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Theorem isvc 25166
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. 
CVecOLD  <->  ( 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 25165 . 2  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
2 elex 3122 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
32adantr 465 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  G  e.  _V )
4 cnex 9572 . . . . . . 7  |-  CC  e.  _V
5 ablogrpo 24978 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
6 isvc.1 . . . . . . . . 9  |-  X  =  ran  G
7 rnexg 6716 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
86, 7syl5eqel 2559 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  e.  _V )
95, 8syl 16 . . . . . . 7  |-  ( G  e.  AbelOp  ->  X  e.  _V )
10 xpexg 6710 . . . . . . 7  |-  ( ( CC  e.  _V  /\  X  e.  _V )  ->  ( CC  X.  X
)  e.  _V )
114, 9, 10sylancr 663 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( CC  X.  X )  e.  _V )
12 fex 6132 . . . . . 6  |-  ( ( S : ( CC 
X.  X ) --> X  /\  ( CC  X.  X )  e.  _V )  ->  S  e.  _V )
1311, 12sylan2 474 . . . . 5  |-  ( ( S : ( CC 
X.  X ) --> X  /\  G  e.  AbelOp )  ->  S  e.  _V )
1413ancoms 453 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  S  e.  _V )
153, 14jca 532 . . 3  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  ( G  e. 
_V  /\  S  e.  _V ) )
16153adant3 1016 . 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 25162 . 2  |-  ( ( G  e.  _V  /\  S  e.  _V )  ->  ( <. G ,  S >.  e.  CVecOLD  <->  ( 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 353 1  |-  ( <. G ,  S >.  e. 
CVecOLD  <->  ( 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033    X. cxp 4997   ran crn 5000   -->wf 5583  (class class class)co 6283   CCcc 9489   1c1 9492    + caddc 9494    x. cmul 9496   GrpOpcgr 24880   AbelOpcablo 24975   CVecOLDcvc 25130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-ablo 24976  df-vc 25131
This theorem is referenced by:  isvci  25167
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