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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rpmaxcl 11801 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
     ![/\](wedge.gif "/\"){kind=small}          | ||
| Theorem | zmaxcl 11802 | The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.) |
 | ||
| Theorem | nn0maxcl 11803 | The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.) |
 | ||
| Theorem | 2zsupmax 11804 | 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 11805* | 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 11806* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
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| Theorem | mincom 11807 | The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf inf
| ||
| Theorem | minmax 11808 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | mincl 11809 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | min1inf 11810 | The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | min2inf 11811 | The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.) |
| inf | ||
| Theorem | lemininf 11812 | 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 11813 | Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.) |
| inf | ||
| Theorem | minabs 11814 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
inf 
     | ||
| Theorem | minclpr 11815 |
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 9523 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 11816 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | bdtrilem 11817 | Lemma for bdtri 11818. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
| ||
| Theorem | bdtri 11818 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf | ||
| Theorem | mul0inf 11819 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11640 and mulap0bd 8837 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
  inf | ||
| Theorem | mingeb 11820 |
Equivalence of
and being equal to the minimum of two reals.
(Contributed by Jim Kingdon, 14-Oct-2024.)
|
| inf
| ||
| Theorem | 2zinfmin 11821 | 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 11822 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
 | ||
| Theorem | xrmaxiflemcl 11823 | Lemma for xrmaxif 11829. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| ||
| Theorem | xrmaxifle 11824 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemab 11825 | Lemma for xrmaxif 11829. A variation of xrmaxleim 11822- 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 11826 |
Lemma for xrmaxif 11829. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemcom 11827 | Lemma for xrmaxif 11829. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
|
| ||
| Theorem | xrmaxiflemval 11828* | Lemma for xrmaxif 11829. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
 
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| Theorem | xrmaxif 11829 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxcl 11830 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| | ||
| Theorem | xrmax1sup 11831 | 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 11832 | 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 11833 | 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 11834 | 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 11835 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
| | ||
| Theorem | xrmaxltsup 11836 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
| | ||
| Theorem | xrmaxlesup 11837 | 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 11838 |
Lemma for xrmaxadd 11839. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
| ||
| Theorem | xrmaxadd 11839 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
| | ||
| Theorem | xrnegiso 11840 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
   | ||
| Theorem | infxrnegsupex 11841* |
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 11842 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
|
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| Theorem | xrminmax 11843 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
| inf
| ||
| Theorem | xrmincl 11844 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin1inf 11845 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin2inf 11846 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmineqinf 11847 | 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 11848 | 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 11849 | 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 11850 | 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 11851 | 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 11852 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
| inf | ||
| Theorem | xrminadd 11853 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
| inf inf | ||
| Theorem | xrbdtri 11854 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf
| ||
| Theorem | iooinsup 11855 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
   inf | ||
| Syntax | cli 11856 | Extend class notation with convergence relation for limits. |
  | ||
| Definition | df-clim 11857* | Define the limit relation for complex number sequences. See clim 11859 for its relational expression. (Contributed by NM, 28-Aug-2005.) |
  
   | ||
| Theorem | climrel 11858 | The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 | ||
| Theorem | clim 11859* |
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 11860 | 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 11861* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 11859. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
 | ||
| Theorem | clim2c 11862* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0 11863* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0c 11864* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | climi 11865* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi2 11866* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi0 11867* | Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climconst 11868* | An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climconst2 11869 | A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | climz 11870 | 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 11871 | 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 11872 | The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
   | ||
| Theorem | climdm 11873 |
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 11874* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
| ||
| Theorem | climreu 11875* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
|
| ||
| Theorem | climmo 11876* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.) |
| ||
| Theorem | climeq 11877* | 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 11878* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
|
|
| ||
| Theorem | 2clim 11879* | 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 11880 | A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.) |
| ||
| Theorem | climres 11881 | 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 11882 | 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 11883 | 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 11884* | 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 11885* | 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 11886* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climcn2 11887* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
 
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| Theorem | addcn2 11888* | 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 15305 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | subcn2 11889* | 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 11890* | Complex number multiplication 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 | reccn2ap 11891* |
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 2231. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
inf #
 #
| ||
| Theorem | cn1lem 11892* | A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | abscn2 11893* | The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | cjcn2 11894* | The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | recn2 11895* | The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | imcn2 11896* | The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | climcn1lem 11897* | The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
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| Theorem | climabs 11898* | Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
| | ||
| Theorem | climcj 11899* | Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
|
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| Theorem | climre 11900* | Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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