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Theorem List for Metamath Proof Explorer - 18001-18100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremblsscls 18001 If two concentric balls have different radii, the closure of the smaller one is contained in the larger one. (Contributed by Mario Carneiro, 5-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( ( D  e.  ( * Met `  X )  /\  P  e.  X )  /\  ( R  e.  RR*  /\  S  e.  RR*  /\  R  <  S ) )  ->  (
 ( cls `  J ) `  ( P ( ball `  D ) R ) )  C_  ( P ( ball `  D ) S ) )
 
Theoremmetss 18002* Two ways of saying that metric  D generates a finer topology than metric  C. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J 
 C_  K  <->  A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( x (
 ball `  D ) s )  C_  ( x ( ball `  C )
 r ) ) )
 
Theoremmetequiv 18003* Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeffrey Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( J  =  K  <->  A. x  e.  X  ( A. r  e.  RR+  E. s  e.  RR+  ( x ( ball `  D )
 s )  C_  ( x ( ball `  C ) r )  /\  A. a  e.  RR+  E. b  e.  RR+  ( x (
 ball `  C ) b )  C_  ( x ( ball `  D )
 a ) ) ) )
 
Theoremmetequiv2 18004* If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  X ) )  ->  ( A. x  e.  X  A. r  e.  RR+  E. s  e.  RR+  ( s  <_  r  /\  ( x ( ball `  C ) s )  =  ( x (
 ball `  D ) s ) )  ->  J  =  K ) )
 
Theoremmetss2lem 18005* Lemma for metss2 18006. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ( ph  /\  ( x  e.  X  /\  S  e.  RR+ ) ) 
 ->  ( x ( ball `  D ) ( S 
 /  R ) ) 
 C_  ( x (
 ball `  C ) S ) )
 
Theoremmetss2 18006* If the metric  D is "strongly finer" than  C (meaning that there is a positive real constant 
R such that  C ( x ,  y )  <_  R  x.  D (
x ,  y )), then  D generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  ( ph  ->  C  e.  ( Met `  X ) )   &    |-  ( ph  ->  D  e.  ( Met `  X )
 )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x C y )  <_  ( R  x.  ( x D y ) ) )   =>    |-  ( ph  ->  J  C_  K )
 
Theoremcomet 18007* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  F : ( 0 [,]  +oo ) --> RR* )   &    |-  ( ( ph  /\  x  e.  ( 0 [,]  +oo ) )  ->  ( ( F `  x )  =  0  <->  x  =  0 ) )   &    |-  ( ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( x  <_  y  ->  ( F `  x ) 
 <_  ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  ( x  e.  ( 0 [,]  +oo )  /\  y  e.  ( 0 [,]  +oo ) ) )  ->  ( F `  ( x + e y ) )  <_  ( ( F `  x ) + e ( F `  y ) ) )   =>    |-  ( ph  ->  ( F  o.  D )  e.  ( * Met `  X )
 )
 
Theoremstdbdmetval 18008* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( R  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  if ( ( A C B )  <_  R ,  ( A C B ) ,  R ) )
 
Theoremstdbdxmet 18009* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( * Met `  X ) )
 
Theoremstdbdmet 18010* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  R  e.  RR+ )  ->  D  e.  ( Met `  X ) )
 
Theoremstdbdbl 18011* The standard bounded metric corresponding to  C generates the same balls as  C for radii less than  R. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  /\  ( P  e.  X  /\  S  e.  RR*  /\  S  <_  R ) )  ->  ( P ( ball `  D ) S )  =  ( P ( ball `  C ) S ) )
 
Theoremstdbdmopn 18012* The standard bounded metric corresponding to  C generates the same topology as  C. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( ( x C y )  <_  R ,  ( x C y ) ,  R ) )   &    |-  J  =  ( MetOpen `  C )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  J  =  ( MetOpen `  D )
 )
 
Theoremmopnex 18013* The topology generated by an extended metric can also be generated by a true metric. Thus "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  E. d  e.  ( Met `  X ) J  =  ( MetOpen `  d
 ) )
 
Theoremmethaus 18014 The topology generated by a metric space is Hausdorff. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
 
Theoremmet1stc 18015 The topology generated by a metric space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  J  e.  1stc )
 
Theoremmet2ndci 18016 A separable metric space (a metric space with a countable dense subset) is second-countable. (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( ( D  e.  ( * Met `  X )  /\  ( A  C_  X  /\  A  ~<_  om  /\  ( ( cls `  J ) `  A )  =  X ) )  ->  J  e.  2ndc )
 
Theoremmet2ndc 18017* A metric space is second-countable iff it is separable (has a countable dense subset). (Contributed by Mario Carneiro, 13-Apr-2015.)
 |-  J  =  ( MetOpen `  D )   =>    |-  ( D  e.  ( * Met `  X )  ->  ( J  e.  2ndc  <->  E. x  e.  ~P  X ( x  ~<_  om  /\  ( ( cls `  J ) `  x )  =  X ) ) )
 
Theoremmetrest 18018 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
 |-  D  =  ( C  |`  ( Y  X.  Y ) )   &    |-  J  =  (
 MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  Y  C_  X )  ->  ( Jt  Y )  =  K )
 
Theoremressxms 18019 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( K  e.  *
 MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
 
Theoremressms 18020 The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( ( K  e.  MetSp  /\  A  e.  V ) 
 ->  ( Ks  A )  e.  MetSp )
 
Theoremprdsmslem1 18021 Lemma for prdsms 18025. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  D  =  ( dist `  Y )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R : I --> MetSp
 )   =>    |-  ( ph  ->  D  e.  ( Met `  B ) )
 
Theoremprdsxmslem1 18022 Lemma for prdsms 18025. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  D  =  ( dist `  Y )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R : I --> * MetSp )   =>    |-  ( ph  ->  D  e.  ( * Met `  B ) )
 
Theoremprdsxmslem2 18023* Lemma for prdsxms 18024. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  W )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  D  =  ( dist `  Y )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  R : I --> * MetSp )   &    |-  J  =  (
 TopOpen `  Y )   &    |-  V  =  ( Base `  ( R `  k ) )   &    |-  E  =  ( ( dist `  ( R `  k ) )  |`  ( V  X.  V ) )   &    |-  K  =  (
 TopOpen `  ( R `  k ) )   &    |-  C  =  { x  |  E. g ( ( g  Fn  I  /\  A. k  e.  I  (
 g `  k )  e.  ( ( TopOpen  o.  R ) `  k )  /\  E. z  e.  Fin  A. k  e.  ( I  \  z
 ) ( g `  k )  =  U. ( ( TopOpen  o.  R ) `  k ) ) 
 /\  x  =  X_ k  e.  I  (
 g `  k )
 ) }   =>    |-  ( ph  ->  J  =  ( MetOpen `  D )
 )
 
Theoremprdsxms 18024 The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   =>    |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp ) 
 ->  Y  e.  * MetSp )
 
Theoremprdsms 18025 The indexed product structure is a metric space when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( S
 X_s
 R )   =>    |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> MetSp )  ->  Y  e.  MetSp )
 
Theorempwsxms 18026 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  *
 MetSp  /\  I  e.  Fin )  ->  Y  e.  * MetSp )
 
Theorempwsms 18027 The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  MetSp  /\  I  e.  Fin )  ->  Y  e.  MetSp )
 
Theoremxpsxms 18028 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  *
 MetSp  /\  S  e.  * MetSp )  ->  T  e.  *
 MetSp )
 
Theoremxpsms 18029 A binary product of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  MetSp  /\  S  e.  MetSp )  ->  T  e.  MetSp )
 
Theoremtmsxps 18030 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   =>    |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )
 
Theoremtmsxpsmopn 18031 Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   &    |-  J  =  ( MetOpen `  M )   &    |-  K  =  (
 MetOpen `  N )   &    |-  L  =  ( MetOpen `  P )   =>    |-  ( ph  ->  L  =  ( J  tX  K )
 )
 
Theoremtmsxpsval 18032 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   =>    |-  ( ph  ->  ( <. A ,  B >. P
 <. C ,  D >. )  =  sup ( {
 ( A M C ) ,  ( B N D ) } ,  RR*
 ,  <  ) )
 
Theoremtmsxpsval2 18033 Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  P  =  ( dist `  ( (toMetSp `  M )  X.s  (toMetSp `  N )
 ) )   &    |-  ( ph  ->  M  e.  ( * Met `  X ) )   &    |-  ( ph  ->  N  e.  ( * Met `  Y )
 )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   =>    |-  ( ph  ->  ( <. A ,  B >. P
 <. C ,  D >. )  =  if ( ( A M C ) 
 <_  ( B N D ) ,  ( B N D ) ,  ( A M C ) ) )
 
11.3.4  Continuity in metric spaces
 
Theoremmetcnp3 18034* Two ways to express that  F is continuous at  P for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  ( F " ( P ( ball `  C )
 z ) )  C_  ( ( F `  P ) ( ball `  D ) y ) ) ) )
 
Theoremmetcnp 18035* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( P C w )  <  z  ->  ( ( F `  P ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnp2 18036* Two ways to say a mapping from metric  C to metric  D is continuous at point  P. The distance arguments are swapped compared to metcnp 18035 (and Munkres' metcn 18037) for compatibility with df-lm 16907. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
 ( F : X --> Y  /\  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( w C P )  <  z  ->  (
 ( F `  w ) D ( F `  P ) )  < 
 y ) ) ) )
 
Theoremmetcn 18037* Two ways to say a mapping from metric  C to metric  D is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon"  y there is a positive "delta"  z such that a distance less than delta in  C maps to a distance less than epsilon in  D. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) )  ->  ( F  e.  ( J  Cn  K )  <->  ( F : X
 --> Y  /\  A. x  e.  X  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  X  ( ( x C w )  <  z  ->  ( ( F `  x ) D ( F `  w ) )  <  y ) ) ) )
 
Theoremmetcnpi 18038* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 18035. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  A ) )
 
Theoremmetcnpi2 18039* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 18036. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <  x  ->  ( ( F `  y ) D ( F `  P ) )  <  A ) )
 
Theoremmetcnpi3 18040* Epsilon-delta property of a metric space function continuous at  P. A variation of metcnpi2 18039 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   =>    |-  ( ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y ) ) 
 /\  ( F  e.  ( ( J  CnP  K ) `  P ) 
 /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( y C P )  <_  x  ->  ( ( F `  y ) D ( F `  P ) )  <_  A )
 )
 
Theoremtxmetcnp 18041* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  E  e.  ( * Met `  Z ) ) 
 /\  ( A  e.  X  /\  B  e.  Y ) )  ->  ( F  e.  ( ( ( J  tX  K )  CnP  L ) `  <. A ,  B >. )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  ( ( ( A C u )  <  w  /\  ( B D v )  <  w )  ->  ( ( A F B ) E ( u F v ) )  <  z ) ) ) )
 
Theoremtxmetcn 18042* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
 |-  J  =  ( MetOpen `  C )   &    |-  K  =  (
 MetOpen `  D )   &    |-  L  =  ( MetOpen `  E )   =>    |-  (
 ( C  e.  ( * Met `  X )  /\  D  e.  ( * Met `  Y )  /\  E  e.  ( * Met `  Z ) ) 
 ->  ( F  e.  (
 ( J  tX  K )  Cn  L )  <->  ( F :
 ( X  X.  Y )
 --> Z  /\  A. x  e.  X  A. y  e.  Y  A. z  e.  RR+  E. w  e.  RR+  A. u  e.  X  A. v  e.  Y  (
 ( ( x C u )  <  w  /\  ( y D v )  <  w ) 
 ->  ( ( x F y ) E ( u F v ) )  <  z ) ) ) )
 
11.3.5  Examples of metric spaces
 
Theoremdscmet 18043* The discrete metric on any set  X. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )   =>    |-  ( X  e.  V  ->  D  e.  ( Met `  X ) )
 
Theoremdscopn 18044* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
 |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )   =>    |-  ( X  e.  V  ->  ( MetOpen `  D )  =  ~P X )
 
Theoremnrmmetd 18045* Show that a group norm generates a metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  F : X --> RR )   &    |-  (
 ( ph  /\  x  e.  X )  ->  (
 ( F `  x )  =  0  <->  x  =  .0.  ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( F `
  ( x  .-  y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) )   =>    |-  ( ph  ->  ( F  o.  .-  )  e.  ( Met `  X )
 )
 
Theoremabvmet 18046 An absolute value  F generates a metric defined by  d (
x ,  y )  =  F ( x  -  y ), analogously to cnmet 18229. (In fact, the ring structure is not needed at all; the group properties abveq0 15539 and abvtri 15543, abvneg 15547 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  A  =  (AbsVal `  R )   &    |-  .-  =  ( -g `  R )   =>    |-  ( F  e.  A  ->  ( F  o.  .-  )  e.  ( Met `  X ) )
 
11.3.6  Normed algebraic structures
 
Syntaxcnm 18047 Norm of a normed ring.
 class  norm
 
Syntaxcngp 18048 The class of all normed groups.
 class NrmGrp
 
Syntaxctng 18049 Make a normed group from a norm and a group.
 class toNrmGrp
 
Syntaxcnrg 18050 Normed ring.
 class NrmRing
 
Syntaxcnlm 18051 Normed module.
 class NrmMod
 
Syntaxcnvc 18052 Normed vector space.
 class NrmVec
 
Definitiondf-nm 18053* Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w ) ( 0g
 `  w ) ) ) )
 
Definitiondf-ngp 18054 Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- NrmGrp  =  { g  e.  ( Grp  i^i  MetSp )  |  ( ( norm `  g )  o.  ( -g `  g
 ) )  C_  ( dist `  g ) }
 
Definitiondf-tng 18055* Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- toNrmGrp  =  ( g  e.  _V ,  f  e.  _V  |->  ( ( g sSet  <. (
 dist `  ndx ) ,  ( f  o.  ( -g `  g ) )
 >. ) sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  ( f  o.  ( -g `  g ) ) ) >. ) )
 
Definitiondf-nrg 18056 A normed ring is a ring with an induced topology and metric such that the metric is translation-invariant and the norm (distance from 0) is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- NrmRing  =  { w  e. NrmGrp  |  (
 norm `  w )  e.  (AbsVal `  w ) }
 
Definitiondf-nlm 18057* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- NrmMod  =  { w  e.  (NrmGrp  i^i  LMod )  |  [. (Scalar `  w )  /  f ]. ( f  e. NrmRing  /\  A. x  e.  ( Base `  f ) A. y  e.  ( Base `  w )
 ( ( norm `  w ) `  ( x ( .s `  w ) y ) )  =  ( ( ( norm `  f ) `  x )  x.  ( ( norm `  w ) `  y
 ) ) ) }
 
Definitiondf-nvc 18058 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |- NrmVec  =  (NrmMod  i^i  LVec )
 
Theoremnmfval 18059* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  W )   &    |-  X  =  (
 Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  D  =  ( dist `  W )   =>    |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
 
Theoremnmval 18060 The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  W )   &    |-  X  =  (
 Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  D  =  ( dist `  W )   =>    |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
 
Theoremnmfval2 18061* The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  W )   &    |-  X  =  (
 Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( D  |`  ( X  X.  X ) )   =>    |-  ( W  e.  Grp 
 ->  N  =  ( x  e.  X  |->  ( x E  .0.  ) ) )
 
Theoremnmval2 18062 The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  W )   &    |-  X  =  (
 Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( D  |`  ( X  X.  X ) )   =>    |-  ( ( W  e.  Grp  /\  A  e.  X )  ->  ( N `
  A )  =  ( A E  .0.  ) )
 
Theoremnmf2 18063 The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  W )   &    |-  X  =  (
 Base `  W )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( D  |`  ( X  X.  X ) )   =>    |-  ( ( W  e.  Grp  /\  E  e.  ( Met `  X ) )  ->  N : X --> RR )
 
Theoremnmpropd 18064 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  ( Base `  K )  =  ( Base `  L )
 )   &    |-  ( ph  ->  ( +g  `  K )  =  ( +g  `  L ) )   &    |-  ( ph  ->  (
 dist `  K )  =  ( dist `  L )
 )   =>    |-  ( ph  ->  ( norm `  K )  =  ( norm `  L )
 )
 
Theoremnmpropd2 18065* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  K  e.  Grp )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ph  ->  ( ( dist `  K )  |`  ( B  X.  B ) )  =  (
 ( dist `  L )  |`  ( B  X.  B ) ) )   =>    |-  ( ph  ->  (
 norm `  K )  =  ( norm `  L )
 )
 
Theoremisngp 18066 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( G  e. NrmGrp  <->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  C_  D ) )
 
Theoremisngp2 18067 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ( D  |`  ( X  X.  X ) )   =>    |-  ( G  e. NrmGrp  <->  ( G  e.  Grp  /\  G  e.  MetSp  /\  ( N  o.  .-  )  =  E ) )
 
Theoremisngp3 18068* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   &    |-  X  =  ( Base `  G )   =>    |-  ( G  e. NrmGrp  <->  ( G  e.  Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  ( x D y )  =  ( N `  ( x 
 .-  y ) ) ) )
 
Theoremngpgrp 18069 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  Grp )
 
Theoremngpms 18070 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
 
Theoremngpxms 18071 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  * MetSp )
 
Theoremngptps 18072 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  TopSp )
 
Theoremngpds 18073 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A  .-  B ) ) )
 
Theoremngpdsr 18074 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( B  .-  A ) ) )
 
Theoremngpds2 18075 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( ( A  .-  B ) D  .0.  ) )
 
Theoremngpds2r 18076 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( ( B  .-  A ) D  .0.  ) )
 
Theoremngpds3 18077 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  (  .0.  D ( A  .-  B ) ) )
 
Theoremngpds3r 18078 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  (  .0.  D ( B  .-  A ) ) )
 
Theoremngprcan 18079 Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  D  =  ( dist `  G )   =>    |-  (
 ( G  e. NrmGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A  .+  C ) D ( B  .+  C ) )  =  ( A D B ) )
 
Theoremngplcan 18080 Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  D  =  ( dist `  G )   =>    |-  (
 ( ( G  e. NrmGrp  /\  G  e.  Abel )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C  .+  A ) D ( C  .+  B ) )  =  ( A D B ) )
 
Theoremisngp4 18081* Express the property of being a normed group purely in terms of right-translation invariance of the metric instead of using the definition of norm (which itself uses the metric). (Contributed by Mario Carneiro, 29-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( G  e. NrmGrp  <->  ( G  e.  Grp  /\  G  e.  MetSp  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x 
 .+  z ) D ( y  .+  z
 ) )  =  ( x D y ) ) )
 
Theoremngpinvds 18082 Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  I  =  ( inv g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  ( ( ( G  e. NrmGrp  /\  G  e.  Abel ) 
 /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( ( I `  A ) D ( I `  B ) )  =  ( A D B ) )
 
Theoremngpsubcan 18083 Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( dist `  G )   =>    |-  (
 ( G  e. NrmGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A  .-  C ) D ( B  .-  C ) )  =  ( A D B ) )
 
Theoremnmf 18084 The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   =>    |-  ( G  e. NrmGrp  ->  N : X --> RR )
 
Theoremnmcl 18085 The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
 
Theoremnmge0 18086 The norm of a normed group is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X )  ->  0  <_  ( N `  A ) )
 
Theoremnmeq0 18087 The identity is the only element of the group with zero norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X ) 
 ->  ( ( N `  A )  =  0  <->  A  =  .0.  ) )
 
Theoremnmne0 18088 The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  A  =/=  .0.  )  ->  ( N `  A )  =/=  0 )
 
Theoremnmrpcl 18089 The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  A  =/=  .0.  )  ->  ( N `  A )  e.  RR+ )
 
Theoremnminv 18090 The norm of a negated element is the same as the norm of the original element. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X ) 
 ->  ( N `  ( I `  A ) )  =  ( N `  A ) )
 
Theoremnmmtri 18091 The triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .-  B ) ) 
 <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnmsub 18092 The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .-  B ) )  =  ( N `  ( B  .-  A ) ) )
 
Theoremnmrtri 18093 Reverse triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( abs `  (
 ( N `  A )  -  ( N `  B ) ) ) 
 <_  ( N `  ( A  .-  B ) ) )
 
Theoremnm2dif 18094 Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  A )  -  ( N `  B ) ) 
 <_  ( N `  ( A  .-  B ) ) )
 
Theoremnmtri 18095 The triangle inequality for the norm of a sum. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  (
 norm `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e. NrmGrp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `
  ( A  .+  B ) )  <_  ( ( N `  A )  +  ( N `  B ) ) )
 
Theoremnm0 18096 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e. NrmGrp  ->  ( N `  .0.  )  =  0 )
 
Theoremsubgnm 18097 The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  H  =  ( Gs  A )   &    |-  N  =  (
 norm `  G )   &    |-  M  =  ( norm `  H )   =>    |-  ( A  e.  (SubGrp `  G )  ->  M  =  ( N  |`  A )
 )
 
Theoremsubgnm2 18098 A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  H  =  ( Gs  A )   &    |-  N  =  (
 norm `  G )   &    |-  M  =  ( norm `  H )   =>    |-  (
 ( A  e.  (SubGrp `  G )  /\  X  e.  A )  ->  ( M `  X )  =  ( N `  X ) )
 
Theoremsubgngp 18099 A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  H  =  ( Gs  A )   =>    |-  ( ( G  e. NrmGrp  /\  A  e.  (SubGrp `  G ) )  ->  H  e. NrmGrp )
 
Theoremngptgp 18100 A normed abelian group is a topological group (with the topology induced by the metric induced by the norm). (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ( G  e. NrmGrp  /\  G  e.  Abel )  ->  G  e.  TopGrp )
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