Type | Label | Description |
Statement |
|
Theorem | absdifled 11201 |
The absolute value of a difference and 'less than or equal to' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
|
                
      |
|
Theorem | icodiamlt 11202 |
Two elements in a half-open interval have separation strictly less than
the difference between the endpoints. (Contributed by Stefan O'Rear,
12-Sep-2014.)
|
    
                    |
|
Theorem | abscld 11203 |
Real closure of absolute value. (Contributed by Mario Carneiro,
29-May-2016.)
|
         |
|
Theorem | absvalsqd 11204 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                   |
|
Theorem | absvalsq2d 11205 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                               |
|
Theorem | absge0d 11206 |
Absolute value is nonnegative. (Contributed by Mario Carneiro,
29-May-2016.)
|
         |
|
Theorem | absval2d 11207 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by Mario Carneiro, 29-May-2016.)
|
                               |
|
Theorem | abs00d 11208 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
           |
|
Theorem | absne0d 11209 |
The absolute value of a number is zero iff the number is zero.
Proposition 10-3.7(c) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
           |
|
Theorem | absrpclapd 11210 |
The absolute value of a complex number apart from zero is a positive
real. (Contributed by Jim Kingdon, 13-Aug-2021.)
|
   #         |
|
Theorem | absnegd 11211 |
Absolute value of negative. (Contributed by Mario Carneiro,
29-May-2016.)
|
              |
|
Theorem | abscjd 11212 |
The absolute value of a number and its conjugate are the same.
Proposition 10-3.7(b) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                 |
|
Theorem | releabsd 11213 |
The real part of a number is less than or equal to its absolute value.
Proposition 10-3.7(d) of [Gleason] p.
133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
             |
|
Theorem | absexpd 11214 |
Absolute value of positive integer exponentiation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                       |
|
Theorem | abssubd 11215 |
Swapping order of subtraction doesn't change the absolute value.
Example of [Apostol] p. 363.
(Contributed by Mario Carneiro,
29-May-2016.)
|
                   |
|
Theorem | absmuld 11216 |
Absolute value distributes over multiplication. Proposition 10-3.7(f)
of [Gleason] p. 133. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                       |
|
Theorem | absdivapd 11217 |
Absolute value distributes over division. (Contributed by Jim
Kingdon, 13-Aug-2021.)
|
     #
                    |
|
Theorem | abstrid 11218 |
Triangle inequality for absolute value. Proposition 10-3.7(h) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
               
       |
|
Theorem | abs2difd 11219 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
              
        |
|
Theorem | abs2dif2d 11220 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
               
       |
|
Theorem | abs2difabsd 11221 |
Absolute value of difference of absolute values. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                           |
|
Theorem | abs3difd 11222 |
Absolute value of differences around common element. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                             |
|
Theorem | abs3lemd 11223 |
Lemma involving absolute value of differences. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                                     |
|
Theorem | qdenre 11224* |
The rational numbers are dense in : any real number can be
approximated with arbitrary precision by a rational number. For order
theoretic density, see qbtwnre 10270. (Contributed by BJ, 15-Oct-2021.)
|
            |
|
4.7.5 The maximum of two real
numbers
|
|
Theorem | maxcom 11225 |
The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
            
  |
|
Theorem | maxabsle 11226 |
An upper bound for    . (Contributed by Jim Kingdon,
20-Dec-2021.)
|
      
          |
|
Theorem | maxleim 11227 |
Value of maximum when we know which number is larger. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
              |
|
Theorem | maxabslemab 11228 |
Lemma for maxabs 11231. A variation of maxleim 11227- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
          
          |
|
Theorem | maxabslemlub 11229 |
Lemma for maxabs 11231. A least upper bound for    .
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
                    
    |
|
Theorem | maxabslemval 11230* |
Lemma for maxabs 11231. Value of the supremum. (Contributed by
Jim
Kingdon, 22-Dec-2021.)
|
       
          
            
              
        |
|
Theorem | maxabs 11231 |
Maximum of two real numbers in terms of absolute value. (Contributed by
Jim Kingdon, 20-Dec-2021.)
|
             
          |
|
Theorem | maxcl 11232 |
The maximum of two real numbers is a real number. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
            |
|
Theorem | maxle1 11233 |
The maximum of two reals is no smaller than the first real. Lemma 3.10 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
            |
|
Theorem | maxle2 11234 |
The maximum of two reals is no smaller than the second real. Lemma 3.10
of [Geuvers], p. 10. (Contributed by Jim
Kingdon, 21-Dec-2021.)
|
            |
|
Theorem | maxleast 11235 |
The maximum of two reals is a least upper bound. Lemma 3.11 of
[Geuvers], p. 10. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
   
            |
|
Theorem | maxleastb 11236 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Jim Kingdon, 31-Jan-2022.)
|
       
        |
|
Theorem | maxleastlt 11237 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
    
             |
|
Theorem | maxleb 11238 |
Equivalence of
and being equal to the maximum of two reals. Lemma
3.12 of [Geuvers], p. 10. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
              |
|
Theorem | dfabsmax 11239 |
Absolute value of a real number in terms of maximum. Definition 3.13 of
[Geuvers], p. 11. (Contributed by BJ and
Jim Kingdon, 21-Dec-2021.)
|
    
   
      |
|
Theorem | maxltsup 11240 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
       
        |
|
Theorem | max0addsup 11241 |
The sum of the positive and negative part functions is the absolute value
function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.)
|
     
                  |
|
Theorem | rexanre 11242* |
Combine two different upper real properties into one. (Contributed by
Mario Carneiro, 8-May-2016.)
|
    
      
         |
|
Theorem | rexico 11243* |
Restrict the base of an upper real quantifier to an upper real set.
(Contributed by Mario Carneiro, 12-May-2016.)
|
          
   
    |
|
Theorem | maxclpr 11244 |
The maximum of two real numbers is one of those numbers if and only if
dichotomy (
) holds. For example, this
can be
combined with zletric 9310 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 1-Feb-2022.)
|
              
    |
|
Theorem | rpmaxcl 11245 |
The maximum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 10-Nov-2023.)
|
            |
|
Theorem | zmaxcl 11246 |
The maximum of two integers is an integer. (Contributed by Jim Kingdon,
27-Sep-2022.)
|
            |
|
Theorem | 2zsupmax 11247 |
Two ways to express the maximum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 22-Jan-2023.)
|
           
 
   |
|
Theorem | fimaxre2 11248* |
A nonempty finite set of real numbers has an upper bound. (Contributed
by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro,
13-Feb-2014.)
|
       |
|
Theorem | negfi 11249* |
The negation of a finite set of real numbers is finite. (Contributed by
AV, 9-Aug-2020.)
|
        |
|
4.7.6 The minimum of two real
numbers
|
|
Theorem | mincom 11250 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
inf      inf  
    |
|
Theorem | minmax 11251 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
   inf                  |
|
Theorem | mincl 11252 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
|
   inf        |
|
Theorem | min1inf 11253 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
|
   inf        |
|
Theorem | min2inf 11254 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
|
   inf        |
|
Theorem | lemininf 11255 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by NM, 3-Aug-2007.)
|
    inf  
   
    |
|
Theorem | ltmininf 11256 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
    inf           |
|
Theorem | minabs 11257 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
|
   inf         
          |
|
Theorem | minclpr 11258 |
The minimum of two real numbers is one of those numbers if and only if
dichotomy (
) holds. For example, this
can be
combined with zletric 9310 if one is dealing with integers, but real
number
dichotomy in general does not follow from our axioms. (Contributed by Jim
Kingdon, 23-May-2023.)
|
   inf  
      
    |
|
Theorem | rpmincl 11259 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
   inf        |
|
Theorem | bdtrilem 11260 |
Lemma for bdtri 11261. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
|
    
                            |
|
Theorem | bdtri 11261 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
    
  inf    
   inf      inf         |
|
Theorem | mul0inf 11262 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 11084 and mulap0bd 8627 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
      inf                 |
|
Theorem | mingeb 11263 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
    inf    
    |
|
Theorem | 2zinfmin 11264 |
Two ways to express the minimum of two integers. Because order of
integers is decidable, we have more flexibility than for real numbers.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
   inf       
 
   |
|
4.7.7 The maximum of two extended
reals
|
|
Theorem | xrmaxleim 11265 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
              |
|
Theorem | xrmaxiflemcl 11266 |
Lemma for xrmaxif 11272. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
        
   
           
       |
|
Theorem | xrmaxifle 11267 |
An upper bound for    in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
  
 
       
                   |
|
Theorem | xrmaxiflemab 11268 |
Lemma for xrmaxif 11272. A variation of xrmaxleim 11265- that is, if we know
which of two real numbers is larger, we know the maximum of the two.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
                    
               |
|
Theorem | xrmaxiflemlub 11269 |
Lemma for xrmaxif 11272. A least upper bound for    .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
                
                       |
|
Theorem | xrmaxiflemcom 11270 |
Lemma for xrmaxif 11272. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
        
   
           
              
                   |
|
Theorem | xrmaxiflemval 11271* |
Lemma for xrmaxif 11272. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
|
 
       
                       
       
    |
|
Theorem | xrmaxif 11272 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
           
           
               |
|
Theorem | xrmaxcl 11273 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
|
            |
|
Theorem | xrmax1sup 11274 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
  
   
     |
|
Theorem | xrmax2sup 11275 |
An extended real is less than or equal to the maximum of it and another.
(Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
30-Apr-2023.)
|
  
   
     |
|
Theorem | xrmaxrecl 11276 |
The maximum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
|
               
   |
|
Theorem | xrmaxleastlt 11277 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
  
 
             |
|
Theorem | xrltmaxsup 11278 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
|
                |
|
Theorem | xrmaxltsup 11279 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
|
                |
|
Theorem | xrmaxlesup 11280 |
Two ways of saying the maximum of two numbers is less than or equal to a
third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 10-May-2023.)
|
                |
|
Theorem | xrmaxaddlem 11281 |
Lemma for xrmaxadd 11282. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
                   
         
    |
|
Theorem | xrmaxadd 11282 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
|
                                  |
|
4.7.8 The minimum of two extended
reals
|
|
Theorem | xrnegiso 11283 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|

          |
|
Theorem | infxrnegsupex 11284* |
The infimum of a set of extended reals is the negative of the
supremum of the negatives of its elements. (Contributed by Jim Kingdon,
2-May-2023.)
|
   
         inf       
   
   |
|
Theorem | xrnegcon1d 11285 |
Contraposition law for extended real unary minus. (Contributed by Jim
Kingdon, 2-May-2023.)
|
        
   |
|
Theorem | xrminmax 11286 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
2-May-2023.)
|
   inf         
          |
|
Theorem | xrmincl 11287 |
The minumum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmin1inf 11288 |
The minimum of two extended reals is less than or equal to the first.
(Contributed by Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmin2inf 11289 |
The minimum of two extended reals is less than or equal to the second.
(Contributed by Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmineqinf 11290 |
The minimum of two extended reals is equal to the second if the first is
bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim
Kingdon, 3-May-2023.)
|
   inf  
     |
|
Theorem | xrltmininf 11291 |
Two ways of saying an extended real is less than the minimum of two
others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon,
3-May-2023.)
|
    inf           |
|
Theorem | xrlemininf 11292 |
Two ways of saying a number is less than or equal to the minimum of two
others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim
Kingdon, 4-May-2023.)
|
    inf           |
|
Theorem | xrminltinf 11293 |
Two ways of saying an extended real is greater than the minimum of two
others. (Contributed by Jim Kingdon, 19-May-2023.)
|
   inf    
      |
|
Theorem | xrminrecl 11294 |
The minimum of two real numbers is the same when taken as extended reals
or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|
   inf      inf        |
|
Theorem | xrminrpcl 11295 |
The minimum of two positive reals is a positive real. (Contributed by Jim
Kingdon, 4-May-2023.)
|
   inf        |
|
Theorem | xrminadd 11296 |
Distributing addition over minimum. (Contributed by Jim Kingdon,
10-May-2023.)
|
   inf                   inf         |
|
Theorem | xrbdtri 11297 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
  
 
 
  inf         
 inf        inf    
    |
|
Theorem | iooinsup 11298 |
Intersection of two open intervals of extended reals. (Contributed by
NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|
  
 
                     inf         |
|
4.8 Elementary limits and
convergence
|
|
4.8.1 Limits
|
|
Syntax | cli 11299 |
Extend class notation with convergence relation for limits.
|
 |
|
Definition | df-clim 11300* |
Define the limit relation for complex number sequences. See clim 11302
for
its relational expression. (Contributed by NM, 28-Aug-2005.)
|
    
                           |