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

Theoremtgpt0 17801 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremdivstgpopn 17802* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        NrmSGrp

Theoremdivstgplem 17803* Lemma for divstgp 17804. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        ~QG        NrmSGrp

Theoremdivstgp 17804 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s ~QG        NrmSGrp

Theoremdivstgphaus 17805 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 17806 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
s                     TopMnd       TopMnd

Theoremprdstgpd 17807 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 17808 Extend class notation to include infinite group sums in a topological group.
tsums

Definitiondf-tsms 17809* 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 12159) and g (df-gsum 13405), 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 17810* 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 17811 The finite partial sums of a function are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                     g

Theoremtsmsval2 17812* 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 17813* 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 17814 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 14398 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
tsums tsums

Theoremeltsms 17815* 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 17816* 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 17817 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums

Theoremhaustsms 17818* A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                          tsums

Theoremhaustsms2 17819 A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 13-Sep-2015.)
CMnd                                          tsums tsums

Theoremtsmscls 17820 One half of tgptsmscls 17832, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmsgsum 17821 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 17822 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 17823 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 17824* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                     tsums

Theoremtsmssubm 17825 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              SubMnd              s        tsums tsums

Theoremtsmsres 17826 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsf1o 17827 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsmhm 17828 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              CMnd              MndHom                             tsums        tsums

Theoremtsmsadd 17829 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                            tsums        tsums        tsums

Theoremtsmsinv 17830 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmssub 17831 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                                   tsums        tsums        tsums

Theoremtgptsmscls 17832 A sum in a topological group is uniquely determined up to a coset of , which is a normal subgroup by clsnsg 17792, 0nsg 14662. (Contributed by Mario Carneiro, 22-Sep-2015.)
CMnd                            tsums        tsums

Theoremtgptsmscld 17833 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 17834 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                     tsums        tsums                      tsums

Theoremtsmsxplem1 17835* Lemma for tsmsxp 17837. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                       g

Theoremtsmsxplem2 17836* Lemma for tsmsxp 17837. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                              g        g        g        g

Theoremtsmsxp 17837* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15227 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 17835 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 17838 The class of all topological division rings.

Syntaxctdrg 17839 The class of all topological division rings.
TopDRing

Syntaxctlm 17840 The class of all topological modules.
TopMod

Syntaxctvc 17841 The class of all topological vector spaces.

Definitiondf-trg 17842 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 17843 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 17844 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 17845 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 17846 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremtrgtmd 17847 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremistdrg 17848 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtdrgunit 17849 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtrgtgp 17850 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgtmd2 17851 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMnd

Theoremtrgtps 17852 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgrng 17853 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrggrp 17854 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtdrgtrg 17855 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgdrng 17856 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgrng 17857 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgtmd 17858 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing TopMnd

Theoremtdrgtps 17859 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremistdrg2 17860 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 17861 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoreminvrcn2 17862 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 17863 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 17864* Continuity of ring multiplication; analogue of cnmpt12f 17360 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2mulr 17865* Continuity of ring multiplication; analogue of cnmpt22f 17369 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremdvrcn 17866 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
/r       Unit       TopDRing t

Theoremistlm 17867 The predicate " is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod TopMnd

Theoremvscacn 17868 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod

Theoremtlmtmd 17869 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 17870 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 17871 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 17872 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 17873 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 17874 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 17875 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 17876* Continuity of scalar multiplication; analogue of cnmpt12f 17360 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 17877* Continuity of scalar multiplication; analogue of cnmpt22f 17369 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 17878 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 17879 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 17880 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 17881 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 17882 Extend class notation with the class of all extended metric spaces.

Syntaxcmt 17883 Extend class notation with the class of all metric spaces.

Syntaxctmt 17884 Extend class notation with the function mapping a metric to a metric space.
toMetSp

Definitiondf-xms 17885 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)

Definitiondf-ms 17886 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)

Definitiondf-tms 17887 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
toMetSp sSet TopSet

Theoremismet 17888* Express the predicate " is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmet 17889* Express the predicate " is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremismeti 17890* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremisxmetd 17891* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremisxmet2d 17892* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetflem 17893* Lemma for metf 17895 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Theoremxmetf 17894 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetf 17895 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)

Theoremxmetcl 17896 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremmetcl 17897 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)

Theoremismet2 17898 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmetxmet 17899 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmetdmdm 17900 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)

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