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Theorem isnv 25169
Description: The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnv.1  |-  X  =  ran  G
isnv.2  |-  Z  =  (GId `  G )
Assertion
Ref Expression
isnv  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 ) ) ) ) )
Distinct variable groups:    x, y, G    x, N, y    x, S, y    x, X, y
Allowed substitution hints:    Z( x, y)

Proof of Theorem isnv
StepHypRef Expression
1 nvex 25168 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
2 vcex 25137 . . . . 5  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
32adantr 465 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 isnv.1 . . . . . . 7  |-  X  =  ran  G
52simpld 459 . . . . . . . 8  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  e.  _V )
6 rnexg 6708 . . . . . . . 8  |-  ( G  e.  _V  ->  ran  G  e.  _V )
75, 6syl 16 . . . . . . 7  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ran  G  e. 
_V )
84, 7syl5eqel 2554 . . . . . 6  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  X  e.  _V )
9 fex 6126 . . . . . 6  |-  ( ( N : X --> RR  /\  X  e.  _V )  ->  N  e.  _V )
108, 9sylan2 474 . . . . 5  |-  ( ( N : X --> RR  /\  <. G ,  S >.  e. 
CVecOLD )  ->  N  e.  _V )
1110ancoms 453 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  N  e.  _V )
12 df-3an 970 . . . 4  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
133, 11, 12sylanbrc 664 . . 3  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
14133adant3 1011 . 2  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  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 ) ) ) )  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
15 isnv.2 . . 3  |-  Z  =  (GId `  G )
164, 15isnvlem 25167 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  ->  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 ) ) ) ) ) )
171, 14, 16pm5.21nii 353 1  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  <->  ( <. G ,  S >.  e.  CVecOLD 
/\  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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108   <.cop 4028   class class class wbr 4442   ran crn 4995   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483    + caddc 9486    x. cmul 9488    <_ cle 9620   abscabs 13019  GIdcgi 24853   CVecOLDcvc 25102   NrmCVeccnv 25141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-vc 25103  df-nv 25149
This theorem is referenced by:  isnvi  25170  nvi  25171
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