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Theorem isnvi 21169
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 423 . . . . 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 439 . . . . . 6  |-  ( ( x  e.  X  /\  y  e.  CC )  ->  ( N `  (
y S x ) )  =  ( ( abs `  y )  x.  ( N `  x ) ) )
87ralrimiva 2626 . . . . 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 2626 . . . . 5  |-  ( x  e.  X  ->  A. y  e.  X  ( N `  ( x G y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) )
115, 8, 103jca 1132 . . . 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 2608 . . 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 21168 . . 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 1134 . 2  |-  <. <. G ,  S >. ,  N >.  e.  NrmCVec
171, 16eqeltri 2353 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719  GIdcgi 20854   CVec OLDcvc 21101   NrmCVeccnv 21140
This theorem is referenced by:  cnnv  21245  hhnv  21744  hhssnv  21841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-vc 21102  df-nv 21148
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