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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | max0addsup 11401 | The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.) |
        
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| Theorem | rexanre 11402* | Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.) |
    
  ![/\](wedge.gif "/\"){kind=small}  
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| Theorem | rexico 11403* | Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.) |
  
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| Theorem | maxclpr 11404 |
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 9387 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 11405 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
 | ||
| Theorem | zmaxcl 11406 | The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.) |
 | ||
| Theorem | nn0maxcl 11407 | The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.) |
 | ||
| Theorem | 2zsupmax 11408 | 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 11409* | 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 11410* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
  | ||
| Theorem | mincom 11411 | The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf inf
| ||
| Theorem | minmax 11412 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | mincl 11413 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | min1inf 11414 | The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | min2inf 11415 | The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.) |
| inf | ||
| Theorem | lemininf 11416 | 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 11417 | Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.) |
| inf | ||
| Theorem | minabs 11418 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
inf
   | ||
| Theorem | minclpr 11419 |
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 9387 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 11420 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | bdtrilem 11421 | Lemma for bdtri 11422. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
|
| ||
| Theorem | bdtri 11422 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf | ||
| Theorem | mul0inf 11423 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11244 and mulap0bd 8701 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
  inf | ||
| Theorem | mingeb 11424 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
| inf
| ||
| Theorem | 2zinfmin 11425 | 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
| ||
| Theorem | xrmaxleim 11426 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
 | ||
| Theorem | xrmaxiflemcl 11427 | Lemma for xrmaxif 11433. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| ||
| Theorem | xrmaxifle 11428 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemab 11429 | Lemma for xrmaxif 11433. A variation of xrmaxleim 11426- 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 11430 |
Lemma for xrmaxif 11433. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemcom 11431 | Lemma for xrmaxif 11433. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
|
| ||
| Theorem | xrmaxiflemval 11432* | Lemma for xrmaxif 11433. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
 
| ||
| Theorem | xrmaxif 11433 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxcl 11434 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| | ||
| Theorem | xrmax1sup 11435 | 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 11436 | 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 11437 | 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 11438 | The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.) |
|
| ||
| Theorem | xrltmaxsup 11439 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
| | ||
| Theorem | xrmaxltsup 11440 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
| | ||
| Theorem | xrmaxlesup 11441 | 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 11442 |
Lemma for xrmaxadd 11443. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
| ||
| Theorem | xrmaxadd 11443 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
| | ||
| Theorem | xrnegiso 11444 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
   | ||
| Theorem | infxrnegsupex 11445* |
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 11446 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
|
| ||
| Theorem | xrminmax 11447 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
| inf
| ||
| Theorem | xrmincl 11448 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin1inf 11449 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin2inf 11450 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmineqinf 11451 | 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 11452 | 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 11453 | 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 11454 | 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 11455 | 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 11456 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
| inf | ||
| Theorem | xrminadd 11457 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
| inf inf | ||
| Theorem | xrbdtri 11458 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf
| ||
| Theorem | iooinsup 11459 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
   inf | ||
| Syntax | cli 11460 | Extend class notation with convergence relation for limits. |
  | ||
| Definition | df-clim 11461* | Define the limit relation for complex number sequences. See clim 11463 for its relational expression. (Contributed by NM, 28-Aug-2005.) |
  
 | ||
| Theorem | climrel 11462 | The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 | ||
| Theorem | clim 11463* |
Express the predicate: The limit of complex number sequence is
, or converges to . This means that for any
real
, no matter how
small, there always exists an integer such
that the absolute difference of any later complex number in the sequence
and the limit is less than . (Contributed by NM, 28-Aug-2005.)
(Revised by Mario Carneiro, 28-Apr-2015.)
|
| ||
| Theorem | climcl 11464 | Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
|
| ||
| Theorem | clim2 11465* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 11463. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
 | ||
| Theorem | clim2c 11466* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0 11467* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0c 11468* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | climi 11469* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi2 11470* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi0 11471* | Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climconst 11472* | An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climconst2 11473 | A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | climz 11474 | The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climuni 11475 | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | fclim 11476 | The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
   | ||
| Theorem | climdm 11477 |
Two ways to express that a function has a limit. (The expression
is sometimes useful as a shorthand for "the unique limit
of the function "). (Contributed by Mario Carneiro,
18-Mar-2014.)
|
| | ||
| Theorem | climeu 11478* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
| ||
| Theorem | climreu 11479* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
|
| ||
| Theorem | climmo 11480* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.) |
| ||
| Theorem | climeq 11481* | Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | climmpt 11482* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
|
|
| ||
| Theorem | 2clim 11483* | If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climshftlemg 11484 | A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.) |
| ||
| Theorem | climres 11485 | A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climshft 11486 | A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | serclim0 11487 | The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climshft2 11488* | A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.) |
 
| ||
| Theorem | climabs0 11489* | Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | climcn1 11490* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climcn2 11491* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
 
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| Theorem | addcn2 11492* | Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn and df-cncf are not yet available to us. See addcncntop 14882 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | subcn2 11493* | Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | mulcn2 11494* | Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | reccn2ap 11495* |
The reciprocal function is continuous. The class  is just for
convenience in writing the proof and typically would be passed in as an
instance of eqid 2196. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
inf #
 #
| ||
| Theorem | cn1lem 11496* | A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
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| Theorem | abscn2 11497* | The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | cjcn2 11498* | The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | recn2 11499* | The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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| Theorem | imcn2 11500* | The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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