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Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtmdlactcn 17801* The left group action of element in a topological monoid is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
TopMnd

Theoremtgplacthmeo 17802* The left group action of element in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremsubmtmd 17803 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
s        TopMnd SubMnd TopMnd

Theoremsubgtgp 17804 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s        SubGrp

Theoremsubgntr 17805 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 17807, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
SubGrp

Theoremopnsubg 17806 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp

Theoremclssubg 17807 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp SubGrp

Theoremclsnsg 17808 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
NrmSGrp NrmSGrp

Theoremcldsubg 17809 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremtgpconcompeqg 17810* The connected component containing is the left coset of the identity component containing . (Contributed by Mario Carneiro, 17-Sep-2015.)
t        ~QG        t

Theoremtgpconcomp 17811* The identity component, the connected component containing the identity element, is a closed (concompcld 17176) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        NrmSGrp

Theoremtgpconcompss 17812* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        SubGrp

Theoremghmcnp 17813 A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
TopMnd TopMnd

Theoremsnclseqg 17814 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
~QG

Theoremtgphaus 17815 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt1 17816 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt0 17817 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremdivstgpopn 17818* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        NrmSGrp

Theoremdivstgplem 17819* Lemma for divstgp 17820. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        ~QG        NrmSGrp

Theoremdivstgp 17820 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s ~QG        NrmSGrp

Theoremdivstgphaus 17821 The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
s ~QG                      NrmSGrp

Theoremprdstmdd 17822 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
s                     TopMnd       TopMnd

Theoremprdstgpd 17823 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
s

11.2.6  Infinite group sum on topological groups

Syntaxctsu 17824 Extend class notation to include infinite group sums in a topological group.
tsums

Definitiondf-tsms 17825* Define the set of limit points of an infinite group sum for the topological group . If is Hausdorff, then there will be at most one element in this set and tsums selects this unique element if it exists. tsums is a way to say that the sum exists and is unique. Note that unlike (df-sum 12175) and g (df-gsum 13421), this does not return the sum itself, but rather the set of all such sums, which is usually either empty or a singleton. (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmsfbas 17826* The collection of all sets of the form , which can be read as the set of all finite subsets of which contain as a subset, for each finite subset of , form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)

Theoremtsmslem1 17827 The finite partial sums of a function are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                     g

Theoremtsmsval2 17828* Definition of the topological group sum(s) of a collection of values in the group with index set . (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmsval 17829* Definition of the topological group sum(s) of a collection of values in the group with index set . (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums g

Theoremtsmspropd 17830 The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14414 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
tsums tsums

Theoremeltsms 17831* The property of being a sum of the sequence in the topological commutative monoid . (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums g

Theoremtsmsi 17832* The property of being a sum of the sequence in the topological commutative monoid . (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums                      g

Theoremtsmscl 17833 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums

Theoremhaustsms 17834* A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                          tsums

Theoremhaustsms2 17835 A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 13-Sep-2015.)
CMnd                                          tsums tsums

Theoremtsmscls 17836 One half of tgptsmscls 17848, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmsgsum 17837 The convergent points of a finite topological group sum are the closure of the finite group sum operation. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd                                          tsums g

Theoremtsmsid 17838 If a sum is finite, the usual sum is always a limit point of the topological sum (although it may not be the only limit point). (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                   g tsums

Theoremhaustsmsid 17839 In a Hausdorff group a finite sum sums to exactly the usual number with no extraneous limit points. By setting the topology to the discrete topology (which is Hausdorff), this theorem can be used to turn any tsums theorem into a g theorem, so that the infinite group sum operation can be viewed as a generalization of the finite group sum. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                                 tsums g

Theoremtsms0 17840* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                     tsums

Theoremtsmssubm 17841 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              SubMnd              s        tsums tsums

Theoremtsmsres 17842 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsf1o 17843 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsmhm 17844 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              CMnd              MndHom                             tsums        tsums

Theoremtsmsadd 17845 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                            tsums        tsums        tsums

Theoremtsmsinv 17846 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmssub 17847 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                                   tsums        tsums        tsums

Theoremtgptsmscls 17848 A sum in a topological group is uniquely determined up to a coset of , which is a normal subgroup by clsnsg 17808, 0nsg 14678. (Contributed by Mario Carneiro, 22-Sep-2015.)
CMnd                            tsums        tsums

Theoremtgptsmscld 17849 The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
CMnd                            tsums

Theoremtsmssplit 17850 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                     tsums        tsums                      tsums

Theoremtsmsxplem1 17851* Lemma for tsmsxp 17853. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                       g

Theoremtsmsxplem2 17852* Lemma for tsmsxp 17853. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                              g        g        g        g

Theoremtsmsxp 17853* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15243 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 17851 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums        tsums tsums

11.2.7  Topological rings, fields, vector spaces

Syntaxctrg 17854 The class of all topological division rings.

Syntaxctdrg 17855 The class of all topological division rings.
TopDRing

Syntaxctlm 17856 The class of all topological modules.
TopMod

Syntaxctvc 17857 The class of all topological vector spaces.

Definitiondf-trg 17858 Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp TopMnd

Definitiondf-tdrg 17859 Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing mulGrps Unit

Definitiondf-tlm 17860 Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd Scalar Scalar

Definitiondf-tvc 17861 Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod Scalar TopDRing

Theoremistrg 17862 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremtrgtmd 17863 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremistdrg 17864 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtdrgunit 17865 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtrgtgp 17866 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgtmd2 17867 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMnd

Theoremtrgtps 17868 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgrng 17869 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrggrp 17870 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtdrgtrg 17871 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgdrng 17872 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgrng 17873 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgtmd 17874 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing TopMnd

Theoremtdrgtps 17875 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremistdrg2 17876 A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp                     TopDRing s

Theoremmulrcn 17877 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoreminvrcn2 17878 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       TopDRing t t

Theoreminvrcn 17879 The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit       TopDRing t

Theoremcnmpt1mulr 17880* Continuity of ring multiplication; analogue of cnmpt12f 17376 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2mulr 17881* Continuity of ring multiplication; analogue of cnmpt22f 17385 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremdvrcn 17882 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
/r       Unit       TopDRing t

Theoremistlm 17883 The predicate " is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod TopMnd

Theoremvscacn 17884 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod

Theoremtlmtmd 17885 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 17886 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 17887 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 17888 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 17889 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 17890 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 17891 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 17892* Continuity of scalar multiplication; analogue of cnmpt12f 17376 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 17893* Continuity of scalar multiplication; analogue of cnmpt22f 17385 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 17894 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 17895 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 17896 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 17897 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)

11.3  Metric spaces

11.3.1  Basic metric space properties

Syntaxcxme 17898 Extend class notation with the class of all extended metric spaces.

Syntaxcmt 17899 Extend class notation with the class of all metric spaces.

Syntaxctmt 17900 Extend class notation with the function mapping a metric to a metric space.
toMetSp

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