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Theorem isnv 25622
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 25621 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
2 vcex 25590 . . . . 5  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
32adantr 463 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 isnv.1 . . . . . . 7  |-  X  =  ran  G
52simpld 457 . . . . . . . 8  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  G  e.  _V )
6 rnexg 6631 . . . . . . . 8  |-  ( G  e.  _V  ->  ran  G  e.  _V )
75, 6syl 16 . . . . . . 7  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ran  G  e. 
_V )
84, 7syl5eqel 2474 . . . . . 6  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  X  e.  _V )
9 fex 6046 . . . . . 6  |-  ( ( N : X --> RR  /\  X  e.  _V )  ->  N  e.  _V )
108, 9sylan2 472 . . . . 5  |-  ( ( N : X --> RR  /\  <. G ,  S >.  e. 
CVecOLD )  ->  N  e.  _V )
1110ancoms 451 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  N  e.  _V )
12 df-3an 973 . . . 4  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
133, 11, 12sylanbrc 662 . . 3  |-  ( (
<. G ,  S >.  e. 
CVecOLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
14133adant3 1014 . 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 25620 . 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 351 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   <.cop 3950   class class class wbr 4367   ran crn 4914   -->wf 5492   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403    + caddc 9406    x. cmul 9408    <_ cle 9540   abscabs 13069  GIdcgi 25306   CVecOLDcvc 25555   NrmCVeccnv 25594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  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 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-vc 25556  df-nv 25602
This theorem is referenced by:  isnvi  25623  nvi  25624
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