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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsszcld 18801 Every subset of the integers are closed in the topology on . (Contributed by Mario Carneiro, 6-Jul-2017.)
fld

Theoremreperflem 18802* 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 18803 The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
fld       t Perf

Theoremcnperf 18804 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
fld       Perf

Theoremiccntr 18805 The interior of a closed interval in the standard topology on is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)

Theoremicccmplem1 18806* Lemma for icccmp 18809. (Contributed by Mario Carneiro, 18-Jun-2014.)
t

Theoremicccmplem2 18807* Lemma for icccmp 18809. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremicccmplem3 18808* Lemma for icccmp 18809. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremicccmp 18809 A closed interval in is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
t

Theoremreconnlem1 18810 Lemma for reconn 18812. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
t

Theoremreconnlem2 18811* Lemma for reconn 18812. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)

Theoremreconn 18812* 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 18813 Corollary of reconn 18812. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)

Theoremiccconn 18814 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
t

Theoremopnreen 18815 Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)

Theoremrectbntr0 18816 A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremxrge0gsumle 18817 A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
s                                    g g

Theoremxrge0tsms 18818* 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.)
s                      g        tsums

Theoremxrge0tsms2 18819 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 18820 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 18821 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 18822 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 18822 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 18823 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 18824 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 18825 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 18826* Continuity of the metric function; analogue of cnmpt12f 17651 which cannot be used directly because is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2ds 18827* Continuity of the metric function; analogue of cnmpt22f 17660 which cannot be used directly because is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremnmcn 18828 The norm of a normed group is a continuous function. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmGrp

Theoremabscn 18829 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 18830* Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetdsf 18831* 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 18832* The distance from the point to the set is greater than iff the -ball around misses . (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetds0 18833* 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 18834* 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 18835* 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 18836* 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 18837* 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 18838* Lemma for metdscn 18839. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremmetdscn 18839* 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 18840* 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 complexes. (Contributed by Mario Carneiro, 5-Sep-2015.)
fld

Theoremmetnrmlem1a 18841* Lemma for metnrm 18845. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetnrmlem1 18842* Lemma for metnrm 18845. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

Theoremmetnrmlem2 18843* Lemma for metnrm 18845. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetnrmlem3 18844* Lemma for metnrm 18845. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetnrm 18845 A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.)

Theoremmetreg 18846 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremaddcnlem 18847* Lemma for addcn 18848, subcn 18849, and mulcn 18850. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
fld

Theoremaddcn 18848 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 18849 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 18850 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 18851 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
fld       t

Theoremcnfldtgp 18852 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 18853* 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 18854* Version of fsumcn 18853 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
fld       TopOn              TopOn

Theoremexpcn 18855* 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 18856* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.)
fld

Theoremsqcn 18857* The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
fld

11.4.11  Topological definitions using the reals

Syntaxcii 18858 Extend class notation with the unit interval.

Syntaxccncf 18859 Extend class notation to include the operation which returns a class of continuous complex functions.

Definitiondf-ii 18860 Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)

Definitiondf-cncf 18861* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)

Theoremiitopon 18862 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
TopOn

Theoremiitop 18863 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiiuni 18864 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)

Theoremdfii2 18865 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
t

Theoremdfii3 18866 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
fld       t

Theoremdfii4 18867 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
flds

Theoremdfii5 18868 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
ordTop

Theoremiicmp 18869 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theoremiicon 18870 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremcncfval 18871* 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 18872* 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 18873* Version of elcncf 18872 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)

Theoremcncfrss 18874 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncfrss2 18875 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Theoremcncff 18876 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremcncfi 18877* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)

Theoremelcncf1di 18878* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremelcncf1ii 18879* Membership in the set of continuous complex functions from to . (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremrescncf 18880 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 18881 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)

Theoremcncfss 18882 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)

Theoremclimcncf 18883 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)

Theoremabscncf 18884 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremrecncf 18885 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremimcncf 18886 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremcjcncf 18887 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

Theoremmulc1cncf 18888* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremdivccncf 18889* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)

Theoremcncfco 18890 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 18891 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfcn 18892 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
fld       t        t

Theoremcncfcn1 18893 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
fld

Theoremcncfmptc 18894* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)

Theoremcncfmptid 18895* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)

Theoremcncfmpt1f 18896* Composition of continuous functions. analog of cnmpt11f 17649. (Contributed by Mario Carneiro, 3-Sep-2014.)

Theoremcncfmpt2f 18897* Composition of continuous functions. analog of cnmpt12f 17651. (Contributed by Mario Carneiro, 3-Sep-2014.)
fld

Theoremcncfmpt2ss 18898* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
fld

Theoremaddccncf 18899* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Theoremcdivcncf 18900* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)

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