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Theorem isvc 25672
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 25671 . 2  |-  ( <. G ,  S >.  e. 
CVecOLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
2 elex 3115 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
32adantr 463 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  G  e.  _V )
4 cnex 9562 . . . . . . 7  |-  CC  e.  _V
5 ablogrpo 25484 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
6 isvc.1 . . . . . . . . 9  |-  X  =  ran  G
7 rnexg 6705 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
86, 7syl5eqel 2546 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  e.  _V )
95, 8syl 16 . . . . . . 7  |-  ( G  e.  AbelOp  ->  X  e.  _V )
10 xpexg 6575 . . . . . . 7  |-  ( ( CC  e.  _V  /\  X  e.  _V )  ->  ( CC  X.  X
)  e.  _V )
114, 9, 10sylancr 661 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( CC  X.  X )  e.  _V )
12 fex 6120 . . . . . 6  |-  ( ( S : ( CC 
X.  X ) --> X  /\  ( CC  X.  X )  e.  _V )  ->  S  e.  _V )
1311, 12sylan2 472 . . . . 5  |-  ( ( S : ( CC 
X.  X ) --> X  /\  G  e.  AbelOp )  ->  S  e.  _V )
1413ancoms 451 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  S  e.  _V )
153, 14jca 530 . . 3  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  ( G  e. 
_V  /\  S  e.  _V ) )
16153adant3 1014 . 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 25668 . 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 351 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   <.cop 4022    X. cxp 4986   ran crn 4989   -->wf 5566  (class class class)co 6270   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486   GrpOpcgr 25386   AbelOpcablo 25481   CVecOLDcvc 25636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-ablo 25482  df-vc 25637
This theorem is referenced by:  isvci  25673
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