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Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rexanre 12001* | Combine two different upper real properties into one. (Contributed by Mario Carneiro, 8-May-2016.) |
          
  ![/\](wedge.gif "/\"){kind=small}   
| ||
| Theorem | rexico 12002* | Restrict the base of an upper real quantifier to an upper real set. (Contributed by Mario Carneiro, 12-May-2016.) |
  
| ||
| Theorem | maxclpr 12003 |
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 9692 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 12004 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
 | ||
| Theorem | zmaxcl 12005 | The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.) |
 | ||
| Theorem | nn0maxcl 12006 | The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.) |
 | ||
| Theorem | 2zsupmax 12007 | 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 12008* | 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 12009* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
   | ||
| Theorem | mincom 12010 | The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf inf
| ||
| Theorem | minmax 12011 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | mincl 12012 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | min1inf 12013 | The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | min2inf 12014 | The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.) |
| inf | ||
| Theorem | lemininf 12015 | 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 12016 | Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.) |
| inf | ||
| Theorem | minabs 12017 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
inf 
     | ||
| Theorem | minclpr 12018 |
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 9692 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 12019 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | zmincl 12020 | The minumum of two integers is an integer. (Contributed by Jim Kingdon, 27-Aug-2026.) |
| inf | ||
| Theorem | bdtrilem 12021 | Lemma for bdtri 12022. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
| ||
| Theorem | bdtri 12022 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf | ||
| Theorem | mul0inf 12023 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11842 and mulap0bd 8987 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
  inf | ||
| Theorem | mingeb 12024 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
| inf
| ||
| Theorem | 2zinfmin 12025 | 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 12026 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
 | ||
| Theorem | xrmaxiflemcl 12027 | Lemma for xrmaxif 12033. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| ||
| Theorem | xrmaxifle 12028 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemab 12029 | Lemma for xrmaxif 12033. A variation of xrmaxleim 12026- 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 12030 |
Lemma for xrmaxif 12033. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemcom 12031 | Lemma for xrmaxif 12033. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
|
| ||
| Theorem | xrmaxiflemval 12032* | Lemma for xrmaxif 12033. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
 
| ||
| Theorem | xrmaxif 12033 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxcl 12034 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| | ||
| Theorem | xrmax1sup 12035 | 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 12036 | 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 12037 | 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 12038 | The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.) |
|
| ||
| Theorem | xrltmaxsup 12039 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
| | ||
| Theorem | xrmaxltsup 12040 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
| | ||
| Theorem | xrmaxlesup 12041 | 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 12042 |
Lemma for xrmaxadd 12043. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
| ||
| Theorem | xrmaxadd 12043 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
| | ||
| Theorem | xrnegiso 12044 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
   | ||
| Theorem | infxrnegsupex 12045* |
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 12046 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
|
| ||
| Theorem | xrminmax 12047 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
| inf
| ||
| Theorem | xrmincl 12048 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin1inf 12049 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin2inf 12050 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmineqinf 12051 | 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 12052 | 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 12053 | 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 12054 | 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 12055 | 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 12056 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
| inf | ||
| Theorem | xrminadd 12057 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
| inf inf | ||
| Theorem | xrbdtri 12058 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf
| ||
| Theorem | iooinsup 12059 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
   inf | ||
| Syntax | cli 12060 | Extend class notation with convergence relation for limits. |
  | ||
| Definition | df-clim 12061* | Define the limit relation for complex number sequences. See clim 12063 for its relational expression. (Contributed by NM, 28-Aug-2005.) |
  
 | ||
| Theorem | climrel 12062 | The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 | ||
| Theorem | clim 12063* |
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 12064 | 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 12065* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 12063. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
 | ||
| Theorem | clim2c 12066* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0 12067* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0c 12068* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | climi 12069* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi2 12070* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi0 12071* | Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climconst 12072* | An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climconst2 12073 | A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | climz 12074 | The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climuni 12075 | 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 12076 | The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
   | ||
| Theorem | climdm 12077 |
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 12078* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
| ||
| Theorem | climreu 12079* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
|
| ||
| Theorem | climmo 12080* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.) |
| ||
| Theorem | climeq 12081* | 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 12082* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
|
|
| ||
| Theorem | 2clim 12083* | 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 12084 | A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.) |
| ||
| Theorem | climres 12085 | 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 12086 | A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | serclim0 12087 | The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climshft2 12088* | 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 12089* | 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.) |
|
| ||
| Theorem | climcn1 12090* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climcn2 12091* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | addcn2 12092* | 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 15712 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | subcn2 12093* | Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | mulcn2 12094* | 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 12095* |
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 2238. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
inf #
 #
| ||
| Theorem | cn1lem 12096* | A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | abscn2 12097* | The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | cjcn2 12098* | The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | recn2 12099* | The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | imcn2 12100* | The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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