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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprdsxmslem1 20901 Lemma for prdsms 20904. 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 20902* Lemma for prdsxms 20903. 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 20903 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 20904 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 20905 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 20906 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 20907 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 20908 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 20909 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 20910 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 20911 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 20912 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 ) ) )
 
12.4.5  Continuity in metric spaces
 
Theoremmetcnp3 20913* 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 20914* 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 20915* 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 20914 (and Munkres' metcn 20916) for compatibility with df-lm 19600. 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 20916* 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 20917* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 20914. (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 20918* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 20915. (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 20919* Epsilon-delta property of a metric space function continuous at  P. A variation of metcnpi2 20918 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 20920* 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 20921* 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 ) ) ) )
 
12.4.6  The uniform structure generated by a metric
 
TheoremmetuvalOLD 20922* Value of the uniform structure generated by metric  D. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD
 `  D )  =  ( ( X  X.  X ) filGen ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) ) )
 
Theoremmetuval 20923* Value of the uniform structure generated by metric  D. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  (metUnif `  D )  =  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
 ) ) ) ) )
 
TheoremmetustelOLD 20924* Define a filter base  F generated by a metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( D  e.  ( *Met `  X )  ->  ( B  e.  F  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a ) ) ) )
 
Theoremmetustel 20925* Define a filter base  F generated by a metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( D  e.  (PsMet `  X )  ->  ( B  e.  F  <->  E. a  e.  RR+  B  =  ( `' D " ( 0 [,) a
 ) ) ) )
 
TheoremmetustssOLD 20926* Range of the elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X ) )
 
Theoremmetustss 20927* Range of the elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  A  C_  ( X  X.  X ) )
 
TheoremmetustrelOLD 20928* Elements of the filter base generated by the metric  D are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  ->  Rel  A )
 
Theoremmetustrel 20929* Elements of the filter base generated by the metric  D are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  Rel  A )
 
TheoremmetusttoOLD 20930* Any two elements of the filter base generated by the metric  D can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F  /\  B  e.  F ) 
 ->  ( A  C_  B  \/  B  C_  A )
 )
 
Theoremmetustto 20931* Any two elements of the filter base generated by the metric  D can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  C_  B  \/  B  C_  A )
 )
 
TheoremmetustidOLD 20932* The identity diagonal is included in all elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  ->  (  _I  |`  X ) 
 C_  A )
 
Theoremmetustid 20933* The identity diagonal is included in all elements of the filter base generated by the metric  D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (  _I  |`  X ) 
 C_  A )
 
TheoremmetustsymOLD 20934* Elements of the filter base generated by the metric  D are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  ->  `' A  =  A )
 
Theoremmetustsym 20935* Elements of the filter base generated by the metric  D are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  `' A  =  A )
 
TheoremmetustexhalfOLD 20936* For any element  A of the filter base generated by the metric  D, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v )  C_  A )
 
Theoremmetustexhalf 20937* For any element  A of the filter base generated by the metric  D, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) ) 
 /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v
 )  C_  A )
 
TheoremmetustfbasOLD 20938* The filter base generated by a metric  D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  F  e.  ( fBas `  ( X  X.  X ) ) )
 
Theoremmetustfbas 20939* The filter base generated by a metric  D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  F  e.  ( fBas `  ( X  X.  X ) ) )
 
TheoremmetustOLD 20940* The uniform structure generated by a metric  D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  ( ( X  X.  X ) filGen F )  e.  (UnifOn `  X ) )
 
Theoremmetust 20941* The uniform structure generated by a metric  D. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  ( ( X  X.  X ) filGen F )  e.  (UnifOn `  X ) )
 
TheoremcfilucfilOLD 20942* Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 21574. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  ( C  e.  (CauFilu `  ( ( X  X.  X ) filGen F ) )  <->  ( C  e.  ( fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y ) ) 
 C_  ( 0 [,) x ) ) ) )
 
Theoremcfilucfil 20943* Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 21574. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  F  =  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  ( C  e.  (CauFilu `  ( ( X  X.  X ) filGen F ) )  <->  ( C  e.  ( fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y ) ) 
 C_  ( 0 [,) x ) ) ) )
 
TheoremmetuustOLD 20944 The uniform structure generated by metric  D is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  (metUnifOLD
 `  D )  e.  (UnifOn `  X )
 )
 
Theoremmetuust 20945 The uniform structure generated by metric  D is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  (metUnif `  D )  e.  (UnifOn `  X )
 )
 
Theoremcfilucfil2OLD 20946* Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 21574. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  ( C  e.  (CauFilu `  (metUnifOLD `  D ) )  <->  ( C  e.  ( fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D " ( y  X.  y ) ) 
 C_  ( 0 [,) x ) ) ) )
 
Theoremcfilucfil2 20947* Given a metric  D and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 21574. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  ( C  e.  (CauFilu `  (metUnif `  D ) )  <-> 
 ( C  e.  ( fBas `  X )  /\  A. x  e.  RR+  E. y  e.  C  ( D "
 ( y  X.  y
 ) )  C_  (
 0 [,) x ) ) ) )
 
Theoremblval2 20948 The ball around a point  P, alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e.  RR+ )  ->  ( P ( ball `  D ) R )  =  ( ( `' D "
 ( 0 [,) R ) ) " { P } ) )
 
Theoremelbl4 20949 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( ( D  e.  (PsMet `  X )  /\  R  e.  RR+ )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( B  e.  ( A (
 ball `  D ) R )  <->  B ( `' D " ( 0 [,) R ) ) A ) )
 
TheoremmetuelOLD 20950* Elementhood in the uniform structure generated by a metric  D (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  ( V  e.  (metUnifOLD `  D ) 
 <->  ( V  C_  ( X  X.  X )  /\  E. w  e.  ran  (
 a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V ) ) )
 
Theoremmetuel 20951* Elementhood in the uniform structure generated by a metric  D (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  ( V  e.  (metUnif `  D )  <->  ( V  C_  ( X  X.  X ) 
 /\  E. w  e.  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) w  C_  V ) ) )
 
Theoremmetuel2 20952* Elementhood in the uniform structure generated by a metric  D (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  U  =  (metUnif `  D )   =>    |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  ( V  e.  U  <->  ( V  C_  ( X  X.  X )  /\  E. d  e.  RR+  A. x  e.  X  A. y  e.  X  ( ( x D y )  < 
 d  ->  x V y ) ) ) )
 
TheoremmetustblOLD 20953* The "section" image of an entourage at a point  P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( D  e.  ( *Met `  X )  /\  V  e.  (metUnifOLD `  D )  /\  P  e.  X )  ->  E. a  e.  ran  ( ball `  D )
 ( P  e.  a  /\  a  C_  ( V
 " { P }
 ) ) )
 
Theoremmetustbl 20954* The "section" image of an entourage at a point  P always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
 |-  ( ( D  e.  (PsMet `  X )  /\  V  e.  (metUnif `  D )  /\  P  e.  X )  ->  E. a  e.  ran  ( ball `  D )
 ( P  e.  a  /\  a  C_  ( V
 " { P }
 ) ) )
 
TheoremmetutopOLD 20955 The topology induced by a uniform structure generated by a metric  D is that metric's open sets. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  (unifTop `  (metUnifOLD
 `  D ) )  =  ( MetOpen `  D ) )
 
Theorempsmetutop 20956 The topology induced by a uniform structure generated by a metric  D is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  ->  (unifTop `  (metUnif `  D ) )  =  ( topGen `
  ran  ( ball `  D ) ) )
 
Theoremxmetutop 20957 The topology induced by a uniform structure generated by an extended metric  D is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018.)
 |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) ) 
 ->  (unifTop `  (metUnif `  D ) )  =  ( MetOpen `  D ) )
 
TheoremxmsuspOLD 20958 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  X  =  ( Base `  F )   &    |-  D  =  ( ( dist `  F )  |`  ( X  X.  X ) )   &    |-  U  =  (UnifSt `  F )   =>    |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnifOLD `  D ) )  ->  F  e. UnifSp )
 
Theoremxmsusp 20959 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.)
 |-  X  =  ( Base `  F )   &    |-  D  =  ( ( dist `  F )  |`  ( X  X.  X ) )   &    |-  U  =  (UnifSt `  F )   =>    |-  ( ( X  =/=  (/)  /\  F  e.  *MetSp  /\  U  =  (metUnif `  D ) )  ->  F  e. UnifSp )
 
Theoremrestmetu 20960 The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
 |-  ( ( A  =/=  (/)  /\  D  e.  (PsMet `  X )  /\  A  C_  X )  ->  ( (metUnif `  D )t  ( A  X.  A ) )  =  (metUnif `  ( D  |`  ( A  X.  A ) ) ) )
 
TheoremmetucnOLD 20961* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 20916. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  U  =  (metUnifOLD `  C )   &    |-  V  =  (metUnifOLD `  D )   &    |-  ( ph  ->  X  =/= 
 (/) )   &    |-  ( ph  ->  Y  =/=  (/) )   &    |-  ( ph  ->  C  e.  ( *Met `  X ) )   &    |-  ( ph  ->  D  e.  ( *Met `  Y )
 )   =>    |-  ( ph  ->  ( F  e.  ( U Cnu V ) 
 <->  ( F : X --> Y  /\  A. d  e.  RR+  E. c  e.  RR+  A. x  e.  X  A. y  e.  X  (
 ( x C y )  <  c  ->  ( ( F `  x ) D ( F `  y ) )  <  d ) ) ) )
 
Theoremmetucn 20962* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 20916. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
 |-  U  =  (metUnif `  C )   &    |-  V  =  (metUnif `  D )   &    |-  ( ph  ->  X  =/= 
 (/) )   &    |-  ( ph  ->  Y  =/=  (/) )   &    |-  ( ph  ->  C  e.  (PsMet `  X ) )   &    |-  ( ph  ->  D  e.  (PsMet `  Y ) )   =>    |-  ( ph  ->  ( F  e.  ( U Cnu V ) 
 <->  ( F : X --> Y  /\  A. d  e.  RR+  E. c  e.  RR+  A. x  e.  X  A. y  e.  X  (
 ( x C y )  <  c  ->  ( ( F `  x ) D ( F `  y ) )  <  d ) ) ) )
 
12.4.7  Examples of metric spaces
 
Theoremdscmet 20963* 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 20964* 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 20965* 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 20966 An absolute value  F generates a metric defined by  d (
x ,  y )  =  F ( x  -  y ), analogously to cnmet 21149. (In fact, the ring structure is not needed at all; the group properties abveq0 17346 and abvtri 17350, abvneg 17354 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 ) )
 
12.4.8  Normed algebraic structures
 
Syntaxcnm 20967 Norm of a normed ring.
 class  norm
 
Syntaxcngp 20968 The class of all normed groups.
 class NrmGrp
 
Syntaxctng 20969 Make a normed group from a norm and a group.
 class toNrmGrp
 
Syntaxcnrg 20970 Normed ring.
 class NrmRing
 
Syntaxcnlm 20971 Normed module.
 class NrmMod
 
Syntaxcnvc 20972 Normed vector space.
 class NrmVec
 
Definitiondf-nm 20973* 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 20974 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 20975* 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 20976 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 20977* 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 20978 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 20979* 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 20980 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 20981* 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 20982 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 20983 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 20984 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 20985* 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 20986 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 20987 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 20988* 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 20989 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  Grp )
 
Theoremngpms 20990 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
 
Theoremngpxms 20991 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  *MetSp )
 
Theoremngptps 20992 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  TopSp )
 
Theoremngpds 20993 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 20994 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 20995 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 20996 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 20997 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 20998 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 20999 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 21000 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 ) )
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