HomeHome Metamath Proof Explorer
Theorem List (p. 179 of 316)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-31527)
 

Theorem List for Metamath Proof Explorer - 17801-17900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtgpt0 17801 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 17802* 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 17803* Lemma for divstgp 17804. (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 17804 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 17805 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 17806 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 17807 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 17808 Extend class notation to include infinite group sums in a topological group.
 class tsums
 
Definitiondf-tsms 17809* 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 12159) and  gsumg (df-gsum 13405), 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 17810* 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 17811 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 17812* 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 17813* 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 17814 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 14398 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 17815* 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 17816* 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 17817 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 17818* 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 17819 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 17820 One half of tgptsmscls 17832, 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 17821 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 17822 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 17823 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 17824* 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 17825 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 17826 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 17827 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 17828 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 17829 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 17830 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 17831 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 17832 A sum in a topological group is uniquely determined up to a coset of  cls ( { 0 } ), which is a normal subgroup by clsnsg 17792, 0nsg 14662. (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 17833 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 17834 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 17835* Lemma for tsmsxp 17837. (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 17836* Lemma for tsmsxp 17837. (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 17837* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 15227 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 17835 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 17838 The class of all topological division rings.
 class  TopRing
 
Syntaxctdrg 17839 The class of all topological division rings.
 class TopDRing
 
Syntaxctlm 17840 The class of all topological modules.
 class TopMod
 
Syntaxctvc 17841 The class of all topological vector spaces.
 class  TopVec
 
Definitiondf-trg 17842 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 17843 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 17844 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 17845 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 17846 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 17847 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 17848 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 17849 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 17850 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopGrp )
 
Theoremtrgtmd2 17851 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e. TopMnd )
 
Theoremtrgtps 17852 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  TopSp )
 
Theoremtrgrng 17853 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Ring )
 
Theoremtrggrp 17854 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e.  TopRing  ->  R  e.  Grp )
 
Theoremtdrgtrg 17855 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 TopRing )
 
Theoremtdrgdrng 17856 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. 
 DivRing )
 
Theoremtdrgrng 17857 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  Ring )
 
Theoremtdrgtmd 17858 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e. TopMnd )
 
Theoremtdrgtps 17859 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. TopDRing  ->  R  e.  TopSp )
 
Theoremistdrg2 17860 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 17861 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 17862 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 17863 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 17864* Continuity of ring multiplication; analogue of cnmpt12f 17360 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 17865* Continuity of ring multiplication; analogue of cnmpt22f 17369 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 17866 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 17867 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 17868 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 ) )
 
Theoremtlmtmd 17869 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e. TopMnd )
 
Theoremtlmtps 17870 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopSp )
 
Theoremtlmlmod 17871 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  LMod )
 
Theoremtlmtrg 17872 The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopRing )
 
Theoremtlmscatps 17873 The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. TopMod  ->  F  e.  TopSp )
 
Theoremistvc 17874 A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  <->  ( W  e. TopMod  /\  F  e. TopDRing ) )
 
Theoremtvctdrg 17875 The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  TopVec  ->  F  e. TopDRing )
 
Theoremcnmpt1vsca 17876* Continuity of scalar multiplication; analogue of cnmpt12f 17360 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B ) )  e.  ( L  Cn  J ) )
 
Theoremcnmpt2vsca 17877* Continuity of scalar multiplication; analogue of cnmpt22f 17369 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  J  =  (
 TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   &    |-  ( ph  ->  W  e. TopMod )   &    |-  ( ph  ->  L  e.  (TopOn `  X ) )   &    |-  ( ph  ->  M  e.  (TopOn `  Y ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L 
 tX  M )  Cn  J ) )   =>    |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A 
 .x.  B ) )  e.  ( ( L  tX  M )  Cn  J ) )
 
Theoremtlmtgp 17878 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e. TopMod  ->  W  e.  TopGrp )
 
Theoremtvctlm 17879 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e. TopMod )
 
Theoremtvclmod 17880 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LMod )
 
Theoremtvclvec 17881 A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( W  e.  TopVec  ->  W  e.  LVec )
 
11.3  Metric spaces
 
11.3.1  Basic metric space properties
 
Syntaxcxme 17882 Extend class notation with the class of all extended metric spaces.
 class  * MetSp
 
Syntaxcmt 17883 Extend class notation with the class of all metric spaces.
 class  MetSp
 
Syntaxctmt 17884 Extend class notation with the function mapping a metric to a metric space.
 class toMetSp
 
Definitiondf-xms 17885 Define the (proper) class of all extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- 
 * MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  ( ( dist `  f
 )  |`  ( ( Base `  f )  X.  ( Base `  f ) ) ) ) }
 
Definitiondf-ms 17886 Define the (proper) class of all metric spaces. (Contributed by NM, 27-Aug-2006.)
 |- 
 MetSp  =  { f  e.  * MetSp  |  (
 ( dist `  f )  |`  ( ( Base `  f
 )  X.  ( Base `  f ) ) )  e.  ( Met `  ( Base `  f ) ) }
 
Definitiondf-tms 17887 Define the function mapping a metric to a metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |- toMetSp  =  ( d  e.  U. ran  * Met  |->  ( { <. ( Base `  ndx ) , 
 dom  dom  d >. ,  <. (
 dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
 
Theoremismet 17888* Express the predicate " D is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( Met `  X )  <->  ( D :
 ( X  X.  X )
 --> RR  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x )  +  ( z D y ) ) ) ) ) )
 
Theoremisxmet 17889* Express the predicate " D is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( X  e.  A  ->  ( D  e.  ( * Met `  X )  <->  ( D : ( X  X.  X ) --> RR*  /\  A. x  e.  X  A. y  e.  X  ( ( ( x D y )  =  0  <->  x  =  y
 )  /\  A. z  e.  X  ( x D y )  <_  (
 ( z D x ) + e ( z D y ) ) ) ) ) )
 
Theoremismeti 17890* Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  X  e.  _V   &    |-  D : ( X  X.  X ) --> RR   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( ( x D y )  =  0  <->  x  =  y
 ) )   &    |-  ( ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( x D y )  <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  D  e.  ( Met `  X )
 
Theoremisxmetd 17891* Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( x D y )  =  0  <-> 
 x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x D y )  <_  ( ( z D x ) + e
 ( z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremisxmet2d 17892* It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample:  D ( x ,  y )  =  if ( x  =  y ,  0 , 
-oo ) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  D : ( X  X.  X ) --> RR* )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
 0  <_  ( x D y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( x D y )  <_ 
 0 
 <->  x  =  y ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  /\  ( ( z D x )  e. 
 RR  /\  ( z D y )  e. 
 RR ) )  ->  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) )   =>    |-  ( ph  ->  D  e.  ( * Met `  X ) )
 
Theoremmetflem 17893* Lemma for metf 17895 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  ( D : ( X  X.  X ) --> RR  /\  A. x  e.  X  A. y  e.  X  (
 ( ( x D y )  =  0  <-> 
 x  =  y ) 
 /\  A. z  e.  X  ( x D y ) 
 <_  ( ( z D x )  +  (
 z D y ) ) ) ) )
 
Theoremxmetf 17894 Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
 
Theoremmetf 17895 Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
 |-  ( D  e.  ( Met `  X )  ->  D : ( X  X.  X ) --> RR )
 
Theoremxmetcl 17896 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( A D B )  e.  RR* )
 
Theoremmetcl 17897 Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
 |-  ( ( D  e.  ( Met `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  RR )
 
Theoremismet2 17898 An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  <->  ( D  e.  ( * Met `  X )  /\  D : ( X  X.  X ) --> RR ) )
 
Theoremmetxmet 17899 A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  ( D  e.  ( Met `  X )  ->  D  e.  ( * Met `  X ) )
 
Theoremxmetdmdm 17900 Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |-  ( D  e.  ( * Met `  X )  ->  X  =  dom  dom  D )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31527
  Copyright terms: Public domain < Previous  Next >