integer (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
  

 |
|
Theorem | numsuc 9396 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
|
    
        |
|
Theorem | deccl 9397 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
|
;  |
|
Theorem | 10nn 9398 |
10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by
AV, 6-Sep-2021.)
|
;  |
|
Theorem | 10pos 9399 |
The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by
AV, 8-Sep-2021.)
|
;  |
|
Theorem | 10nn0 9400 |
10 is a nonnegative integer. (Contributed by Mario Carneiro,
19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
|
;  |