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Theorem isnvi 22130
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnvi.5  |-  X  =  ran  G
isnvi.6  |-  Z  =  (GId `  G )
isnvi.7  |-  <. G ,  S >.  e.  CVec OLD
isnvi.8  |-  N : X
--> RR
isnvi.9  |-  ( ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )
isnvi.10  |-  ( ( y  e.  CC  /\  x  e.  X )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
isnvi.11  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) ) )
isnvi.12  |-  U  = 
<. <. G ,  S >. ,  N >.
Assertion
Ref Expression
isnvi  |-  U  e.  NrmCVec
Distinct variable groups:    x, y, G    x, N, y    x, S, y    x, X, y
Allowed substitution hints:    U( x, y)    Z( x, y)

Proof of Theorem isnvi
StepHypRef Expression
1 isnvi.12 . 2  |-  U  = 
<. <. G ,  S >. ,  N >.
2 isnvi.7 . . 3  |-  <. G ,  S >.  e.  CVec OLD
3 isnvi.8 . . 3  |-  N : X
--> RR
4 isnvi.9 . . . . . 6  |-  ( ( x  e.  X  /\  ( N `  x )  =  0 )  ->  x  =  Z )
54ex 425 . . . . 5  |-  ( x  e.  X  ->  (
( N `  x
)  =  0  ->  x  =  Z )
)
6 isnvi.10 . . . . . . 7  |-  ( ( y  e.  CC  /\  x  e.  X )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
76ancoms 441 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  CC )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
87ralrimiva 2796 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) ) )
9 isnvi.11 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  X )  ->  ( N `  (
x G y ) )  <_  ( ( N `  x )  +  ( N `  y ) ) )
109ralrimiva 2796 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
115, 8, 103jca 1135 . . . 4  |-  ( x  e.  X  ->  (
( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) )
1211rgen 2778 . . 3  |-  A. x  e.  X  ( (
( N `  x
)  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
13 isnvi.5 . . . 4  |-  X  =  ran  G
14 isnvi.6 . . . 4  |-  Z  =  (GId `  G )
1513, 14isnv 22129 . . 3  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVec OLD 
/\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  Z )  /\  A. y  e.  CC  ( N `  ( y S x ) )  =  ( ( abs `  y )  x.  ( N `  x )
)  /\  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
162, 3, 12, 15mpbir3an 1137 . 2  |-  <. <. G ,  S >. ,  N >.  e.  NrmCVec
171, 16eqeltri 2513 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   A.wral 2712   <.cop 3846   class class class wbr 4243   ran crn 4914   -->wf 5485   ` cfv 5489  (class class class)co 6117   CCcc 9026   RRcr 9027   0cc0 9028    + caddc 9031    x. cmul 9033    <_ cle 9159   abscabs 12077  GIdcgi 21813   CVec OLDcvc 22062   NrmCVeccnv 22101
This theorem is referenced by:  cnnv  22206  hhnv  22705  hhssnv  22802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-vc 22063  df-nv 22109
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