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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremreperflem 21301* A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)
fld                     t Perf

Theoremreperf 21302 The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
fld       t Perf

Theoremcnperf 21303 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
fld       Perf

Theoremiccntr 21304 The interior of a closed interval in the standard topology on is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremicccmplem1 21305* Lemma for icccmp 21308. (Contributed by Mario Carneiro, 18-Jun-2014.)
t

Theoremicccmplem2 21306* Lemma for icccmp 21308. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremicccmplem3 21307* Lemma for icccmp 21308. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremicccmp 21308 A closed interval in is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremreconnlem1 21309 Lemma for reconn 21311. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
t

Theoremreconnlem2 21310* Lemma for reconn 21311. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)

Theoremreconn 21311* A subset of the reals is connected iff it has the interval property. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
t

Theoremretopcon 21312 Corollary of reconn 21311. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)

Theoremiccconn 21313 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
t

Theoremopnreen 21314 Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremrectbntr0 21315 A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremxrge0gsumle 21316 A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
s                                    g g

Theoremxrge0tsms 21317* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Proof shortened by AV, 26-Jul-2019.)
s                      g        tsums

Theoremxrge0tsms2 21318 Any finite or infinite sum in the nonnegative extended reals is convergent. This is a rather unique property of the set ; a similar theorem is not true for or or . It is true for , however, or more generally any additive submonoid of with adjoined. (Contributed by Mario Carneiro, 13-Sep-2015.)
s        tsums

Theoremmetdcnlem 21319 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremxmetdcn2 21320 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 21321 we use the metric topology instead of the order topology on , which makes the theorem a bit stronger. Since is an isolated point in the metric topology, this is saying that for any points which are an infinite distance apart, there is a product neighborhood around such that for any near and near , i.e. the distance function is locally constant . (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremxmetdcn 21321 The metric function of an extended metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 4-Sep-2015.)
ordTop

Theoremmetdcn2 21322 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdcn 21323 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
fld

Theoremmsdcn 21324 The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)

Theoremcnmpt1ds 21325* Continuity of the metric function; analogue of cnmpt12f 20145 which cannot be used directly because is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2ds 21326* Continuity of the metric function; analogue of cnmpt22f 20154 which cannot be used directly because is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremnmcn 21327 The norm of a normed group is a continuous function. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremabscn 21328 The absolute value function on complex numbers is continuous. (Contributed by NM, 22-Aug-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.)
fld

Theoremmetdsval 21329* Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdsf 21330* The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdsge 21331* The distance from the point to the set is greater than iff the -ball around misses . (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetds0 21332* If a point is in a set, its distance to the set is zero. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdstri 21333* A generalization of the triangle inequality to the point-set distance function. Under the usual notation where the same symbol denotes the point-point and point-set distance functions, this theorem would be written . (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetdsle 21334* The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremmetdsre 21335* The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremmetdseq0 21336* The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremmetdscnlem 21337* Lemma for metdscn 21338. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetdscn 21338* The function which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdscn2 21339* The function which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complex numbers. (Contributed by Mario Carneiro, 5-Sep-2015.)
fld

Theoremmetnrmlem1a 21340* Lemma for metnrm 21344. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetnrmlem1 21341* Lemma for metnrm 21344. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetnrmlem2 21342* Lemma for metnrm 21344. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetnrmlem3 21343* Lemma for metnrm 21344. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetnrm 21344 A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetreg 21345 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremaddcnlem 21346* Lemma for addcn 21347, subcn 21348, and mulcn 21349. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
fld

Theoremaddcn 21347 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
fld

Theoremsubcn 21348 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
fld

Theoremmulcn 21349 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
fld

Theoremdivcn 21350 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
fld       t

Theoremcnfldtgp 21351 The complex numbers form a topological group under addition, with the standard topology induced by the absolute value metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld

Theoremfsumcn 21352* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for normally contains free variables and to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
fld       TopOn

Theoremfsum2cn 21353* Version of fsumcn 21352 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
fld       TopOn              TopOn

Theoremexpcn 21354* The power function on complex numbers, for fixed exponent , is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
fld

Theoremdivccn 21355* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.)
fld

Theoremsqcn 21356* The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
fld

12.4.11  Topological definitions using the reals

Syntaxcii 21357 Extend class notation with the unit interval.

Syntaxccncf 21358 Extend class notation to include the operation which returns a class of continuous complex functions.

Definitiondf-ii 21359 Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Definitiondf-cncf 21360* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)

Theoremiitopon 21361 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn

Theoremiitop 21362 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiiuni 21363 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)

Theoremdfii2 21364 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremdfii3 21365 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
fld       t

Theoremdfii4 21366 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
flds

Theoremdfii5 21367 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremiicmp 21368 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theoremiicon 21369 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcncfval 21370* The value of the continuous complex function operation is the set of continuous functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf 21371* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)

Theoremelcncf2 21372* Version of elcncf 21371 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremcncfrss 21373 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncfrss2 21374 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncff 21375 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfi 21376* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremelcncf1di 21377* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremelcncf1ii 21378* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremrescncf 21379 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncffvrn 21380 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)

Theoremcncfss 21381 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremclimcncf 21382 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremabscncf 21383 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremrecncf 21384 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremimcncf 21385 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremcjcncf 21386 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremmulc1cncf 21387* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremdivccncf 21388* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)

Theoremcncfco 21389 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfmet 21390 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfcn 21391 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
fld       t        t

Theoremcncfcn1 21392 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
fld

Theoremcncfmptc 21393* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfmptid 21394* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Theoremcncfmpt1f 21395* Composition of continuous functions. analog of cnmpt11f 20143. (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremcncfmpt2f 21396* Composition of continuous functions. analog of cnmpt12f 20145. (Contributed by Mario Carneiro, 3-Sep-2014.)
fld

Theoremcncfmpt2ss 21397* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
fld

Theoremaddccncf 21398* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremcdivcncf 21399* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)

Theoremnegcncf 21400* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)

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