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Theorem isvc 21137
Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
isvc.1  |-  X  =  ran  G
Assertion
Ref Expression
isvc  |-  ( <. G ,  S >.  e. 
CVec OLD  <->  ( G  e. 
AbelOp  /\  S : ( CC  X.  X ) --> X  /\  A. x  e.  X  ( (
1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) )
Distinct variable groups:    x, y,
z, G    x, S, y, z    x, X, z
Allowed substitution hint:    X( y)

Proof of Theorem isvc
StepHypRef Expression
1 vcex 21136 . 2  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
2 elex 2796 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
32adantr 451 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  G  e.  _V )
4 cnex 8818 . . . . . . 7  |-  CC  e.  _V
5 ablogrpo 20951 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
6 isvc.1 . . . . . . . . 9  |-  X  =  ran  G
7 rnexg 4940 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
86, 7syl5eqel 2367 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  e.  _V )
95, 8syl 15 . . . . . . 7  |-  ( G  e.  AbelOp  ->  X  e.  _V )
10 xpexg 4800 . . . . . . 7  |-  ( ( CC  e.  _V  /\  X  e.  _V )  ->  ( CC  X.  X
)  e.  _V )
114, 9, 10sylancr 644 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( CC  X.  X )  e.  _V )
12 fex 5749 . . . . . 6  |-  ( ( S : ( CC 
X.  X ) --> X  /\  ( CC  X.  X )  e.  _V )  ->  S  e.  _V )
1311, 12sylan2 460 . . . . 5  |-  ( ( S : ( CC 
X.  X ) --> X  /\  G  e.  AbelOp )  ->  S  e.  _V )
1413ancoms 439 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  S  e.  _V )
153, 14jca 518 . . 3  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  ( G  e. 
_V  /\  S  e.  _V ) )
16153adant3 975 . 2  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X  /\  A. x  e.  X  ( ( 1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) )  -> 
( G  e.  _V  /\  S  e.  _V )
)
176isvclem 21133 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( <. G ,  S >.  e.  CVec OLD  <->  ( G  e. 
AbelOp  /\  S : ( CC  X.  X ) --> X  /\  A. x  e.  X  ( (
1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) ) )
181, 16, 17pm5.21nii 342 1  |-  ( <. G ,  S >.  e. 
CVec OLD  <->  ( G  e. 
AbelOp  /\  S : ( CC  X.  X ) --> X  /\  A. x  e.  X  ( (
1 S x )  =  x  /\  A. y  e.  CC  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  CC  (
( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643    X. cxp 4687   ran crn 4690   -->wf 5251  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   GrpOpcgr 20853   AbelOpcablo 20948   CVec
OLDcvc 21101
This theorem is referenced by:  isvci  21138  tcnvec  25690
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-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793
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-pw 3627  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-ablo 20949  df-vc 21102
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