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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtxflf 20801* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  Z ) )   &    |-  ( ph  ->  F : Z --> X )   &    |-  ( ph  ->  G : Z
 --> Y )   &    |-  H  =  ( n  e.  Z  |->  <.
 ( F `  n ) ,  ( G `  n ) >. )   =>    |-  ( ph  ->  (
 <. R ,  S >.  e.  ( ( ( J 
 tX  K )  fLimf  L ) `  H )  <-> 
 ( R  e.  (
 ( J  fLimf  L ) `
  F )  /\  S  e.  ( ( K  fLimf  L ) `  G ) ) ) )
 
Theoremflfcnp2 20802* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  ( ph  ->  L  e.  ( Fil `  Z ) )   &    |-  ( ( ph  /\  x  e.  Z ) 
 ->  A  e.  X )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  Y )   &    |-  ( ph  ->  R  e.  ( ( J 
 fLimf  L ) `  ( x  e.  Z  |->  A ) ) )   &    |-  ( ph  ->  S  e.  ( ( K 
 fLimf  L ) `  ( x  e.  Z  |->  B ) ) )   &    |-  ( ph  ->  O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. ) )   =>    |-  ( ph  ->  ( R O S )  e.  ( ( N 
 fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
 
Theoremfclsval 20803* The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  ( J  fClus  F )  =  if ( X  =  Y ,  |^|_ t  e.  F  ( ( cls `  J ) `  t ) ,  (/) ) )
 
Theoremisfcls 20804* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  <->  ( J  e.  Top  /\  F  e.  ( Fil `  X )  /\  A. s  e.  F  A  e.  ( ( cls `  J ) `  s ) ) )
 
Theoremfclsfil 20805 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  X )
 )
 
Theoremfclstop 20806 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
 
Theoremfclstopon 20807 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X ) ) )
 
Theoremisfcls2 20808* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  A. s  e.  F  A  e.  ( ( cls `  J ) `  s ) ) )
 
Theoremfclsopn 20809* Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  F  ( o  i^i  s )  =/=  (/) ) ) ) )
 
Theoremfclsopni 20810 An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( A  e.  ( J  fClus  F ) 
 /\  ( U  e.  J  /\  A  e.  U  /\  S  e.  F ) )  ->  ( U  i^i  S )  =/=  (/) )
 
Theoremfclselbas 20811 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( A  e.  ( J  fClus  F )  ->  A  e.  X )
 
Theoremfclsneii 20812 A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( A  e.  ( J  fClus  F ) 
 /\  N  e.  (
 ( nei `  J ) `  { A } )  /\  S  e.  F ) 
 ->  ( N  i^i  S )  =/=  (/) )
 
Theoremfclssscls 20813 The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( S  e.  F  ->  ( J  fClus  F ) 
 C_  ( ( cls `  J ) `  S ) )
 
Theoremfclsnei 20814* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) A. s  e.  F  ( n  i^i  s )  =/=  (/) ) ) )
 
Theoremsupnfcls 20815* The filter of supersets of  X  \  U does not cluster at any point of the open set  U. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  A  e.  U )  ->  -.  A  e.  ( J  fClus  { x  e. 
 ~P X  |  ( X  \  U ) 
 C_  x } )
 )
 
Theoremfclsbas 20816* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  F  =  ( X
 filGen B )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  ->  ( A  e.  ( J  fClus  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  B  ( o  i^i  s )  =/=  (/) ) ) ) )
 
Theoremfclsss1 20817 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  J  C_  K )  ->  ( K  fClus  F )  C_  ( J  fClus  F ) )
 
Theoremfclsss2 20818 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  F  C_  G )  ->  ( J  fClus  G )  C_  ( J  fClus  F ) )
 
Theoremfclsrest 20819 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X )  /\  Y  e.  F )  ->  (
 ( Jt  Y )  fClus  ( Ft  Y ) )  =  ( ( J  fClus  F )  i^i  Y ) )
 
Theoremfclscf 20820* Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X ) )  ->  ( J 
 C_  K  <->  A. f  e.  ( Fil `  X ) ( K  fClus  f )  C_  ( J  fClus  f ) ) )
 
Theoremflimfcls 20821 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( J  fLim  F ) 
 C_  ( J  fClus  F )
 
Theoremfclsfnflim 20822* A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( A  e.  ( J  fClus  F )  <->  E. g  e.  ( Fil `  X ) ( F  C_  g  /\  A  e.  ( J  fLim  g ) ) ) )
 
Theoremflimfnfcls 20823* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 20822, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( F  e.  ( Fil `  X )  ->  ( A  e.  ( J  fLim  F )  <->  A. g  e.  ( Fil `  X ) ( F  C_  g  ->  A  e.  ( J  fClus  g ) ) ) )
 
Theoremfclscmpi 20824 Forward direction of fclscmp 20825. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Comp  /\  F  e.  ( Fil `  X ) )  ->  ( J  fClus  F )  =/=  (/) )
 
Theoremfclscmp 20825* A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Comp  <->  A. f  e.  ( Fil `  X ) ( J  fClus  f )  =/=  (/) ) )
 
Theoremuffclsflim 20826 The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  ( J  fClus  F )  =  ( J  fLim  F ) )
 
Theoremufilcmp 20827* A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( X  e. UFL  /\  J  e.  (TopOn `  X ) )  ->  ( J  e.  Comp  <->  A. f  e.  ( UFil `  X ) ( J  fLim  f )  =/= 
 (/) ) )
 
Theoremfcfval 20828 The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fClusf  L ) `  F )  =  ( J  fClus  ( ( X  FilMap  F ) `
  L ) ) )
 
Theoremisfcf 20829* The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fClusf  L ) `
  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  A. s  e.  L  ( o  i^i  ( F
 " s ) )  =/=  (/) ) ) ) )
 
Theoremfcfnei 20830* The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( A  e.  (
 ( J  fClusf  L ) `
  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `  { A }
 ) A. s  e.  L  ( n  i^i  ( F
 " s ) )  =/=  (/) ) ) )
 
Theoremfcfelbas 20831 A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  A  e.  (
 ( J  fClusf  L ) `
  F ) ) 
 ->  A  e.  X )
 
Theoremfcfneii 20832 A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X ) 
 /\  ( A  e.  ( ( J  fClusf  L ) `  F ) 
 /\  N  e.  (
 ( nei `  J ) `  { A } )  /\  S  e.  L ) )  ->  ( N  i^i  ( F " S ) )  =/=  (/) )
 
Theoremflfssfcf 20833 A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  ( ( J  fLimf  L ) `  F ) 
 C_  ( ( J 
 fClusf  L ) `  F ) )
 
Theoremuffcfflf 20834 If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( UFil `  Y )  /\  F : Y --> X )  ->  ( ( J  fClusf  L ) `  F )  =  ( ( J 
 fLimf  L ) `  F ) )
 
Theoremcnpfcfi 20835 Lemma for cnpfcf 20836. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( K  e.  Top  /\  A  e.  ( J 
 fClus  L )  /\  F  e.  ( ( J  CnP  K ) `  A ) )  ->  ( F `  A )  e.  (
 ( K  fClusf  L ) `
  F ) )
 
Theoremcnpfcf 20836* A function  F is continuous at point  A iff  F respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )  /\  A  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  A )  <-> 
 ( F : X --> Y  /\  A. f  e.  ( Fil `  X ) ( A  e.  ( J  fClus  f ) 
 ->  ( F `  A )  e.  ( ( K  fClusf  f ) `  F ) ) ) ) )
 
Theoremcnfcf 20837* Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. f  e.  ( Fil `  X ) A. x  e.  ( J  fClus  f ) ( F `  x )  e.  ( ( K 
 fClusf  f ) `  F ) ) ) )
 
Theoremalexsublem 20838* Lemma for alexsub 20839. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ph  ->  X  e. UFL )   &    |-  ( ph  ->  X  =  U. B )   &    |-  ( ph  ->  J  =  ( topGen `  ( fi `  B ) ) )   &    |-  ( ( ph  /\  ( x  C_  B  /\  X  =  U. x ) ) 
 ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )   &    |-  ( ph  ->  F  e.  ( UFil `  X )
 )   &    |-  ( ph  ->  ( J  fLim  F )  =  (/) )   =>    |- 
 -.  ph
 
Theoremalexsub 20839* The Alexander Subbase Theorem: If 
B is a subbase for the topology  J, and any cover taken from  B has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 20845 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  ( ph  ->  X  e. UFL )   &    |-  ( ph  ->  X  =  U. B )   &    |-  ( ph  ->  J  =  ( topGen `  ( fi `  B ) ) )   &    |-  ( ( ph  /\  ( x  C_  B  /\  X  =  U. x ) ) 
 ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y )   =>    |-  ( ph  ->  J  e.  Comp
 )
 
Theoremalexsubb 20840* Biconditional form of the Alexander Subbase Theorem alexsub 20839. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( X  e. UFL  /\  X  =  U. B )  ->  ( ( topGen `  ( fi `  B ) )  e.  Comp  <->  A. x  e.  ~P  B ( X  =  U. x  ->  E. y  e.  ( ~P x  i^i  Fin ) X  =  U. y ) ) )
 
TheoremalexsubALTlem1 20841* Lemma for alexsubALT 20845. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
 |-  X  =  U. J   =>    |-  ( J  e.  Comp  ->  E. x ( J  =  ( topGen `
  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) ) )
 
TheoremalexsubALTlem2 20842* Lemma for alexsubALT 20845. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
 |-  X  =  U. J   =>    |-  (
 ( ( J  =  ( topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 /\  a  e.  ~P ( fi `  x ) )  /\  A. b  e.  ( ~P a  i^i 
 Fin )  -.  X  =  U. b )  ->  E. u  e.  ( { z  e.  ~P ( fi `  x )  |  ( a  C_  z  /\  A. b  e.  ( ~P z  i^i 
 Fin )  -.  X  =  U. b ) }  u.  { (/) } ) A. v  e.  ( {
 z  e.  ~P ( fi `  x )  |  ( a  C_  z  /\  A. b  e.  ( ~P z  i^i  Fin )  -.  X  =  U. b
 ) }  u.  { (/)
 } )  -.  u  C.  v )
 
TheoremalexsubALTlem3 20843* Lemma for alexsubALT 20845. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  X  =  U. J   =>    |-  (
 ( ( ( ( J  =  ( topGen `  ( fi `  x ) )  /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 /\  a  e.  ~P ( fi `  x ) )  /\  ( u  e.  ~P ( fi
 `  x )  /\  ( a  C_  u  /\  A. b  e.  ( ~P u  i^i  Fin )  -.  X  =  U. b
 ) ) )  /\  w  e.  u )  /\  ( ( t  e.  ( ~P x  i^i  Fin )  /\  w  = 
 |^| t )  /\  ( y  e.  w  /\  -.  y  e.  U. ( x  i^i  u ) ) ) )  ->  E. s  e.  t  A. n  e.  ( ~P ( u  u.  {
 s } )  i^i 
 Fin )  -.  X  =  U. n )
 
TheoremalexsubALTlem4 20844* Lemma for alexsubALT 20845. If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  X  =  U. J   =>    |-  ( J  =  ( topGen `  ( fi `  x ) )  ->  ( A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i  Fin ) X  =  U. d ) 
 ->  A. a  e.  ~P  ( fi `  x ) ( X  =  U. a  ->  E. b  e.  ( ~P a  i^i  Fin ) X  =  U. b ) ) )
 
TheoremalexsubALT 20845* The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  X  =  U. J   =>    |-  ( J  e.  Comp  <->  E. x ( J  =  ( topGen `  ( fi `  x ) ) 
 /\  A. c  e.  ~P  x ( X  =  U. c  ->  E. d  e.  ( ~P c  i^i 
 Fin ) X  =  U. d ) ) )
 
Theoremptcmplem1 20846* Lemma for ptcmp 20852. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   =>    |-  ( ph  ->  ( X  =  U. ( ran 
 S  u.  { X } )  /\  ( Xt_ `  F )  =  (
 topGen `  ( fi `  ( ran  S  u.  { X } ) ) ) ) )
 
Theoremptcmplem2 20847* Lemma for ptcmp 20852. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   =>    |-  ( ph  ->  U_ k  e. 
 { n  e.  A  |  -.  U. ( F `
  n )  ~~  1o } U. ( F `
  k )  e. 
 dom  card )
 
Theoremptcmplem3 20848* Lemma for ptcmp 20852. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   &    |-  K  =  { u  e.  ( F `  k
 )  |  ( `' ( w  e.  X  |->  ( w `  k ) ) " u )  e.  U }   =>    |-  ( ph  ->  E. f ( f  Fn  A  /\  A. k  e.  A  ( f `  k )  e.  ( U. ( F `  k
 )  \  U. K ) ) )
 
Theoremptcmplem4 20849* Lemma for ptcmp 20852. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   &    |-  ( ph  ->  U 
 C_  ran  S )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  -. 
 E. z  e.  ( ~P U  i^i  Fin ) X  =  U. z )   &    |-  K  =  { u  e.  ( F `  k
 )  |  ( `' ( w  e.  X  |->  ( w `  k ) ) " u )  e.  U }   =>    |-  -.  ph
 
Theoremptcmplem5 20850* Lemma for ptcmp 20852. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  S  =  ( k  e.  A ,  u  e.  ( F `  k
 )  |->  ( `' ( w  e.  X  |->  ( w `
  k ) )
 " u ) )   &    |-  X  =  X_ n  e.  A  U. ( F `
  n )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> Comp )   &    |-  ( ph  ->  X  e.  (UFL  i^i  dom  card
 ) )   =>    |-  ( ph  ->  ( Xt_ `  F )  e. 
 Comp )
 
Theoremptcmpg 20851 Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if  ~P ~P X is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 20852). (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  J  =  ( Xt_ `  F )   &    |-  X  =  U. J   =>    |-  ( ( A  e.  V  /\  F : A --> Comp  /\  X  e.  (UFL  i^i  dom  card ) )  ->  J  e.  Comp )
 
Theoremptcmp 20852 Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  ( ( A  e.  V  /\  F : A --> Comp )  ->  ( Xt_ `  F )  e.  Comp )
 
12.2.5  Extension by continuity
 
Syntaxccnext 20853 Extend class notation with the continuous extension operation.
 class CnExt
 
Definitiondf-cnext 20854* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |- CnExt  =  ( j  e.  Top ,  k  e.  Top  |->  ( f  e.  ( U. k  ^pm  U. j )  |->  U_ x  e.  ( ( cls `  j ) `  dom  f ) ( { x }  X.  (
 ( k  fLimf  ( ( ( nei `  j
 ) `  { x } )t  dom  f ) ) `
  f ) ) ) )
 
Theoremcnextval 20855* The function applying continuous extension to a given function  f. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( JCnExt K )  =  ( f  e.  ( U. K  ^pm  U. J )  |->  U_ x  e.  ( ( cls `  J ) `  dom  f ) ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 dom  f ) ) `
  f ) ) ) )
 
Theoremcnextfval 20856* The continuous extension of a given function  F. (Contributed by Thierry Arnoux, 1-Dec-2017.)
 |-  X  =  U. J   &    |-  B  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  X ) ) 
 ->  ( ( JCnExt K ) `  F )  = 
 U_ x  e.  (
 ( cls `  J ) `  A ) ( { x }  X.  (
 ( K  fLimf  ( ( ( nei `  J ) `  { x }
 )t 
 A ) ) `  F ) ) )
 
Theoremcnextrel 20857 In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
 |-  C  =  U. J   &    |-  B  =  U. K   =>    |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C ) ) 
 ->  Rel  ( ( JCnExt
 K ) `  F ) )
 
Theoremcnextfun 20858 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 20859* 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 20860* 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 20861* 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 ) )
 
Theoremcnextfres 20862*  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 )
 
12.2.6  Topological groups
 
Syntaxctmd 20863 Extend class notation with the class of all topological monoids.
 class TopMnd
 
Syntaxctgp 20864 Extend class notation with the class of all topological groups.
 class  TopGrp
 
Definitiondf-tmd 20865* 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 20866* 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 20867 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 20868 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  Mnd )
 
Theoremtmdtps 20869 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e. TopMnd  ->  G  e.  TopSp )
 
Theoremistgp 20870 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 20871 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 20872 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
 |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
 
Theoremtgptps 20873 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 20874 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 20875 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 20876 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 20877 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 20878 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 20879 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 20880* Continuity of the group sum; analogue of cnmpt12f 20461 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 20881* Continuity of the group sum; analogue of cnmpt22f 20470 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 20882* 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 20883 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 20884 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 20885* 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 20886* 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 20887 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 21615 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 20888* 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 20889* 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 20890 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 20891 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 20892 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 20893 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 20894 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  TopGrp )
 
Theoremtmdlactcn 20895* 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 20896* 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 20897 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 20898 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 20899 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 20901, 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 20900 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 ) )
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