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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremcfilucfilOLD 18601* 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 19220. (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 18602* 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 19220. (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 18603 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 18604 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 18605* 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 19220. (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 18606* 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 19220. (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 18607 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 18608 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 18609* 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 18610* 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 18611* 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 18612* 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 18613* 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 18614 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 18615 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 18616 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 18617 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 18618 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 18619 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 18620* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 18575. (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 18621* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 18575. (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 ) ) ) )
 
11.4.7  Examples of metric spaces
 
Theoremdscmet 18622* 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 18623* 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 18624* 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 18625 An absolute value  F generates a metric defined by  d (
x ,  y )  =  F ( x  -  y ), analogously to cnmet 18808. (In fact, the ring structure is not needed at all; the group properties abveq0 15916 and abvtri 15920, abvneg 15924 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 ) )
 
11.4.8  Normed algebraic structures
 
Syntaxcnm 18626 Norm of a normed ring.
 class  norm
 
Syntaxcngp 18627 The class of all normed groups.
 class NrmGrp
 
Syntaxctng 18628 Make a normed group from a norm and a group.
 class toNrmGrp
 
Syntaxcnrg 18629 Normed ring.
 class NrmRing
 
Syntaxcnlm 18630 Normed module.
 class NrmMod
 
Syntaxcnvc 18631 Normed vector space.
 class NrmVec
 
Definitiondf-nm 18632* 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 18633 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 18634* 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 18635 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 18636* 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 18637 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 18638* 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 18639 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 18640* 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 18641 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 18642 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 18643 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 18644* 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 18645 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 18646 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 18647* 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 18648 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  Grp )
 
Theoremngpms 18649 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
 
Theoremngpxms 18650 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  * MetSp )
 
Theoremngptps 18651 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  TopSp )
 
Theoremngpds 18652 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 18653 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 18654 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 18655 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 18656 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 18657 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 18658 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 18659 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 18660* 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 18661 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 18662 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 18663 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 18664 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 18665 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 18666 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 18667 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 18668 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 18669 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 18670 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 18671 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 18672 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 18673 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 18674 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 18675 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 18676 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 18677 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 18678 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 18679 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 )
 
Theoremngppropd 18680* Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( 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  ->  (
 TopOpen `  K )  =  ( TopOpen `  L )
 )   =>    |-  ( ph  ->  ( K  e. NrmGrp  <->  L  e. NrmGrp ) )
 
Theoremreldmtng 18681 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |- 
 Rel  dom toNrmGrp
 
Theoremtngval 18682 Value of the function which augments a given structure  G with a norm  N. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .-  =  ( -g `  G )   &    |-  D  =  ( N  o.  .-  )   &    |-  J  =  ( MetOpen `  D )   =>    |-  ( ( G  e.  V  /\  N  e.  W )  ->  T  =  ( ( G sSet  <. ( dist ` 
 ndx ) ,  D >. ) sSet  <. (TopSet `  ndx ) ,  J >. ) )
 
Theoremtnglem 18683 Lemma for tngbas 18684 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  E  = Slot  K   &    |-  K  e.  NN   &    |-  K  <  9   =>    |-  ( N  e.  V  ->  ( E `  G )  =  ( E `  T ) )
 
Theoremtngbas 18684 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  B  =  (
 Base `  G )   =>    |-  ( N  e.  V  ->  B  =  (
 Base `  T ) )
 
Theoremtngplusg 18685 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( N  e.  V  ->  .+  =  ( +g  `  T ) )
 
Theoremtng0 18686 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( N  e.  V  ->  .0.  =  ( 0g
 `  T ) )
 
Theoremtngmulr 18687 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .x.  =  ( .r `  G )   =>    |-  ( N  e.  V  ->  .x.  =  ( .r `  T ) )
 
Theoremtngsca 18688 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  F  =  (Scalar `  G )   =>    |-  ( N  e.  V  ->  F  =  (Scalar `  T ) )
 
Theoremtngvsca 18689 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .x.  =  ( .s `  G )   =>    |-  ( N  e.  V  ->  .x.  =  ( .s `  T ) )
 
Theoremtngip 18690 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .,  =  ( .i `  G )   =>    |-  ( N  e.  V  ->  .,  =  ( .i `  T ) )
 
Theoremtngds 18691 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  .-  =  ( -g `  G )   =>    |-  ( N  e.  V  ->  ( N  o.  .-  )  =  ( dist `  T ) )
 
Theoremtngtset 18692 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  D  =  (
 dist `  T )   &    |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( G  e.  V  /\  N  e.  W ) 
 ->  J  =  (TopSet `  T ) )
 
Theoremtngtopn 18693 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  D  =  (
 dist `  T )   &    |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( G  e.  V  /\  N  e.  W ) 
 ->  J  =  ( TopOpen `  T ) )
 
Theoremtngnm 18694 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  A  e.  _V   =>    |-  ( ( G  e.  Grp  /\  N : X --> A ) 
 ->  N  =  ( norm `  T ) )
 
Theoremtngngp2 18695 A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  D  =  ( dist `  T )   =>    |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e.  Grp  /\  D  e.  ( Met `  X ) ) ) )
 
Theoremtngngpd 18696* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  N : X --> RR )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  ( ( N `  x )  =  0  <->  x  =  .0.  ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( N `  ( x  .-  y
 ) )  <_  (
 ( N `  x )  +  ( N `  y ) ) )   =>    |-  ( ph  ->  T  e. NrmGrp )
 
Theoremtngngp 18697* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( G toNrmGrp  N )   &    |-  X  =  (
 Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( N : X --> RR  ->  ( T  e. NrmGrp  <->  ( G  e.  Grp  /\  A. x  e.  X  (
 ( ( N `  x )  =  0  <->  x  =  .0.  )  /\  A. y  e.  X  ( N `  ( x 
 .-  y ) ) 
 <_  ( ( N `  x )  +  ( N `  y ) ) ) ) ) )
 
Theoremisnrg 18698 A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. NrmRing  <->  ( R  e. NrmGrp  /\  N  e.  A ) )
 
Theoremnrgabv 18699 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( R  e. NrmRing  ->  N  e.  A )
 
Theoremnrgngp 18700 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
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