MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnv Unicode version

Theorem isnv 21602
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.  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 ) ) ) ) )
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 21601 . 2  |-  ( <. <. G ,  S >. ,  N >.  e.  NrmCVec  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
2 vcex 21570 . . . . 5  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
32adantr 451 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVec OLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V ) )
4 isnv.1 . . . . . . 7  |-  X  =  ran  G
52simpld 445 . . . . . . . 8  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  G  e.  _V )
6 rnexg 5043 . . . . . . . 8  |-  ( G  e.  _V  ->  ran  G  e.  _V )
75, 6syl 15 . . . . . . 7  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ran  G  e. 
_V )
84, 7syl5eqel 2450 . . . . . 6  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  X  e.  _V )
9 fex 5869 . . . . . 6  |-  ( ( N : X --> RR  /\  X  e.  _V )  ->  N  e.  _V )
108, 9sylan2 460 . . . . 5  |-  ( ( N : X --> RR  /\  <. G ,  S >.  e. 
CVec OLD )  ->  N  e.  _V )
1110ancoms 439 . . . 4  |-  ( (
<. G ,  S >.  e. 
CVec OLD  /\  N : X
--> RR )  ->  N  e.  _V )
12 df-3an 937 . . . 4  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  <->  ( ( G  e.  _V  /\  S  e.  _V )  /\  N  e.  _V ) )
133, 11, 12sylanbrc 645 . . 3  |-  ( (
<. G ,  S >.  e. 
CVec OLD  /\  N : X
--> RR )  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
14133adant3 976 . 2  |-  ( (
<. 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 ) ) ) )  ->  ( G  e.  _V  /\  S  e.  _V  /\  N  e. 
_V ) )
15 isnv.2 . . 3  |-  Z  =  (GId `  G )
164, 15isnvlem 21600 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V  /\  N  e.  _V )  ->  ( <. <. 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 ) ) ) ) ) )
171, 14, 16pm5.21nii 342 1  |-  ( <. <. 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873   <.cop 3732   class class class wbr 4125   ran crn 4793   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884    + caddc 8887    x. cmul 8889    <_ cle 9015   abscabs 11926  GIdcgi 21286   CVec OLDcvc 21535   NrmCVeccnv 21574
This theorem is referenced by:  isnvi  21603  nvi  21604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-vc 21536  df-nv 21582
  Copyright terms: Public domain W3C validator