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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnmge0 18101 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 18102 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 18103 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 18104 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 18105 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 18106 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 18107 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 18108 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 18109 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 18110 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 18111 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 18112 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 18113 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 18114 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 18115 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 18116* 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 18117 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |- 
 Rel  dom toNrmGrp
 
Theoremtngval 18118 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 18119 Lemma for tngbas 18120 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 18120 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 18121 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 18122 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 18123 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 18124 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 18125 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 18126 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 18127 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 18128 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 18129 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 18130 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 18131 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 18132* 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 18133* 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 18134 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 18135 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 18136 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
 
Theoremnrgrng 18137 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  Ring )
 
Theoremnmmul 18138 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 18139 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 18140 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 18141 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 18142 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 18143 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 18144 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 18145 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  ( R  e. Domn  <->  R  e. NzRing ) )
 
Theoremnrgtgp 18146 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e.  TopGrp )
 
Theoremsubrgnrg 18147 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 18148 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 18149* 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 18150 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 18151 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
 
Theoremnlmlmod 18152 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e.  LMod )
 
Theoremnlmnrg 18153 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 18154 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 18155 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 18156 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 18157 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 18158 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 18159 Lemma for nlmvscn 18161. Compare this proof with the similar elementary proof mulcn2 12035 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 18160* Lemma for nlmvscn 18161. (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 18161 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 18163 and nlmtlm 18167. (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 ) )
 
Theoremrlmnlm 18162 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  (ringLMod `  R )  e. NrmMod )
 
Theoremnrgtrg 18163 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e. NrmRing  ->  R  e. 
 TopRing )
 
Theoremnrginvrcnlem 18164* Lemma for nrginvrcn 18165. Compare this proof with reccn2 12036, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  N  =  ( norm `  R )   &    |-  D  =  ( dist `  R )   &    |-  ( ph  ->  R  e. NrmRing )   &    |-  ( ph  ->  R  e. NzRing )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  T  =  ( if ( 1  <_  ( ( N `  A )  x.  B ) ,  1 ,  ( ( N `  A )  x.  B ) )  x.  (
 ( N `  A )  /  2 ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. y  e.  U  ( ( A D y )  <  x  ->  ( ( I `  A ) D ( I `  y ) )  <  B ) )
 
Theoremnrginvrcn 18165 The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  X  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  ( R  e. NrmRing  ->  I  e.  ( ( Jt  U )  Cn  ( Jt  U ) ) )
 
Theoremnrgtdrg 18166 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( ( R  e. NrmRing  /\  R  e.  DivRing )  ->  R  e. TopDRing )
 
Theoremnlmtlm 18167 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( W  e. NrmMod  ->  W  e. TopMod )
 
Theoremisnvc 18168 A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  <->  ( W  e. NrmMod  /\  W  e.  LVec )
 )
 
Theoremnvcnlm 18169 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e. NrmMod )
 
Theoremnvclvec 18170 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LVec )
 
Theoremnvclmod 18171 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  LMod )
 
Theoremisnvc2 18172 A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. NrmVec  <->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
 
Theoremnvctvc 18173 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( W  e. NrmVec  ->  W  e.  TopVec )
 
Theoremlssnlm 18174 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. NrmMod  /\  U  e.  S )  ->  X  e. NrmMod )
 
Theoremlssnvc 18175 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e. NrmVec  /\  U  e.  S )  ->  X  e. NrmVec )
 
Theoremrlmnvc 18176 The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ( R  e. NrmRing  /\  R  e.  DivRing )  ->  (ringLMod `  R )  e. NrmVec )
 
11.3.7  Normed space homomorphisms (bounded linear operators)
 
Syntaxcnmo 18177 The operator norm function.
 class  normOp
 
Syntaxcnghm 18178 The class of normed group homomorphisms.
 class NGHom
 
Syntaxcnmhm 18179 The class of normed module homomorphisms.
 class NMHom
 
Definitiondf-nmo 18180* Define the norm of an operator between two normed groups (usually vector spaces). This definition produces an operator norm function for each pair of groups  <. s ,  t
>.. Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  normOp  =  ( s  e. NrmGrp ,  t  e. NrmGrp  |->  ( f  e.  ( s  GrpHom  t )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  ( Base `  s )
 ( ( norm `  t
 ) `  ( f `  x ) )  <_  ( r  x.  (
 ( norm `  s ) `  x ) ) } ,  RR* ,  `'  <  ) ) )
 
Definitiondf-nghm 18181* Define the set of normed group homomorphisms between two normed groups. A normed group homomorphism is a group homomorphism which additionally bounds the increase of norm by a fixed real operator. In vector spaces these are also known as bounded linear operators. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- NGHom  =  ( s  e. NrmGrp ,  t  e. NrmGrp 
 |->  ( `' ( s
 normOp t ) " RR ) )
 
Definitiondf-nmhm 18182* Define a normed module homomorphism, also known as a bounded linear operator. This is a module homomorphism (a linear function) such that the operator norm is finite, or equivalently there is a constant  c such that (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- NMHom  =  ( s  e. NrmMod ,  t  e. NrmMod 
 |->  ( ( s LMHom  t
 )  i^i  ( s NGHom  t ) ) )
 
Theoremnmoffn 18183 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  normOp  Fn  (NrmGrp  X. NrmGrp )
 
Theoremreldmnghm 18184 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NGHom
 
Theoremreldmnmhm 18185 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |- 
 Rel  dom NMHom
 
Theoremnmofval 18186* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N  =  ( f  e.  ( S  GrpHom  T )  |->  sup ( { r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( f `  x ) )  <_  ( r  x.  ( L `  x ) ) } ,  RR* ,  `'  <  ) ) )
 
Theoremnmoval 18187* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) ) 
 ->  ( N `  F )  =  sup ( {
 r  e.  ( 0 [,)  +oo )  |  A. x  e.  V  ( M `  ( F `  x ) )  <_  ( r  x.  ( L `  x ) ) } ,  RR* ,  `'  <  ) )
 
Theoremnmogelb 18188* Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  A  e.  RR* )  ->  ( A  <_  ( N `  F ) 
 <-> 
 A. r  e.  (
 0 [,)  +oo ) (
 A. x  e.  V  ( M `  ( F `
  x ) ) 
 <_  ( r  x.  ( L `  x ) ) 
 ->  A  <_  r )
 ) )
 
Theoremnmolb 18189* Any upper bound on the values of a linear operator translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  A  e.  RR  /\  0  <_  A )  ->  ( A. x  e.  V  ( M `  ( F `  x ) )  <_  ( A  x.  ( L `  x ) )  ->  ( N `
  F )  <_  A ) )
 
Theoremnmolb2d 18190* Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  ( ph  ->  S  e. NrmGrp )   &    |-  ( ph  ->  T  e. NrmGrp )   &    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ( ph  /\  ( x  e.  V  /\  x  =/=  .0.  ) ) 
 ->  ( M `  ( F `  x ) ) 
 <_  ( A  x.  ( L `  x ) ) )   =>    |-  ( ph  ->  ( N `  F )  <_  A )
 
Theoremnmof 18191 The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  ->  N : ( S  GrpHom  T ) --> RR* )
 
Theoremnmocl 18192 The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( N `  F )  e.  RR* )
 
Theoremnmoge0 18193 The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  0  <_  ( N `  F ) )
 
Theoremnghmfval 18194 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( S NGHom  T )  =  ( `' N " RR )
 
Theoremisnghm 18195 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NGHom  T )  <->  ( ( S  e. NrmGrp  /\  T  e. NrmGrp )  /\  ( F  e.  ( S  GrpHom  T )  /\  ( N `  F )  e.  RR ) ) )
 
Theoremisnghm2 18196 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
 ( N `  F )  e.  RR )
 )
 
Theoremisnghm3 18197 A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  ->  ( F  e.  ( S NGHom  T )  <-> 
 ( N `  F )  <  +oo ) )
 
Theorembddnghm 18198 A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( ( ( S  e. NrmGrp  /\  T  e. NrmGrp  /\  F  e.  ( S  GrpHom  T ) )  /\  ( A  e.  RR  /\  ( N `  F )  <_  A ) )  ->  F  e.  ( S NGHom  T ) )
 
Theoremnghmcl 18199 A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   =>    |-  ( F  e.  ( S NGHom  T )  ->  ( N `  F )  e. 
 RR )
 
Theoremnmoi 18200 The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   =>    |-  (
 ( F  e.  ( S NGHom  T )  /\  X  e.  V )  ->  ( M `  ( F `  X ) )  <_  ( ( N `  F )  x.  ( L `  X ) ) )
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