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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
11.3.6  Normed algebraic structures
 
Syntaxcnm 18101 Norm of a normed ring.
 class  norm
 
Syntaxcngp 18102 The class of all normed groups.
 class NrmGrp
 
Syntaxctng 18103 Make a normed group from a norm and a group.
 class toNrmGrp
 
Syntaxcnrg 18104 Normed ring.
 class NrmRing
 
Syntaxcnlm 18105 Normed module.
 class NrmMod
 
Syntaxcnvc 18106 Normed vector space.
 class NrmVec
 
Definitiondf-nm 18107* 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 18108 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 18109* 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 18110 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 18111* 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 18112 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 18113* 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 18114 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 18115* 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 18116 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 18117 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 18118 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 18119* 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 18120 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 18121 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 18122* 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 18123 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  Grp )
 
Theoremngpms 18124 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  MetSp )
 
Theoremngpxms 18125 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  * MetSp )
 
Theoremngptps 18126 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( G  e. NrmGrp  ->  G  e.  TopSp )
 
Theoremngpds 18127 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 18128 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 18129 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 18130 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 18131 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 18132 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 18133 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 18134 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 18135* 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 18136 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 18137 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 18138 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 18139 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 18140 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 18141 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 18142 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 18143 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 18144 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 18145 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 18146 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 18147 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 18148 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 18149 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 18150 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 18151 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 18152 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 18153 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 18154 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 18155* 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 18156 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |- 
 Rel  dom toNrmGrp
 
Theoremtngval 18157 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 18158 Lemma for tngbas 18159 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 18159 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 18160 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 18161 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 18162 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 18163 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 18164 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 18165 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 18166 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 18167 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 18168 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 18169 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 18170 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 18171* 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 18172* 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 18173 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 18174 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 18175 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
 
Theoremnrgrng 18176 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  Ring )
 
Theoremnmmul 18177 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `
  A )  x.  ( N `  B ) ) )
 
Theoremnrgdsdi 18178 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  ( dist `  R )   =>    |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( N `  A )  x.  ( B D C ) )  =  (
 ( A  .x.  B ) D ( A  .x.  C ) ) )
 
Theoremnrgdsdir 18179 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  ( dist `  R )   =>    |-  ( ( R  e. NrmRing  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B )  x.  ( N `  C ) )  =  (
 ( A  .x.  C ) D ( B  .x.  C ) ) )
 
Theoremnm1 18180 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing )  ->  ( N `  .1.  )  =  1 )
 
Theoremunitnmn0 18181 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  A  e.  U )  ->  ( N `
  A )  =/=  0 )
 
Theoremnminvr 18182 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  N  =  ( norm `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  A  e.  U )  ->  ( N `  ( I `  A ) )  =  ( 1  /  ( N `  A ) ) )
 
Theoremnmdvr 18183 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  N  =  (
 norm `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  ( N `  ( A  ./  B ) )  =  (
 ( N `  A )  /  ( N `  B ) ) )
 
Theoremnrgdomn 18184 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  ( R  e. Domn  <->  R  e. NzRing ) )
 
Theoremnrgtgp 18185 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  TopGrp )
 
Theoremsubrgnrg 18186 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  H  =  ( Gs  A )   =>    |-  ( ( G  e. NrmRing  /\  A  e.  (SubRing `  G ) )  ->  H  e. NrmRing )
 
Theoremtngnrg 18187 Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  T  =  ( R toNrmGrp  F )   &    |-  A  =  (AbsVal `  R )   =>    |-  ( F  e.  A  ->  T  e. NrmRing )
 
Theoremisnlm 18188* 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.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  A  =  (
 norm `  F )   =>    |-  ( W  e. NrmMod  <->  (
 ( W  e. NrmGrp  /\  W  e.  LMod  /\  F  e. NrmRing ) 
 /\  A. x  e.  K  A. y  e.  V  ( N `  ( x 
 .x.  y ) )  =  ( ( A `
  x )  x.  ( N `  y
 ) ) ) )
 
Theoremnmvs 18189 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 norm `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  A  =  (
 norm `  F )   =>    |-  ( ( W  e. NrmMod  /\  X  e.  K  /\  Y  e.  V ) 
 ->  ( N `  ( X  .x.  Y ) )  =  ( ( A `
  X )  x.  ( N `  Y ) ) )
 
Theoremnlmngp 18190 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
 
Theoremnlmlmod 18191 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e.  LMod )
 
Theoremnlmnrg 18192 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmMod  ->  F  e. NrmRing )
 
Theoremnlmngp2 18193 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
 
Theoremnlmdsdi 18194 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  A  =  ( norm `  F )   =>    |-  (
 ( W  e. NrmMod  /\  ( X  e.  K  /\  Y  e.  V  /\  Z  e.  V )
 )  ->  ( ( A `  X )  x.  ( Y D Z ) )  =  (
 ( X  .x.  Y ) D ( X  .x.  Z ) ) )
 
Theoremnlmdsdir 18195 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  N  =  ( norm `  W )   &    |-  E  =  ( dist `  F )   =>    |-  (
 ( W  e. NrmMod  /\  ( X  e.  K  /\  Y  e.  K  /\  Z  e.  V )
 )  ->  ( ( X E Y )  x.  ( N `  Z ) )  =  (
 ( X  .x.  Z ) D ( Y  .x.  Z ) ) )
 
Theoremnlmmul0or 18196 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  O  =  ( 0g `  F )   =>    |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  ( ( A  .x.  B )  =  .0.  <->  ( A  =  O  \/  B  =  .0.  ) ) )
 
Theoremsranlm 18197 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( ( W  e. NrmRing  /\  S  e.  (SubRing `  W ) )  ->  A  e. NrmMod )
 
Theoremnlmvscnlem2 18198 Lemma for nlmvscn 18200. Compare this proof with the similar elementary proof mulcn2 12071 for continuity of multiplication on  CC. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  (
 Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( dist `  F )   &    |-  N  =  ( norm `  W )   &    |-  A  =  ( norm `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  T  =  ( ( R  /  2 )  /  ( ( A `  B )  +  1
 ) )   &    |-  U  =  ( ( R  /  2
 )  /  ( ( N `  X )  +  T ) )   &    |-  ( ph  ->  W  e. NrmMod )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( B E C )  <  U )   &    |-  ( ph  ->  ( X D Y )  <  T )   =>    |-  ( ph  ->  ( ( B  .x.  X ) D ( C  .x.  Y ) )  <  R )
 
Theoremnlmvscnlem1 18199* Lemma for nlmvscn 18200. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  (
 Base `  F )   &    |-  D  =  ( dist `  W )   &    |-  E  =  ( dist `  F )   &    |-  N  =  ( norm `  W )   &    |-  A  =  ( norm `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  T  =  ( ( R  /  2 )  /  ( ( A `  B )  +  1
 ) )   &    |-  U  =  ( ( R  /  2
 )  /  ( ( N `  X )  +  T ) )   &    |-  ( ph  ->  W  e. NrmMod )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  E. r  e.  RR+  A. x  e.  K  A. y  e.  V  ( ( ( B E x )  <  r  /\  ( X D y )  < 
 r )  ->  (
 ( B  .x.  X ) D ( x  .x.  y ) )  <  R ) )
 
Theoremnlmvscn 18200 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18202 and nlmtlm 18206. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s f `  W )   &    |-  J  =  ( TopOpen `  W )   &    |-  K  =  ( TopOpen `  F )   =>    |-  ( W  e. NrmMod  ->  .x.  e.  ( ( K  tX  J )  Cn  J ) )
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