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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxctgp 18101 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 18102* 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 18103* 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 ]. ( inv g `  f
 )  e.  ( j  Cn  j ) }
 
Theoremistmd 18104 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 18105 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 18106 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 18107 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  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  <->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J ) ) )
 
Theoremtgpgrp 18108 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 18109 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 18110 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 18111 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 18112 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 18113 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 18114 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 18115 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  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J  Cn  J ) )
 
Theoremgrpinvhmeo 18116 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  =  ( inv g `  G )   =>    |-  ( G  e.  TopGrp  ->  I  e.  ( J  Homeo  J ) )
 
Theoremcnmpt1plusg 18117* Continuity of the group sum; analogue of cnmpt12f 17698 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 18118* Continuity of the group sum; analogue of cnmpt22f 17707 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 18119* 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 18120 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 18121 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 18122* 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 18123* 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 18124 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 18847 to write the left topology as a subset of the complexes. (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 18125* 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.)
 |-  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  ^ k o  ~P A )  Cn  J ) )
 
Theoremtmdgsum2 18126* 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 18127 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 18128 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 18129 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 18130 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 18131 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
 
Theoremtmdlactcn 18132* 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 18133* 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 18134 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 18135 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 18136 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 18138, 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 18137 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 18138 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 18139 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 18140 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 18141* 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 18142* The identity component, the connected component containing the identity element, is a closed (concompcld 17497) 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 18143* 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 18144 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 18145 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 18146 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 18147 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 18148 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 )
 )
 
Theoremdivstgpopn 18149* 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 )
 
Theoremdivstgplem 18150* Lemma for divstgp 18151. (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 )
 
Theoremdivstgp 18151 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 )
 
Theoremdivstgphaus 18152 The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  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 18153 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 18154 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 )
 
11.2.7  Infinite group sum on topological groups
 
Syntaxctsu 18155 Extend class notation to include infinite group sums in a topological group.
 class tsums
 
Definitiondf-tsms 18156* 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 12480) and  gsumg (df-gsum 13728), 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 18157* 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 18158 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 18159* 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 18160* 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 18161 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 14721 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 18162* 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 18163* 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 18164 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 18165* A Hausdorff group has at most one limit point for a given sum. (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 18166 A Hausdorff group has at most one limit point for a given sum. (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 18167 One half of tgptsmscls 18179, 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 18168 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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   &    |-  J  =  (
 TopOpen `  G )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { ( G 
 gsumg  F ) } )
 )
 
Theoremtsmsid 18169 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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  e.  ( G tsums  F ) )
 
Theoremhaustsmsid 18170 In a Hausdorff group a finite sum sums to exactly the usual number with no extraneous limit points. By setting the topology to the discrete topology (which is Hausdorff), this theorem can be used to turn any tsums theorem into a 
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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   &    |-  J  =  (
 TopOpen `  G )   &    |-  ( ph  ->  J  e.  Haus )   =>    |-  ( ph  ->  ( G tsums  F )  =  { ( G  gsumg 
 F ) } )
 
Theoremtsms0 18171* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |- 
 .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 18172 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 18173 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  G  e.  TopSp
 )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  W )   =>    |-  ( ph  ->  ( G tsums  ( F  |`  W ) )  =  ( G tsums  F ) )
 
Theoremtsmsf1o 18174 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 18175 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 18176 The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  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  o F  .+  H ) ) )
 
Theoremtsmsinv 18177 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  I  =  ( inv g `  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 18178 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  o F  .-  H ) ) )
 
Theoremtgptsmscls 18179 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 18139, 0nsg 14985. (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  ->  X  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  ( ( cls `  J ) `  { X } )
 )
 
Theoremtgptsmscld 18180 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 18181 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  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 18182* Lemma for tsmsxp 18184. (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 18183* Lemma for tsmsxp 18184. (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 18184* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15550 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 18182 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 ) )
 
11.2.8  Topological rings, fields, vector spaces
 
Syntaxctrg 18185 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 18186 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 18187 The class of all topological modules.
 class TopMod
 
Syntaxctvc 18188 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 18189 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 18190 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 18191 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  /\  ( .s f `  w )  e.  ( (
 ( TopOpen `  (Scalar `  w ) )  tX  ( TopOpen `  w ) )  Cn  ( TopOpen `  w )
 ) ) }
 
Definitiondf-tvc 18192 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 18193 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 18194 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 18195 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 18196 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 18197 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 18198 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 18199 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgrng 18200 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
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