Type | Label | Description |
Statement |
|
Theorem | sqrtltd 11201 |
Square root is strictly monotonic. (Contributed by Mario Carneiro,
29-May-2016.)
|
                     |
|
Theorem | sqr11d 11202 |
The square root function is one-to-one. (Contributed by Mario Carneiro,
29-May-2016.)
|
                     |
|
Theorem | absltd 11203 |
Absolute value and 'less than' relation. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                |
|
Theorem | absled 11204 |
Absolute value and 'less than or equal to' relation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                |
|
Theorem | abssubge0d 11205 |
Absolute value of a nonnegative difference. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                 |
|
Theorem | abssuble0d 11206 |
Absolute value of a nonpositive difference. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                 |
|
Theorem | absdifltd 11207 |
The absolute value of a difference and 'less than' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
|
                
      |
|
Theorem | absdifled 11208 |
The absolute value of a difference and 'less than or equal to' relation.
(Contributed by Mario Carneiro, 29-May-2016.)
|
                
      |
|
Theorem | icodiamlt 11209 |
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 11210 |
Real closure of absolute value. (Contributed by Mario Carneiro,
29-May-2016.)
|
         |
|
Theorem | absvalsqd 11211 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                   |
|
Theorem | absvalsq2d 11212 |
Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                               |
|
Theorem | absge0d 11213 |
Absolute value is nonnegative. (Contributed by Mario Carneiro,
29-May-2016.)
|
         |
|
Theorem | absval2d 11214 |
Value of absolute value function. Definition 10.36 of [Gleason] p. 133.
(Contributed by Mario Carneiro, 29-May-2016.)
|
                               |
|
Theorem | abs00d 11215 |
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 11216 |
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 11217 |
The absolute value of a complex number apart from zero is a positive
real. (Contributed by Jim Kingdon, 13-Aug-2021.)
|
   #         |
|
Theorem | absnegd 11218 |
Absolute value of negative. (Contributed by Mario Carneiro,
29-May-2016.)
|
              |
|
Theorem | abscjd 11219 |
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 11220 |
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 11221 |
Absolute value of positive integer exponentiation. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                       |
|
Theorem | abssubd 11222 |
Swapping order of subtraction doesn't change the absolute value.
Example of [Apostol] p. 363.
(Contributed by Mario Carneiro,
29-May-2016.)
|
                   |
|
Theorem | absmuld 11223 |
Absolute value distributes over multiplication. Proposition 10-3.7(f)
of [Gleason] p. 133. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                       |
|
Theorem | absdivapd 11224 |
Absolute value distributes over division. (Contributed by Jim
Kingdon, 13-Aug-2021.)
|
     #
                    |
|
Theorem | abstrid 11225 |
Triangle inequality for absolute value. Proposition 10-3.7(h) of
[Gleason] p. 133. (Contributed by Mario
Carneiro, 29-May-2016.)
|
               
       |
|
Theorem | abs2difd 11226 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
              
        |
|
Theorem | abs2dif2d 11227 |
Difference of absolute values. (Contributed by Mario Carneiro,
29-May-2016.)
|
               
       |
|
Theorem | abs2difabsd 11228 |
Absolute value of difference of absolute values. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                           |
|
Theorem | abs3difd 11229 |
Absolute value of differences around common element. (Contributed by
Mario Carneiro, 29-May-2016.)
|
                             |
|
Theorem | abs3lemd 11230 |
Lemma involving absolute value of differences. (Contributed by Mario
Carneiro, 29-May-2016.)
|
                                     |
|
Theorem | qdenre 11231* |
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 10277. (Contributed by BJ, 15-Oct-2021.)
|
            |
|
4.7.5 The maximum of two real
numbers
|
|
Theorem | maxcom 11232 |
The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10.
(Contributed by Jim Kingdon, 21-Dec-2021.)
|
            
  |
|
Theorem | maxabsle 11233 |
An upper bound for    . (Contributed by Jim Kingdon,
20-Dec-2021.)
|
      
          |
|
Theorem | maxleim 11234 |
Value of maximum when we know which number is larger. (Contributed by
Jim Kingdon, 21-Dec-2021.)
|
              |
|
Theorem | maxabslemab 11235 |
Lemma for maxabs 11238. A variation of maxleim 11234- 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 11236 |
Lemma for maxabs 11238. A least upper bound for    .
(Contributed by Jim Kingdon, 20-Dec-2021.)
|
                    
    |
|
Theorem | maxabslemval 11237* |
Lemma for maxabs 11238. Value of the supremum. (Contributed by
Jim
Kingdon, 22-Dec-2021.)
|
       
          
            
              
        |
|
Theorem | maxabs 11238 |
Maximum of two real numbers in terms of absolute value. (Contributed by
Jim Kingdon, 20-Dec-2021.)
|
             
          |
|
Theorem | maxcl 11239 |
The maximum of two real numbers is a real number. (Contributed by Jim
Kingdon, 22-Dec-2021.)
|
            |
|
Theorem | maxle1 11240 |
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 11241 |
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 11242 |
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 11243 |
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 11244 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
    
             |
|
Theorem | maxleb 11245 |
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 11246 |
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 11247 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
       
        |
|
Theorem | max0addsup 11248 |
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 11249* |
Combine two different upper real properties into one. (Contributed by
Mario Carneiro, 8-May-2016.)
|
    
      
         |
|
Theorem | rexico 11250* |
Restrict the base of an upper real quantifier to an upper real set.
(Contributed by Mario Carneiro, 12-May-2016.)
|
          
   
    |
|
Theorem | maxclpr 11251 |
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 9317 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 11252 |
The maximum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 10-Nov-2023.)
|
            |
|
Theorem | zmaxcl 11253 |
The maximum of two integers is an integer. (Contributed by Jim Kingdon,
27-Sep-2022.)
|
            |
|
Theorem | 2zsupmax 11254 |
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 11255* |
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 11256* |
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 11257 |
The minimum of two reals is commutative. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
inf      inf  
    |
|
Theorem | minmax 11258 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
8-Feb-2021.)
|
   inf                  |
|
Theorem | mincl 11259 |
The minumum of two real numbers is a real number. (Contributed by Jim
Kingdon, 25-Apr-2023.)
|
   inf        |
|
Theorem | min1inf 11260 |
The minimum of two numbers is less than or equal to the first.
(Contributed by Jim Kingdon, 8-Feb-2021.)
|
   inf        |
|
Theorem | min2inf 11261 |
The minimum of two numbers is less than or equal to the second.
(Contributed by Jim Kingdon, 9-Feb-2021.)
|
   inf        |
|
Theorem | lemininf 11262 |
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 11263 |
Two ways of saying a number is less than the minimum of two others.
(Contributed by Jim Kingdon, 10-Feb-2022.)
|
    inf           |
|
Theorem | minabs 11264 |
The minimum of two real numbers in terms of absolute value. (Contributed
by Jim Kingdon, 15-May-2023.)
|
   inf         
          |
|
Theorem | minclpr 11265 |
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 9317 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 11266 |
The minumum of two positive real numbers is a positive real number.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
   inf        |
|
Theorem | bdtrilem 11267 |
Lemma for bdtri 11268. (Contributed by Steven Nguyen and Jim
Kingdon,
17-May-2023.)
|
    
                            |
|
Theorem | bdtri 11268 |
Triangle inequality for bounded values. (Contributed by Jim Kingdon,
15-May-2023.)
|
    
  inf    
   inf      inf         |
|
Theorem | mul0inf 11269 |
Equality of a product with zero. A bit of a curiosity, in the sense that
theorems like abs00ap 11091 and mulap0bd 8634 may better express the ideas behind
it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|
      inf                 |
|
Theorem | mingeb 11270 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
    inf    
    |
|
Theorem | 2zinfmin 11271 |
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 11272 |
Value of maximum when we know which extended real is larger.
(Contributed by Jim Kingdon, 25-Apr-2023.)
|
              |
|
Theorem | xrmaxiflemcl 11273 |
Lemma for xrmaxif 11279. Closure. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
        
   
           
       |
|
Theorem | xrmaxifle 11274 |
An upper bound for    in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
  
 
       
                   |
|
Theorem | xrmaxiflemab 11275 |
Lemma for xrmaxif 11279. A variation of xrmaxleim 11272- 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 11276 |
Lemma for xrmaxif 11279. A least upper bound for    .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
                
                       |
|
Theorem | xrmaxiflemcom 11277 |
Lemma for xrmaxif 11279. Commutativity of an expression which we
will
later show to be the supremum. (Contributed by Jim Kingdon,
29-Apr-2023.)
|
        
   
           
              
                   |
|
Theorem | xrmaxiflemval 11278* |
Lemma for xrmaxif 11279. Value of the supremum. (Contributed by
Jim
Kingdon, 29-Apr-2023.)
|
 
       
                       
       
    |
|
Theorem | xrmaxif 11279 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
           
           
               |
|
Theorem | xrmaxcl 11280 |
The maximum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 29-Apr-2023.)
|
            |
|
Theorem | xrmax1sup 11281 |
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 11282 |
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 11283 |
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 11284 |
The maximum as a least upper bound, in terms of less than. (Contributed
by Jim Kingdon, 9-Feb-2022.)
|
  
 
             |
|
Theorem | xrltmaxsup 11285 |
The maximum as a least upper bound. (Contributed by Jim Kingdon,
10-May-2023.)
|
                |
|
Theorem | xrmaxltsup 11286 |
Two ways of saying the maximum of two numbers is less than a third.
(Contributed by Jim Kingdon, 30-Apr-2023.)
|
                |
|
Theorem | xrmaxlesup 11287 |
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 11288 |
Lemma for xrmaxadd 11289. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
                   
         
    |
|
Theorem | xrmaxadd 11289 |
Distributing addition over maximum. (Contributed by Jim Kingdon,
11-May-2023.)
|
                                  |
|
4.7.8 The minimum of two extended
reals
|
|
Theorem | xrnegiso 11290 |
Negation is an order anti-isomorphism of the extended reals, which is
its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|

          |
|
Theorem | infxrnegsupex 11291* |
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 11292 |
Contraposition law for extended real unary minus. (Contributed by Jim
Kingdon, 2-May-2023.)
|
        
   |
|
Theorem | xrminmax 11293 |
Minimum expressed in terms of maximum. (Contributed by Jim Kingdon,
2-May-2023.)
|
   inf         
          |
|
Theorem | xrmincl 11294 |
The minumum of two extended reals is an extended real. (Contributed by
Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmin1inf 11295 |
The minimum of two extended reals is less than or equal to the first.
(Contributed by Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmin2inf 11296 |
The minimum of two extended reals is less than or equal to the second.
(Contributed by Jim Kingdon, 3-May-2023.)
|
   inf        |
|
Theorem | xrmineqinf 11297 |
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 11298 |
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 11299 |
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 11300 |
Two ways of saying an extended real is greater than the minimum of two
others. (Contributed by Jim Kingdon, 19-May-2023.)
|
   inf    
      |