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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpinvhmeo 17601 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 17602* Continuity of the group sum; analogue of cnmpt12f 17192 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 17603* Continuity of the group sum; analogue of cnmpt22f 17201 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 17604* 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 17605 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 17606 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 17607* 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 17608* 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 17609 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 18154 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 17610* 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 17611* 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 17612 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 17613 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 17614 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 17615 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 17616 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
 
Theoremtmdlactcn 17617* 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 17618* 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 17619 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 17620 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 17621 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 17623, 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 17622 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 17623 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 17624 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 17625 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 17626* 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 17627* The identity component, the connected component containing the identity element, is a closed (concompcld 16992) 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 17628* 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 17629 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 17630 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 17631 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 17632 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 17633 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 17634* 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 17635* Lemma for divstgp 17636. (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 17636 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 17637 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 17638 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 17639 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 )
 
11.2.6  Infinite group sum on topological groups
 
Syntaxctsu 17640 Extend class notation to include infinite group sums in a topological group.
 class tsums
 
Definitiondf-tsms 17641* 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 12036) and  gsumg (df-gsum 13279), 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 17642* 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 17643 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 17644* 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 17645* 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 17646 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 14233 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 17647* 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 17648* 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 17649 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 17650* 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 17651 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 17652 One half of tgptsmscls 17664, 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 17653 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 17654 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 17655 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 17656* 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 17657 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 17658 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 17659 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 17660 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 17661 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 17662 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 17663 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 17664 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 17624, 0nsg 14497. (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 17665 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 17666 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 17667* Lemma for tsmsxp 17669. (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 17668* Lemma for tsmsxp 17669. (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 17669* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15062 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 17667 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 ) )
 
11.2.7  Topological rings, fields, vector spaces
 
Syntaxctrg 17670 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 17671 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 17672 The class of all topological modules.
 class TopMod
 
Syntaxctvc 17673 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 17674 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 17675 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 17676 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 17677 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 17678 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 17679 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 17680 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 17681 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 17682 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 17683 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 17684 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgrng 17685 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
 
Theoremtrggrp 17686 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Grp )
 
Theoremtdrgtrg 17687 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 TopRing )
 
Theoremtdrgdrng 17688 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 DivRing )
 
Theoremtdrgrng 17689 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  Ring )
 
Theoremtdrgtmd 17690 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. TopMnd )
 
Theoremtdrgtps 17691 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  TopSp )
 
Theoremistdrg2 17692 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 17693 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 17694 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 17695 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 17696* Continuity of ring multiplication; analogue of cnmpt12f 17192 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 17697* Continuity of ring multiplication; analogue of cnmpt22f 17201 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 17698 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 17699 The predicate " W is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  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 17700 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. TopMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
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