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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnextfun 21001 If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Haus
 )  /\  ( F : A --> B  /\  A  C_  C ) )  ->  Fun  ( ( JCnExt K ) `  F ) )
 
Theoremcnextfvval 21002* The value of the continuous extension of a given function  F at a point  X. (Contributed by Thierry Arnoux, 21-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A
 --> B )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  C )   &    |-  (
 ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   =>    |-  ( ( ph  /\  X  e.  C ) 
 ->  ( ( ( JCnExt
 K ) `  F ) `  X )  = 
 U. ( ( K 
 fLimf  ( ( ( nei `  J ) `  { X } )t  A ) ) `  F ) )
 
Theoremcnextf 21003* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A
 --> B )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  C )   &    |-  (
 ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   =>    |-  ( ph  ->  ( ( JCnExt K ) `
  F ) : C --> B )
 
Theoremcnextcn 21004* Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology  J on  C, a subset  A dense in  C, this states a condition for  F from  A to a regular space  K to be extensible by continuity (Contributed by Thierry Arnoux, 1-Jan-2018.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A
 --> B )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  C )   &    |-  (
 ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   &    |-  ( ph  ->  K  e.  Reg )   =>    |-  ( ph  ->  (
 ( JCnExt K ) `  F )  e.  ( J  Cn  K ) )
 
Theoremcnextfres1 21005*  F and its extension by continuity agree on the domain of 
F. (Contributed by Thierry Arnoux, 17-Jan-2018.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F : A
 --> B )   &    |-  ( ph  ->  A 
 C_  C )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  C )   &    |-  (
 ( ph  /\  x  e.  C )  ->  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F )  =/=  (/) )   &    |-  ( ph  ->  K  e.  Reg )   &    |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K ) )   =>    |-  ( ph  ->  ( (
 ( JCnExt K ) `  F )  |`  A )  =  F )
 
Theoremcnextfres 21006  F and its extension by continuity agree on the domain of 
F. (Contributed by Thierry Arnoux, 29-Aug-2020.)
 |-  C  =  U. J   &    |-  B  =  U. K   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  F  e.  ( ( Jt  A )  Cn  K ) )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  (
 ( ( JCnExt K ) `  F ) `  X )  =  ( F `  X ) )
 
12.2.6  Topological groups
 
Syntaxctmd 21007 Extend class notation with the class of all topological monoids.
 class TopMnd
 
Syntaxctgp 21008 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 21009* Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |- TopMnd  =  { f  e.  ( Mnd  i^i  TopSp )  |  [. ( TopOpen `  f )  /  j ]. ( +f `  f )  e.  ( ( j 
 tX  j )  Cn  j ) }
 
Definitiondf-tgp 21010* Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
 |-  TopGrp  =  { f  e.  ( Grp  i^i TopMnd )  | 
 [. ( TopOpen `  f
 )  /  j ]. ( invg `  f
 )  e.  ( j  Cn  j ) }
 
Theoremistmd 21011 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  F  =  ( +f `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  ( G  e. TopMnd  <->  ( G  e.  Mnd  /\  G  e.  TopSp  /\  F  e.  ( ( J  tX  J )  Cn  J ) ) )
 
Theoremtmdmnd 21012 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 21013 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 21014 The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( G  e.  TopGrp  <->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J ) ) )
 
Theoremtgpgrp 21015 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( G  e.  TopGrp  ->  G  e.  Grp )
 
Theoremtgptmd 21016 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 21017 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( G  e.  TopGrp  ->  G  e.  TopSp )
 
Theoremtmdtopon 21018 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  X ) )
 
Theoremtgptopon 21019 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( G  e.  TopGrp  ->  J  e.  (TopOn `  X ) )
 
Theoremtmdcn 21020 In a topological monoid, the operation  F representing the functionalization of the operator slot  +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  F  =  ( +f `  G )   =>    |-  ( G  e. TopMnd  ->  F  e.  ( ( J 
 tX  J )  Cn  J ) )
 
Theoremtgpcn 21021 In a topological group, the operation  F representing the functionalization of the operator slot  +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  F  =  ( +f `  G )   =>    |-  ( G  e.  TopGrp  ->  F  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremtgpinv 21022 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
 
Theoremgrpinvhmeo 21023 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J Homeo J ) )
 
Theoremcnmpt1plusg 21024* Continuity of the group sum; analogue of cnmpt12f 20603 which cannot be used directly because 
+g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( 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  .+  B ) )  e.  ( K  Cn  J ) )
 
Theoremcnmpt2plusg 21025* Continuity of the group sum; analogue of cnmpt22f 20612 which cannot be used directly because  +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( 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 
 .+  B ) )  e.  ( ( K 
 tX  L )  Cn  J ) )
 
Theoremtmdcn2 21026* Write out the definition of continuity of  +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( ( G  e. TopMnd  /\  U  e.  J ) 
 /\  ( X  e.  B  /\  Y  e.  B  /\  ( X  .+  Y )  e.  U )
 )  ->  E. u  e.  J  E. v  e.  J  ( X  e.  u  /\  Y  e.  v  /\  A. x  e.  u  A. y  e.  v  ( x  .+  y )  e.  U ) )
 
Theoremtgpsubcn 21027 In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1 (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  TopGrp  -> 
 .-  e.  ( ( J  tX  J )  Cn  J ) )
 
Theoremistgp2 21028 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  TopGrp  <->  ( G  e.  Grp  /\  G  e.  TopSp  /\  .-  e.  ( ( J  tX  J )  Cn  J ) ) )
 
Theoremtmdmulg 21029* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e. TopMnd  /\  N  e.  NN0 )  ->  ( x  e.  B  |->  ( N  .x.  x ) )  e.  ( J  Cn  J ) )
 
Theoremtgpmulg 21030* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e.  TopGrp  /\  N  e.  ZZ )  ->  ( x  e.  B  |->  ( N 
 .x.  x ) )  e.  ( J  Cn  J ) )
 
Theoremtgpmulg2 21031 In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 21736 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e.  TopGrp  ->  .x.  e.  ( ( ~P ZZ  tX  J )  Cn  J ) )
 
Theoremtmdgsum 21032* In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when  A is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
 |-  J  =  ( TopOpen `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ( G  e. CMnd  /\  G  e. TopMnd  /\  A  e.  Fin )  ->  ( x  e.  ( B  ^m  A )  |->  ( G  gsumg  x ) )  e.  ( ( J  ^ko  ~P A )  Cn  J ) )
 
Theoremtmdgsum2 21033* For any neighborhood  U of  n X, there is a neighborhood  u of  X such that any sum of  n elements in  u sums to an element of  U. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   &    |-  B  =  (
 Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e. TopMnd )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  U  e.  J )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  ( ( # `  A )  .x.  X )  e.  U )   =>    |-  ( ph  ->  E. u  e.  J  ( X  e.  u  /\  A. f  e.  ( u  ^m  A ) ( G  gsumg  f )  e.  U ) )
 
Theoremoppgtmd 21034 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e. TopMnd  ->  O  e. TopMnd )
 
Theoremoppgtgp 21035 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e.  TopGrp  ->  O  e.  TopGrp )
 
Theoremdistgp 21036 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( ( G  e.  Grp  /\  J  =  ~P B )  ->  G  e.  TopGrp )
 
Theoremindistgp 21037 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( ( G  e.  Grp  /\  J  =  { (/) ,  B }
 )  ->  G  e.  TopGrp )
 
Theoremsymgtgp 21038 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
 
Theoremtmdlactcn 21039* The left group action of element  A in a topological monoid 
G is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  ( A 
 .+  x ) )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  (
 ( G  e. TopMnd  /\  A  e.  X )  ->  F  e.  ( J  Cn  J ) )
 
Theoremtgplacthmeo 21040* The left group action of element  A in a topological group 
G is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  ( A 
 .+  x ) )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  J  =  ( TopOpen `  G )   =>    |-  (
 ( G  e.  TopGrp  /\  A  e.  X ) 
 ->  F  e.  ( J
 Homeo J ) )
 
Theoremsubmtmd 21041 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. TopMnd  /\  S  e.  (SubMnd `  G ) )  ->  H  e. TopMnd )
 
Theoremsubgtgp 21042 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G ) )  ->  H  e.  TopGrp )
 
Theoremsubgntr 21043 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21045, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  A  e.  (
 ( int `  J ) `  S ) )  ->  S  e.  J )
 
Theoremopnsubg 21044 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  J  =  ( TopOpen `  G )   =>    |-  ( ( G  e.  TopGrp  /\  S  e.  (SubGrp `  G )  /\  S  e.  J )  ->  S  e.  ( Clsd `  J ) )
 
Theoremclssubg 21045 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 21046 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 21047 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 21048* 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 21049* The identity component, the connected component containing the identity element, is a closed (concompcld 20371) 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 21050* 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 21051 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 21052 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 21053 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 21054 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 21055 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 21056* 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 21057* Lemma for qustgp 21058. (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 21058 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 21059 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 21060 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 21061 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 21062 Extend class notation to include infinite group sums in a topological group.
 class tsums
 
Definitiondf-tsms 21063* 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 13731) and  gsumg (df-gsum 15291), 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 21064* 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 21065 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 21066* 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 21067* 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 21068 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 16504 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 21069* 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 21070* 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 21071 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 21072* 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 21073 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 21074 One half of tgptsmscls 21086, 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 21075 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 21076 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 21077 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 21078* 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 21079 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 21080 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 21081 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 21082 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 21083 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 21084 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 21085 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 21086 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 21046, 0nsg 16804. (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 21087 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 21088 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 21089* Lemma for tsmsxp 21091. (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 21090* Lemma for tsmsxp 21091. (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 21091* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 17534 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21089 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 21092 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 21093 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 21094 The class of all topological modules.
 class TopMod
 
Syntaxctvc 21095 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 21096 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 21097 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 21098 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 21099 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 21100 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 ) )
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