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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | max0addsup 11101 | 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 11102* | 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 11103* | 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 11104 |
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 9194 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 11105 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
 | ||
| Theorem | zmaxcl 11106 | The maximum of two integers is an integer. (Contributed by Jim Kingdon, 27-Sep-2022.) |
 | ||
| Theorem | 2zsupmax 11107 | 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 11108* | 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 11109* | The negation of a finite set of real numbers is finite. (Contributed by AV, 9-Aug-2020.) |
  | ||
| Theorem | mincom 11110 | The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf inf
| ||
| Theorem | minmax 11111 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | mincl 11112 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | min1inf 11113 | The minimum of two numbers is less than or equal to the first. (Contributed by Jim Kingdon, 8-Feb-2021.) |
| inf | ||
| Theorem | min2inf 11114 | The minimum of two numbers is less than or equal to the second. (Contributed by Jim Kingdon, 9-Feb-2021.) |
| inf | ||
| Theorem | lemininf 11115 | 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 11116 | Two ways of saying a number is less than the minimum of two others. (Contributed by Jim Kingdon, 10-Feb-2022.) |
| inf | ||
| Theorem | minabs 11117 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
inf
   | ||
| Theorem | minclpr 11118 |
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 9194 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 11119 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
| inf | ||
| Theorem | bdtrilem 11120 | Lemma for bdtri 11121. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
|
| ||
| Theorem | bdtri 11121 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf | ||
| Theorem | mul0inf 11122 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 10944 and mulap0bd 8514 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
  inf | ||
| Theorem | xrmaxleim 11123 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
 | ||
| Theorem | xrmaxiflemcl 11124 | Lemma for xrmaxif 11130. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| ||
| Theorem | xrmaxifle 11125 |
An upper bound for in the extended reals. (Contributed by
Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemab 11126 | Lemma for xrmaxif 11130. A variation of xrmaxleim 11123- 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 11127 |
Lemma for xrmaxif 11130. A least upper bound for .
(Contributed by Jim Kingdon, 28-Apr-2023.)
|
|
| ||
| Theorem | xrmaxiflemcom 11128 | Lemma for xrmaxif 11130. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
|
| ||
| Theorem | xrmaxiflemval 11129* | Lemma for xrmaxif 11130. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
 
| ||
| Theorem | xrmaxif 11130 |
Maximum of two extended reals in terms of expressions.
(Contributed by Jim Kingdon, 26-Apr-2023.)
|
|
| ||
| Theorem | xrmaxcl 11131 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
| | ||
| Theorem | xrmax1sup 11132 | 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 11133 | 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 11134 | 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 11135 | The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.) |
|
| ||
| Theorem | xrltmaxsup 11136 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
| | ||
| Theorem | xrmaxltsup 11137 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
| | ||
| Theorem | xrmaxlesup 11138 | 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 11139 |
Lemma for xrmaxadd 11140. The case where is real. (Contributed by
Jim Kingdon, 11-May-2023.)
|
| ||
| Theorem | xrmaxadd 11140 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
| | ||
| Theorem | xrnegiso 11141 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
   | ||
| Theorem | infxrnegsupex 11142* |
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 11143 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
|
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| Theorem | xrminmax 11144 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
| inf
| ||
| Theorem | xrmincl 11145 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin1inf 11146 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmin2inf 11147 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
| inf | ||
| Theorem | xrmineqinf 11148 | 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 11149 | 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 11150 | 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 11151 | 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 11152 | 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 11153 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
| inf | ||
| Theorem | xrminadd 11154 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
| inf inf | ||
| Theorem | xrbdtri 11155 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
|
inf
inf inf
| ||
| Theorem | iooinsup 11156 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
   inf | ||
| Syntax | cli 11157 | Extend class notation with convergence relation for limits. |
  | ||
| Definition | df-clim 11158* | Define the limit relation for complex number sequences. See clim 11160 for its relational expression. (Contributed by NM, 28-Aug-2005.) |
  
 | ||
| Theorem | climrel 11159 | The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 | ||
| Theorem | clim 11160* |
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 11161 | 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 11162* |
Express the predicate: The limit of complex number sequence is
, or converges to , with more general
quantifier
restrictions than clim 11160. (Contributed by NM, 6-Jan-2007.) (Revised
by Mario Carneiro, 31-Jan-2014.)
|
 | ||
| Theorem | clim2c 11163* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0 11164* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | clim0c 11165* |
Express the predicate
converges to .
(Contributed by NM,
24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
|
|
| ||
| Theorem | climi 11166* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi2 11167* | Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climi0 11168* | Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.) |
| | ||
| Theorem | climconst 11169* | An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climconst2 11170 | A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
 
| ||
| Theorem | climz 11171 | The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.) |
|
| ||
| Theorem | climuni 11172 | 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 11173 | The limit relation is function-like, and with range the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.) |
   | ||
| Theorem | climdm 11174 |
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 11175* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
| ||
| Theorem | climreu 11176* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.) |
|
| ||
| Theorem | climmo 11177* | An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.) |
| ||
| Theorem | climeq 11178* | 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 11179* |
Exhibit a function
with the same convergence properties as the
not-quite-function . (Contributed by Mario Carneiro,
31-Jan-2014.)
|
|
| ||
| Theorem | 2clim 11180* | 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 11181 | A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.) |
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| Theorem | climres 11182 | 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 11183 | 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 11184 | 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 11185* | 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 11186* | 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 11187* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
| ||
| Theorem | climcn2 11188* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
 
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| Theorem | addcn2 11189* | 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 12912 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.) |
|
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| Theorem | subcn2 11190* | 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 11191* | 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 11192* |
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 2157. (Contributed by Mario Carneiro,
9-Feb-2014.)
Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|
inf #
 #
| ||
| Theorem | cn1lem 11193* | A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
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| Theorem | abscn2 11194* | The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
| ||
| Theorem | cjcn2 11195* | The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | recn2 11196* | The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | imcn2 11197* | The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
| ||
| Theorem | climcn1lem 11198* | The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.) |
|
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| Theorem | climabs 11199* | 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 11200* | 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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