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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrggrp 18201 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtdrgtrg 18202 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgdrng 18203 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgrng 18204 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgtmd 18205 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing TopMnd

Theoremtdrgtps 18206 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremistdrg2 18207 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 18208 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoreminvrcn2 18209 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 18210 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 18211* Continuity of ring multiplication; analogue of cnmpt12f 17698 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2mulr 18212* Continuity of ring multiplication; analogue of cnmpt22f 17707 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremdvrcn 18213 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
/r       Unit       TopDRing t

Theoremistlm 18214 The predicate " is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod TopMnd

Theoremvscacn 18215 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod

Theoremtlmtmd 18216 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 18217 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 18218 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 18219 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 18220 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 18221 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 18222 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 18223* Continuity of scalar multiplication; analogue of cnmpt12f 17698 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 18224* Continuity of scalar multiplication; analogue of cnmpt22f 17707 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 18225 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 18226 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 18227 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 18228 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 18229 Extend class notation with the class function of uniform structures.
UnifOn

Definitiondf-ust 18230* 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 18231 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustval 18232* The class of all uniform structures for a base . (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremisust 18233* The predicate " is a uniform structure with base ." (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustssxp 18234 Entourages are subsets of the cross product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustssel 18235 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustbasel 18236 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustincl 18237 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn

Theoremustdiag 18238 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 18239 If is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn

Theoremustexhalf 18240* 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 18241 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustfilxp 18242 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 18243 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustssco 18244 In an uniform structure, any entourage is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn

Theoremustexsym 18245* In an uniform structure, for any entourage , there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
UnifOn

Theoremustex2sym 18246* In an uniform structure, for any entourage , there exists a symmetrical entourage smaller than half . (Contributed by Thierry Arnoux, 16-Jan-2018.)
UnifOn

Theoremustex3sym 18247* 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

Theoremustref 18248 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremust0 18249 The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustn0 18250 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
UnifOn

Theoremustund 18251 If two intersecting sets and are both small in , their union is small in . Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Theoremustelimasn 18252 Any point is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
UnifOn

Theoremustneism 18253 For a point in , is small enough in . This proposition actually does not require any axiom of the defintion of unifom structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Theoremelrnust 18254 First direction for ustbas 18257. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn

Theoremustbas2 18255 Second direction for ustbas 18257. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn

Theoremustuni 18256 The set union of a uniform structure is the cross product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremustbas 18257 Recover the base of an uniform structure . UnifOn is to UnifOn what is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
UnifOn UnifOn

Theoremustimasn 18258 Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn

Theoremtrust 18259 The trace of a uniform structure on a subset is a uniform structure on . Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
UnifOn t UnifOn

11.3.2  The topology induced by an uniform structure

Syntaxcutop 18260 Extend class notation with the function inducing a topology from a uniform structure.
unifTop

Definitiondf-utop 18261* Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
unifTop UnifOn

Theoremutopval 18262* The topology induced by a uniform structure . (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremelutop 18263* Open sets in the topology induced by an uniform structure on (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutoptop 18264 The topology induced by a uniform structure is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn unifTop

Theoremutopbas 18265 The base of the topology induced by a uniform structure . (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn unifTop

Theoremutoptopon 18266 Topology induced by a uniform structure with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
UnifOn unifTop TopOn

Theoremrestutop 18267 Restriction of a topology induced by an uniform structure (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifOn unifTopt unifTopt

Theoremrestutopopn 18268 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
UnifOn unifTop unifTopt unifTopt

Theoremustuqtoplem 18269* Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop0 18270* Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop1 18271* Lemma for ustuqtop 18276, similar to ssnei2 17180 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop2 18272* Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop3 18273* Lemma for ustuqtop 18276, similar to elnei 17175 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop4 18274* Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop5 18275* Lemma for ustuqtop 18276 (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn

Theoremustuqtop 18276* For a given uniform structure on a set , there is a unique topology such that the set is the filter of the neighbourhoods of for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
UnifOn TopOn

Theoremutopsnneiplem 18277* The neighborhoods of a point for the topology induced by an uniform space . (Contributed by Thierry Arnoux, 11-Jan-2018.)
unifTop                     UnifOn

Theoremutopsnneip 18278* The neighborhoods of a point for the topology induced by an uniform space . (Contributed by Thierry Arnoux, 13-Jan-2018.)
unifTop       UnifOn

Theoremutopsnnei 18279 Images of singletons by entourages are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
unifTop       UnifOn

Theoremutop2nei 18280 For any symmetrical entourage and any relation , build a neighborhood of . First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
unifTop       UnifOn

Theoremutop3cls 18281 Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
unifTop       UnifOn

Theoremutopreg 18282 All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)
unifTop       UnifOn

11.3.3  Uniform Spaces

Syntaxcuss 18283 Extend class notation with the Uniform Structure extractor function.
UnifSt

Syntaxcusp 18284 Extend class notation with the class of uniform spaces.
UnifSp

Syntaxctus 18285 Extend class notation with the function mapping a uniform structure to a uniform space.
toUnifSp

Definitiondf-uss 18286 Define the uniform structure extractor function. Similarly with df-topn 13651 this differs from df-unif 13552 when a structure has been restricted using df-ress 13476; in this case the component will still have a uniform set over the larger set, and this function fixes this by restricting the uniform set as well. (Contributed by Thierry Arnoux, 1-Dec-2017.)
UnifSt t

Definitiondf-usp 18287 Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
UnifSp UnifSt UnifOn unifTopUnifSt

Definitiondf-tus 18288 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
toUnifSp UnifOn sSet TopSet unifTop

Theoremussval 18289 The uniform structure on uniform space . This proof uses a trick with fvprc 5722 to avoid requiring to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
t UnifSt

Theoremussid 18290 In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt

Theoremisusp 18291 The predicate is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
UnifSt              UnifSp UnifOn unifTop

Theoremressunif 18292 is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
s

Theoremressuss 18293 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
UnifSts UnifStt

Theoremressust 18294 The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
UnifSts        UnifSp UnifOn

Theoremressusp 18295 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
UnifSp s UnifSp

Theoremtusval 18296 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
UnifOn toUnifSp sSet TopSet unifTop

Theoremtuslem 18297 Lemma for tusbas 18298, tusunif 18299, and tustopn 18301. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn unifTop

Theoremtusbas 18298 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtusunif 18299 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
toUnifSp       UnifOn

Theoremtususs 18300 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
toUnifSp       UnifOn UnifSt

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