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Theorem isvc 22060
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 22059 . 2  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
2 elex 2964 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
32adantr 452 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  G  e.  _V )
4 cnex 9071 . . . . . . 7  |-  CC  e.  _V
5 ablogrpo 21872 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
6 isvc.1 . . . . . . . . 9  |-  X  =  ran  G
7 rnexg 5131 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
86, 7syl5eqel 2520 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  e.  _V )
95, 8syl 16 . . . . . . 7  |-  ( G  e.  AbelOp  ->  X  e.  _V )
10 xpexg 4989 . . . . . . 7  |-  ( ( CC  e.  _V  /\  X  e.  _V )  ->  ( CC  X.  X
)  e.  _V )
114, 9, 10sylancr 645 . . . . . 6  |-  ( G  e.  AbelOp  ->  ( CC  X.  X )  e.  _V )
12 fex 5969 . . . . . 6  |-  ( ( S : ( CC 
X.  X ) --> X  /\  ( CC  X.  X )  e.  _V )  ->  S  e.  _V )
1311, 12sylan2 461 . . . . 5  |-  ( ( S : ( CC 
X.  X ) --> X  /\  G  e.  AbelOp )  ->  S  e.  _V )
1413ancoms 440 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  S  e.  _V )
153, 14jca 519 . . 3  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  ( G  e. 
_V  /\  S  e.  _V ) )
16153adant3 977 . 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 22056 . 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 343 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   <.cop 3817    X. cxp 4876   ran crn 4879   -->wf 5450  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995   GrpOpcgr 21774   AbelOpcablo 21869   CVec
OLDcvc 22024
This theorem is referenced by:  isvci  22061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-ablo 21870  df-vc 22025
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