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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | maxle2 12001 | 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.) |
     ![/\](wedge.gif "/\"){kind=small}          | ||
| Theorem | maxleast 12002 | The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.) |
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| Theorem | maxleastb 12003 | Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Jim Kingdon, 31-Jan-2022.) |
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| Theorem | maxleastlt 12004 | The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.) |
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| Theorem | maxleb 12005 |
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 12006 | 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.) |
 
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| Theorem | maxltsup 12007 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.) |
|
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| Theorem | max0addsup 12008 | 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 12009* | Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.) |
    
 
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| Theorem | rexico 12010* | 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 12011 |
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 9699 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 12012 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
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| Theorem | zmaxcl 12013 | The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.) |
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| Theorem | nn0maxcl 12014 | The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.) |
 | ||
| Theorem | 2zsupmax 12015 | 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.) |
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| Theorem | fimaxre2 12016* | 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 12017* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
  | ||
| Theorem | fiidxsupcl 12018* | A set of integers indexed by a finite set has an upper bound. (Contributed by Jim Kingdon, 8-Sep-2026.) |
    
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| Theorem | mincom 12019 | The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf inf
| ||
| Theorem | minmax 12020 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | mincl 12021 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | min1inf 12022 | The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | min2inf 12023 | The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.) |
| inf | ||
| Theorem | lemininf 12024 | 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 12025 | Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.) |
| inf | ||
| Theorem | minabs 12026 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
inf
   | ||
| Theorem | minclpr 12027 |
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 9699 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 12028 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | zmincl 12029 | The minumum of two integers is an integer. (Contributed by Jim Kingdon, 27-Aug-2026.) |
| inf | ||
| Theorem | bdtrilem 12030 | Lemma for bdtri 12031. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
|
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| Theorem | bdtri 12031 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf | ||
| Theorem | mul0inf 12032 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11850 and mulap0bd 8993 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
  inf | ||
| Theorem | mingeb 12033 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
| inf
| ||
| Theorem | 2zinfmin 12034 | 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 12035 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
 | ||
| Theorem | xrmaxiflemcl 12036 | Lemma for xrmaxif 12042. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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| Theorem | xrmaxifle 12037 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemab 12038 | Lemma for xrmaxif 12042. A variation of xrmaxleim 12035- 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.) |
|
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| Theorem | xrmaxiflemlub 12039 |
Lemma for xrmaxif 12042. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
|
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| Theorem | xrmaxiflemcom 12040 | Lemma for xrmaxif 12042. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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| Theorem | xrmaxiflemval 12041* | Lemma for xrmaxif 12042. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
 
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| Theorem | xrmaxif 12042 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
|
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| Theorem | xrmaxcl 12043 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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| Theorem | xrmax1sup 12044 | 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.) |
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| Theorem | xrmax2sup 12045 | 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.) |
|
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| Theorem | xrmaxrecl 12046 | The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.) |
|
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| Theorem | xrmaxleastlt 12047 | The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.) |
|
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| Theorem | xrltmaxsup 12048 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
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| Theorem | xrmaxltsup 12049 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
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| Theorem | xrmaxlesup 12050 | 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 12051 |
Lemma for xrmaxadd 12052. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
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| Theorem | xrmaxadd 12052 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
| | ||
| Theorem | xrnegiso 12053 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
   | ||
| Theorem | infxrnegsupex 12054* |
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 12055 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
|
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| Theorem | xrminmax 12056 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
| inf
| ||
| Theorem | xrmincl 12057 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin1inf 12058 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin2inf 12059 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmineqinf 12060 | 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 12061 | 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 12062 | 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 12063 | 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 12064 | 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 12065 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
| inf | ||
| Theorem | xrminadd 12066 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
| inf inf | ||
| Theorem | xrbdtri 12067 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf
| ||
| Theorem | iooinsup 12068 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
   inf | ||
| Syntax | cli 12069 | Extend class notation with convergence relation for limits. |
  | ||
| Definition | df-clim 12070* | Define the limit relation for complex number sequences. See clim 12072 for its relational expression. (Contributed by NM, 28-Aug-2005.) |
  
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| Theorem | climrel 12071 | The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | clim 12072* |
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.)
|
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| Theorem | climcl 12073 | Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
|
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| Theorem | clim2 12074* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 12072. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
 | ||
| Theorem | clim2c 12075* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
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| Theorem | clim0 12076* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
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| Theorem | clim0c 12077* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
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| Theorem | climi 12078* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climi2 12079* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi0 12080* | Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climconst 12081* | An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | climconst2 12082 | A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
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| Theorem | climz 12083 | The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climuni 12084 | 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.) |
|
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| Theorem | fclim 12085 | The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climdm 12086 |
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 12087* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
| ||
| Theorem | climreu 12088* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
|
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| Theorem | climmo 12089* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.) |
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| Theorem | climeq 12090* | 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.) |
 
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| Theorem | climmpt 12091* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
|
|
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| Theorem | 2clim 12092* | 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.) |
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| Theorem | climshftlemg 12093 | A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.) |
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| Theorem | climres 12094 | 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.) |
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| Theorem | climshft 12095 | A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | serclim0 12096 | The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climshft2 12097* | A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.) |
 
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| Theorem | climabs0 12098* | 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 12099* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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| Theorem | climcn2 12100* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
 
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