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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.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  ->  ( ( cls `  J ) `  S )  e.  (SubGrp `  G ) )
 
Theoremclsnsg 21202 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (NrmSGrp `  G ) )  ->  ( ( cls `  J ) `  S )  e.  (NrmSGrp `  G ) )
 
Theoremcldsubg 21203 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  R  =  ( G ~QG 
 S )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  ( X /. R )  e.  Fin )  ->  ( S  e.  ( Clsd `  J )  <->  S  e.  J ) )
 
Theoremtgpconcompeqg 21204* The connected component containing 
A is the left coset of the identity component containing  A. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   &    |-  .~  =  ( G ~QG  S )   =>    |-  ( ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  [ A ]  .~  =  U. { x  e. 
 ~P X  |  ( A  e.  x  /\  ( Jt  x )  e.  Con ) } )
 
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.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  ( G  e.  TopGrp  ->  S  e.  (NrmSGrp `  G )
 )
 
Theoremtgpconcompss 21206* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  S  =  U. { x  e.  ~P X  |  (  .0.  e.  x  /\  ( Jt  x )  e.  Con ) }   =>    |-  (
 ( G  e.  TopGrp  /\  T  e.  (SubGrp `  G )  /\  T  e.  J )  ->  S  C_  T )
 
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.)
 |-  X  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   =>    |-  (
 ( G  e. TopMnd  /\  H  e. TopMnd  /\  F  e.  ( G  GrpHom  H ) ) 
 ->  ( F  e.  (
 ( J  CnP  K ) `  A )  <->  ( A  e.  X  /\  F  e.  ( J  Cn  K ) ) ) )
 
Theoremsnclseqg 21208 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |- 
 .~  =  ( G ~QG  S )   &    |-  S  =  ( ( cls `  J ) `  {  .0.  } )   =>    |-  (
 ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  [ A ]  .~  =  ( ( cls `  J ) `  { A }
 ) )
 
Theoremtgphaus 21209 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( G  e.  TopGrp  ->  ( J  e.  Haus  <->  {  .0.  }  e.  ( Clsd `  J ) ) )
 
Theoremtgpt1 21210 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( G  e.  TopGrp  ->  ( J  e.  Haus  <->  J  e.  Fre ) )
 
Theoremtgpt0 21211 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( G  e.  TopGrp  ->  ( J  e.  Haus  <->  J  e.  Kol2 )
 )
 
Theoremqustgpopn 21212* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  Y ) )   &    |-  X  =  ( Base `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   &    |-  F  =  ( x  e.  X  |->  [ x ] ( G ~QG  Y ) )   =>    |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  S  e.  J )  ->  ( F " S )  e.  K )
 
Theoremqustgplem 21213* Lemma for qustgp 21214. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  Y ) )   &    |-  X  =  ( Base `  G )   &    |-  J  =  ( TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   &    |-  F  =  ( x  e.  X  |->  [ x ] ( G ~QG  Y ) )   &    |-  .-  =  (
 z  e.  X ,  w  e.  X  |->  [ (
 z ( -g `  G ) w ) ] ( G ~QG  Y ) )   =>    |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G ) )  ->  H  e.  TopGrp )
 
Theoremqustgp 21214 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  Y ) )   =>    |-  ( ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G ) ) 
 ->  H  e.  TopGrp )
 
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.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  Y ) )   &    |-  J  =  ( TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   =>    |-  (
 ( G  e.  TopGrp  /\  Y  e.  (NrmSGrp `  G )  /\  Y  e.  ( Clsd `  J ) ) 
 ->  K  e.  Haus )
 
Theoremprdstmdd 21216 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I -->TopMnd )   =>    |-  ( ph  ->  Y  e. TopMnd )
 
Theoremprdstgpd 21217 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> TopGrp )   =>    |-  ( ph  ->  Y  e.  TopGrp )
 
12.2.7  Infinite group sum on topological groups
 
Syntaxctsu 21218 Extend class notation to include infinite group sums in a topological group.
 class tsums
 
Definitiondf-tsms 21219* Define the set of limit points of an infinite group sum for the topological group  G. If  G is Hausdorff, then there will be at most one element in this set and  U. ( W tsums  F ) selects this unique element if it exists. 
( W tsums  F )  ~~  1o is a way to say that the sum exists and is unique. Note that unlike  sum_ (df-sum 13830) and  gsumg (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  =  ( w  e.  _V ,  f  e.  _V  |->  [_ ( ~P dom  f  i^i  Fin )  /  s ]_ ( ( ( TopOpen `  w )  fLimf  ( s
 filGen ran  ( z  e.  s  |->  { y  e.  s  |  z  C_  y }
 ) ) ) `  ( y  e.  s  |->  ( w  gsumg  ( f  |`  y ) ) ) ) )
 
Theoremtsmsfbas 21220* The collection of all sets of the form  F ( z )  =  { y  e.  S  |  z 
C_  y }, which can be read as the set of all finite subsets of  A which contain  z as a subset, for each finite subset  z of  A, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  S  =  ( ~P A  i^i  Fin )   &    |-  F  =  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )   &    |-  L  =  ran  F   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  L  e.  ( fBas `  S ) )
 
Theoremtsmslem1 21221 The finite partial sums of a function  F are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  S  =  ( ~P A  i^i  Fin )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  X  e.  S )  ->  ( G  gsumg  ( F  |`  X ) )  e.  B )
 
Theoremtsmsval2 21222* Definition of the topological group sum(s) of a collection  F
( x ) of values in the group with index set  A. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  S  =  ( ~P A  i^i  Fin )   &    |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  dom 
 F  =  A )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( J  fLimf  ( S filGen L ) ) `  (
 y  e.  S  |->  ( G  gsumg  ( F  |`  y ) ) ) ) )
 
Theoremtsmsval 21223* Definition of the topological group sum(s) of a collection  F
( x ) of values in the group with index set  A. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  S  =  ( ~P A  i^i  Fin )   &    |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( J  fLimf  ( S filGen L ) ) `  (
 y  e.  S  |->  ( G  gsumg  ( F  |`  y ) ) ) ) )
 
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.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  ( +g  `  G )  =  ( +g  `  H ) )   &    |-  ( ph  ->  (
 TopOpen `  G )  =  ( TopOpen `  H )
 )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( H tsums  F ) )
 
Theoremeltsms 21225* The property of being a sum of the sequence  F in the topological commutative monoid  G. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  S  =  ( ~P A  i^i  Fin )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( C  e.  ( G tsums  F )  <->  ( C  e.  B  /\  A. u  e.  J  ( C  e.  u  ->  E. z  e.  S  A. y  e.  S  ( z  C_  y  ->  ( G  gsumg  ( F  |`  y ) )  e.  u ) ) ) ) )
 
Theoremtsmsi 21226* The property of being a sum of the sequence  F in the topological commutative monoid  G. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  S  =  ( ~P A  i^i  Fin )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  e.  ( G tsums  F ) )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  E. z  e.  S  A. y  e.  S  ( z  C_  y  ->  ( G  gsumg  ( F  |`  y ) )  e.  U ) )
 
Theoremtsmscl 21227 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G tsums  F )  C_  B )
 
Theoremhaustsms 21228* In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  J  e.  Haus )   =>    |-  ( ph  ->  E* x  x  e.  ( G tsums  F ) )
 
Theoremhaustsms2 21229 In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  J  e.  Haus )   =>    |-  ( ph  ->  ( X  e.  ( G tsums  F ) 
 ->  ( G tsums  F )  =  { X }
 ) )
 
Theoremtsmscls 21230 One half of tgptsmscls 21242, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( ( cls `  J ) `  { X } )  C_  ( G tsums  F ) )
 
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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  F finSupp  .0.  )   &    |-  J  =  ( TopOpen `  G )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { ( G  gsumg  F ) } ) )
 
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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  F finSupp  .0.  )   =>    |-  ( ph  ->  ( G  gsumg  F )  e.  ( G tsums  F ) )
 
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 
gsumg 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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  F finSupp  .0.  )   &    |-  J  =  ( TopOpen `  G )   &    |-  ( ph  ->  J  e.  Haus )   =>    |-  ( ph  ->  ( G tsums  F )  =  { ( G  gsumg 
 F ) } )
 
Theoremtsms0 21234* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  .0. 
 e.  ( G tsums  ( x  e.  A  |->  .0.  )
 ) )
 
Theoremtsmssubm 21235 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  S  e.  (SubMnd `  G ) )   &    |-  ( ph  ->  F : A --> S )   &    |-  H  =  ( Gs  S )   =>    |-  ( ph  ->  ( H tsums  F )  =  ( ( G tsums  F )  i^i  S ) )
 
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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( F supp  .0.  )  C_  W )   =>    |-  ( ph  ->  ( G tsums  ( F  |`  W ) )  =  ( G tsums  F ) )
 
Theoremtsmsf1o 21237 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : C -1-1-onto-> A )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( G tsums  ( F  o.  H ) ) )
 
Theoremtsmsmhm 21238 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  K  =  ( TopOpen `  H )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp )   &    |-  ( ph  ->  H  e. CMnd )   &    |-  ( ph  ->  H  e.  TopSp
 )   &    |-  ( ph  ->  C  e.  ( G MndHom  H )
 )   &    |-  ( ph  ->  C  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( C `  X )  e.  ( H tsums  ( C  o.  F ) ) )
 
Theoremtsmsadd 21239 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   &    |-  ( ph  ->  Y  e.  ( G tsums  H ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( G tsums  ( F  oF  .+  H ) ) )
 
Theoremtsmsinv 21240 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  I  =  ( invg `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( I `  X )  e.  ( G tsums  ( I  o.  F ) ) )
 
Theoremtsmssub 21241 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   &    |-  ( ph  ->  Y  e.  ( G tsums  H ) )   =>    |-  ( ph  ->  ( X  .-  Y )  e.  ( G tsums  ( F  oF  .-  H ) ) )
 
Theoremtgptsmscls 21242 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 21202, 0nsg 16940. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } )
 )
 
Theoremtgptsmscld 21243 The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G tsums  F )  e.  ( Clsd `  J ) )
 
Theoremtsmssplit 21244 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  X  e.  ( G tsums  ( F  |`  C ) ) )   &    |-  ( ph  ->  Y  e.  ( G tsums  ( F  |`  D ) ) )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( G tsums  F ) )
 
Theoremtsmsxplem1 21245* Lemma for tsmsxp 21247. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   &    |-  J  =  ( TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  .0.  e.  L )   &    |-  ( ph  ->  K  e.  ( ~P A  i^i  Fin ) )   &    |-  ( ph  ->  dom 
 D  C_  K )   &    |-  ( ph  ->  D  e.  ( ~P ( A  X.  C )  i^i  Fin ) )   =>    |-  ( ph  ->  E. n  e.  ( ~P C  i^i  Fin )
 ( ran  D  C_  n  /\  A. x  e.  K  ( ( H `  x )  .-  ( G 
 gsumg  ( F  |`  ( { x }  X.  n ) ) ) )  e.  L ) )
 
Theoremtsmsxplem2 21246* Lemma for tsmsxp 21247. (Contributed by Mario Carneiro, 21-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   &    |-  J  =  ( TopOpen `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  .0.  e.  L )   &    |-  ( ph  ->  K  e.  ( ~P A  i^i  Fin ) )   &    |-  ( ph  ->  A. c  e.  S  A. d  e.  T  (
 c  .+  d )  e.  U )   &    |-  ( ph  ->  N  e.  ( ~P C  i^i  Fin ) )   &    |-  ( ph  ->  D  C_  ( K  X.  N ) )   &    |-  ( ph  ->  A. x  e.  K  ( ( H `
  x )  .-  ( G  gsumg  ( F  |`  ( { x }  X.  N ) ) ) )  e.  L )   &    |-  ( ph  ->  ( G  gsumg  ( F  |`  ( K  X.  N ) ) )  e.  S )   &    |-  ( ph  ->  A. g  e.  ( L  ^m  K ) ( G  gsumg  g )  e.  T )   =>    |-  ( ph  ->  ( G  gsumg  ( H  |`  K ) )  e.  U )
 
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.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopGrp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  H : A
 --> B )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  ( H `  j
 )  e.  ( G tsums 
 ( k  e.  C  |->  ( j F k ) ) ) )   =>    |-  ( ph  ->  ( G tsums  F )  C_  ( G tsums  H ) )
 
12.2.8  Topological rings, fields, vector spaces
 
Syntaxctrg 21248 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 21249 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 21250 The class of all topological modules.
 class TopMod
 
Syntaxctvc 21251 The class of all topological vector spaces.
 class  TopVec
 
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.)
 |-  TopRing  =  { r  e.  ( TopGrp  i^i  Ring )  |  (mulGrp `  r )  e. 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  =  { r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
 )  e.  TopGrp }
 
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  =  { w  e.  (TopMnd  i^i  LMod )  |  ( (Scalar `  w )  e.  TopRing  /\  ( .sf `  w )  e.  ( (
 ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w ) )  Cn  ( TopOpen `  w )
 ) ) }
 
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.)
 |-  TopVec  =  { w  e. TopMod  |  (Scalar `  w )  e. TopDRing }
 
Theoremistrg 21256 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  <->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
 
Theoremtrgtmd 21257 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  TopRing  ->  M  e. TopMnd )
 
Theoremistdrg 21258 Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
 
Theoremtdrgunit 21259 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ( Ms  U )  e.  TopGrp )
 
Theoremtrgtgp 21260 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 21261 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 21262 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgring 21263 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
 
Theoremtrggrp 21264 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Grp )
 
Theoremtdrgtrg 21265 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 TopRing )
 
Theoremtdrgdrng 21266 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 DivRing )
 
Theoremtdrgring 21267 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  Ring )
 
Theoremtdrgtmd 21268 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. TopMnd )
 
Theoremtdrgtps 21269 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  TopSp )
 
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.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e. TopDRing  <->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
 
Theoremmulrcn 21271 The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  T  =  ( +f `  (mulGrp `  R ) )   =>    |-  ( R  e.  TopRing  ->  T  e.  ( ( J  tX  J )  Cn  J ) )
 
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.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
 
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.)
 |-  J  =  ( TopOpen `  R )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  I  e.  ( ( Jt  U )  Cn  J ) )
 
Theoremcnmpt1mulr 21274* Continuity of ring multiplication; analogue of cnmpt12f 20758 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( K  Cn  J ) )
 
Theoremcnmpt2mulr 21275* Continuity of ring multiplication; analogue of cnmpt22f 20767 which cannot be used directly because 
.r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  TopRing )   &    |-  ( ph  ->  K  e.  (TopOn `  X ) )   &    |-  ( ph  ->  L  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( K  tX  L )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( K 
 tX  L )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( K  tX  L )  Cn  J ) )
 
Theoremdvrcn 21276 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  J  =  ( TopOpen `  R )   &    |-  ./  =  (/r `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( R  e. TopDRing  ->  ./  e.  ( ( J  tX  ( Jt  U ) )  Cn  J ) )
 
Theoremistlm 21277 The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .sf `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  <->  ( ( W  e. TopMnd  /\  W  e.  LMod  /\  F  e.  TopRing )  /\  .x. 
 e.  ( ( K 
 tX  J )  Cn  J ) ) )
 
Theoremvscacn 21278 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .sf `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
 
Theoremtlmtmd 21279 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e. TopMnd )
 
Theoremtlmtps 21280 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopSp )
 
Theoremtlmlmod 21281 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  LMod )
 
Theoremtlmtrg 21282 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopRing )
 
Theoremtlmscatps 21283 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopSp )
 
Theoremistvc 21284 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  <->  ( W  e. TopMod  /\  F  e. TopDRing ) )
 
Theoremtvctdrg 21285 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  ->  F  e. TopDRing )
 
Theoremcnmpt1vsca 21286* Continuity of scalar multiplication; analogue of cnmpt12f 20758 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( L  Cn  J ) )
 
Theoremcnmpt2vsca 21287* Continuity of scalar multiplication; analogue of cnmpt22f 20767 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L 
 tX  M )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( L  tX  M )  Cn  J ) )
 
Theoremtlmtgp 21288 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopGrp )
 
Theoremtvctlm 21289 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e. TopMod )
 
Theoremtvclmod 21290 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LMod )
 
Theoremtvclvec 21291 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LVec )
 
12.3  Uniform Structures and Spaces
 
12.3.1  Uniform structures
 
Syntaxcust 21292 Extend class notation with the class function of uniform structures.
 class 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  =  ( x  e.  _V  |->  { u  |  ( u 
 C_  ~P ( x  X.  x )  /\  ( x  X.  x )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( x  X.  x ) ( v  C_  w  ->  w  e.  u )  /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  x ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremustfn 21294 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |- UnifOn  Fn  _V
 
Theoremustval 21295* The class of all uniform structures for a base  X. (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  (UnifOn `  X )  =  { u  |  ( u  C_  ~P ( X  X.  X )  /\  ( X  X.  X )  e.  u  /\  A. v  e.  u  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  u ) 
 /\  A. w  e.  u  ( v  i^i  w )  e.  u  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  u  /\  E. w  e.  u  ( w  o.  w ) 
 C_  v ) ) ) } )
 
Theoremisust 21296* The predicate " U is a uniform structure with base  X." (Contributed by Thierry Arnoux, 15-Nov-2017.)
 |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X )  <->  ( U  C_  ~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U ) 
 /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
 (  _I  |`  X ) 
 C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
 C_  v ) ) ) ) )
 
Theoremustssxp 21297 Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X ) )
 
Theoremustssel 21298 A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X ) )  ->  ( V  C_  W  ->  W  e.  U ) )
 
Theoremustbasel 21299 The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
 |-  ( U  e.  (UnifOn `  X )  ->  ( X  X.  X )  e.  U )
 
Theoremustincl 21300 A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
 |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  e.  U )  ->  ( V  i^i  W )  e.  U )
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