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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclssubg 21201 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
SubGrp SubGrp

Theoremclsnsg 21202 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
NrmSGrp NrmSGrp

Theoremcldsubg 21203 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
~QG               SubGrp

Theoremtgpconcompeqg 21204* The connected component containing is the left coset of the identity component containing . (Contributed by Mario Carneiro, 17-Sep-2015.)
t        ~QG        t

Theoremtgpconcomp 21205* The identity component, the connected component containing the identity element, is a closed (concompcld 20526) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        NrmSGrp

Theoremtgpconcompss 21206* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
t        SubGrp

Theoremghmcnp 21207 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 21208 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
~QG

Theoremtgphaus 21209 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt1 21210 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremtgpt0 21211 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)

Theoremqustgpopn 21212* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        NrmSGrp

Theoremqustgplem 21213* Lemma for qustgp 21214. (Contributed by Mario Carneiro, 18-Sep-2015.)
s ~QG                             ~QG        ~QG        NrmSGrp

Theoremqustgp 21214 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
s ~QG        NrmSGrp

Theoremqustgphaus 21215 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 topological 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 21216 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
s                     TopMnd       TopMnd

Theoremprdstgpd 21217 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
s

12.2.7  Infinite group sum on topological groups

Syntaxctsu 21218 Extend class notation to include infinite group sums in a topological group.
tsums

Definitiondf-tsms 21219* 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 13830) and g (df-gsum 15419), 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 21220* 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 21221 The finite partial sums of a function are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                     g

Theoremtsmsval2 21222* 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 21223* 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 21224 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 16640 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
tsums tsums

Theoremeltsms 21225* 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 21226* 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 21227 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                            tsums

Theoremhaustsms 21228* In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.)
CMnd                                          tsums

Theoremhaustsms2 21229 In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.)
CMnd                                          tsums tsums

Theoremtsmscls 21230 One half of tgptsmscls 21242, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmsgsum 21231 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.) (Revised by AV, 24-Jul-2019.)
CMnd                            finSupp               tsums g

Theoremtsmsid 21232 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.) (Revised by AV, 24-Jul-2019.)
CMnd                            finSupp        g tsums

Theoremhaustsmsid 21233 In a Hausdorff topological 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.) (Revised by AV, 24-Jul-2019.)
CMnd                            finSupp                      tsums g

Theoremtsms0 21234* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
CMnd                     tsums

Theoremtsmssubm 21235 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              SubMnd              s        tsums tsums

Theoremtsmsres 21236 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.)
CMnd                            supp        tsums tsums

Theoremtsmsf1o 21237 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd                                   tsums tsums

Theoremtsmsmhm 21238 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
CMnd              CMnd              MndHom                             tsums        tsums

Theoremtsmsadd 21239 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
CMnd       TopMnd                            tsums        tsums        tsums

Theoremtsmsinv 21240 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                            tsums        tsums

Theoremtsmssub 21241 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
CMnd                                   tsums        tsums        tsums

Theoremtgptsmscls 21242 A sum in a topological group is uniquely determined up to a coset of , which is a normal subgroup by clsnsg 21202, 0nsg 16940. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
CMnd                            tsums        tsums

Theoremtgptsmscld 21243 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 21244 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
CMnd       TopMnd                     tsums        tsums                      tsums

Theoremtsmsxplem1 21245* Lemma for tsmsxp 21247. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                       g

Theoremtsmsxplem2 21246* Lemma for tsmsxp 21247. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums                                                                              g        g        g        g

Theoremtsmsxp 21247* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 17686 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21245 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
CMnd                                          tsums        tsums tsums

12.2.8  Topological rings, fields, vector spaces

Syntaxctrg 21248 The class of all topological division rings.

Syntaxctdrg 21249 The class of all topological division rings.
TopDRing

Syntaxctlm 21250 The class of all topological modules.
TopMod

Syntaxctvc 21251 The class of all topological vector spaces.

Definitiondf-trg 21252 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 21253 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 21254 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 21255 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 21256 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremtrgtmd 21257 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       TopMnd

Theoremistdrg 21258 Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtdrgunit 21259 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp       Unit       TopDRing s

Theoremtrgtgp 21260 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgtmd2 21261 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMnd

Theoremtrgtps 21262 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrgring 21263 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtrggrp 21264 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtdrgtrg 21265 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgdrng 21266 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgring 21267 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremtdrgtmd 21268 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing TopMnd

Theoremtdrgtps 21269 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing

Theoremistdrg2 21270 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 21271 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
mulGrp

Theoreminvrcn2 21272 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 21273 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 21274* Continuity of ring multiplication; analogue of cnmpt12f 20758 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn

Theoremcnmpt2mulr 21275* Continuity of ring multiplication; analogue of cnmpt22f 20767 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopOn       TopOn

Theoremdvrcn 21276 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
/r       Unit       TopDRing t

Theoremistlm 21277 The predicate " is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod TopMnd

Theoremvscacn 21278 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar              TopMod

Theoremtlmtmd 21279 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod TopMnd

Theoremtlmtps 21280 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmlmod 21281 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtlmtrg 21282 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremtlmscatps 21283 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod

Theoremistvc 21284 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopMod TopDRing

Theoremtvctdrg 21285 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar       TopDRing

Theoremcnmpt1vsca 21286* Continuity of scalar multiplication; analogue of cnmpt12f 20758 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn

Theoremcnmpt2vsca 21287* Continuity of scalar multiplication; analogue of cnmpt22f 20767 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Scalar                            TopMod       TopOn       TopOn

Theoremtlmtgp 21288 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvctlm 21289 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod

Theoremtvclmod 21290 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)

Theoremtvclvec 21291 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)

12.3  Uniform Structures and Spaces

12.3.1  Uniform structures

Syntaxcust 21292 Extend class notation with the class function of uniform structures.
UnifOn

Definitiondf-ust 21293* 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 definition 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 21294 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustval 21295* The class of all uniform structures for a base . (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremisust 21296* The predicate " is a uniform structure with base ." (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn

Theoremustssxp 21297 Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustssel 21298 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustbasel 21299 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
UnifOn

Theoremustincl 21300 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
UnifOn

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
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