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Theorem isnv 23989
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 23988 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
2 vcex 23957 . . . . 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 6509 . . . . . . . 8  |-  ( G  e.  _V  ->  ran  G  e.  _V )
75, 6syl 16 . . . . . . 7  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ran  G  e. 
_V )
84, 7syl5eqel 2526 . . . . . 6  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  X  e.  _V )
9 fex 5949 . . . . . 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 967 . . . 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 1008 . 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 23987 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971   <.cop 3882   class class class wbr 4291   ran crn 4840   -->wf 5413   ` cfv 5417  (class class class)co 6090   CCcc 9279   RRcr 9280   0cc0 9281    + caddc 9284    x. cmul 9286    <_ cle 9418   abscabs 12722  GIdcgi 23673   CVecOLDcvc 23922   NrmCVeccnv 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-vc 23923  df-nv 23969
This theorem is referenced by:  isnvi  23990  nvi  23991
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