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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtgpt0 18101 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremdivstgpopn 18102* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        NrmSGrp

Theoremdivstgplem 18103* Lemma for divstgp 18104. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        ~QG        NrmSGrp

Theoremdivstgp 18104 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s ~QG        NrmSGrp

Theoremdivstgphaus 18105 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 18106 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
s                     TopMnd       TopMnd

Theoremprdstgpd 18107 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
s

11.2.7  Infinite group sum on topological groups

Syntaxctsu 18108 Extend class notation to include infinite group sums in a topological group.
tsums

Definitiondf-tsms 18109* 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 12435) and g (df-gsum 13683), 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 18110* 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 18111 The finite partial sums of a function are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                     g

Theoremtsmsval2 18112* 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 18113* 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 18114 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 14676 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
tsums tsums

Theoremeltsms 18115* 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 18116* 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 18117 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums

Theoremhaustsms 18118* A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                          tsums

Theoremhaustsms2 18119 A Hausdorff group has at most one limit point for a given sum. (Contributed by Mario Carneiro, 13-Sep-2015.)
CMnd                                          tsums tsums

Theoremtsmscls 18120 One half of tgptsmscls 18132, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmsgsum 18121 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 18122 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 18123 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 18124* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                     tsums

Theoremtsmssubm 18125 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              SubMnd              s        tsums tsums

Theoremtsmsres 18126 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsf1o 18127 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsmhm 18128 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              CMnd              MndHom                             tsums        tsums

Theoremtsmsadd 18129 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                            tsums        tsums        tsums

Theoremtsmsinv 18130 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmssub 18131 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                                   tsums        tsums        tsums

Theoremtgptsmscls 18132 A sum in a topological group is uniquely determined up to a coset of , which is a normal subgroup by clsnsg 18092, 0nsg 14940. (Contributed by Mario Carneiro, 22-Sep-2015.)
CMnd                            tsums        tsums

Theoremtgptsmscld 18133 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 18134 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.)
CMnd       TopMnd                     tsums        tsums                      tsums

Theoremtsmsxplem1 18135* Lemma for tsmsxp 18137. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                       g

Theoremtsmsxplem2 18136* Lemma for tsmsxp 18137. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                              g        g        g        g

Theoremtsmsxp 18137* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15505 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 18135 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums        tsums tsums

11.2.8  Topological rings, fields, vector spaces

Syntaxctrg 18138 The class of all topological division rings.

Syntaxctdrg 18139 The class of all topological division rings.
TopDRing

Syntaxctlm 18140 The class of all topological modules.
TopMod

Syntaxctvc 18141 The class of all topological vector spaces.

Definitiondf-trg 18142 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 18143 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 18144 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 18145 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 18146 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremtrgtmd 18147 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremistdrg 18148 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtdrgunit 18149 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtrgtgp 18150 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgtmd2 18151 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMnd

Theoremtrgtps 18152 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgrng 18153 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrggrp 18154 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtdrgtrg 18155 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgdrng 18156 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgrng 18157 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgtmd 18158 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing TopMnd

Theoremtdrgtps 18159 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremistdrg2 18160 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 18161 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoreminvrcn2 18162 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 18163 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 18164* Continuity of ring multiplication; analogue of cnmpt12f 17651 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2mulr 18165* Continuity of ring multiplication; analogue of cnmpt22f 17660 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremdvrcn 18166 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
/r       Unit       TopDRing t

Theoremistlm 18167 The predicate " is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod TopMnd

Theoremvscacn 18168 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod

Theoremtlmtmd 18169 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 18170 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 18171 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 18172 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 18173 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 18174 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 18175 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 18176* Continuity of scalar multiplication; analogue of cnmpt12f 17651 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 18177* Continuity of scalar multiplication; analogue of cnmpt22f 17660 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 18178 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 18179 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 18180 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 18181 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)

11.3  Uniform Stuctures and Spaces

11.3.1  Uniform structures

Syntaxcust 18182 Extend class notation with the class function of uniform structures.
UnifOn

Definitiondf-ust 18183* Definition of a uniform structure. Definition 1 of [BourbakiTop1] p. II.1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. This defintion is analogous to TopOn. Elements of an uniform structure are called entourages. (Contributed by FL, 29-May-2014.) (Revised by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustfn 18184 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustval 18185* The class of all uniform structures for a base . (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremisust 18186* The predicate " is a uniform structure with base ." (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustssxp 18187 Entourages are subsets of the cross product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustssel 18188 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustbasel 18189 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustincl 18190 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn

Theoremustdiag 18191 The diagonal set is included in any entourage, i.e. any point is -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustinvel 18192 If is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustexhalf 18193* For each entourage there is an entourage that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustrel 18194 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustfilxp 18195 A uniform structure on a non-empty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustne0 18196 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustssco 18197 In an uniform structure, any entourage is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn

Theoremustexsym 18198* In an uniform structure, for any entourage , there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
UnifOn

Theoremustex2sym 18199* In an uniform structure, for any entourage , there exists a symmetrical entourage smaller than half . (Contributed by Thierry Arnoux, 16-Jan-2018.)
UnifOn

Theoremustex3sym 18200* In an uniform structure, for any entourage , there exists a symmetrical entourage smaller than a third of . (Contributed by Thierry Arnoux, 16-Jan-2018.)
UnifOn

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