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Theorem isvc 21192
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 21191 . 2  |-  ( <. G ,  S >.  e. 
CVec OLD  ->  ( G  e.  _V  /\  S  e. 
_V ) )
2 elex 2830 . . . . 5  |-  ( G  e.  AbelOp  ->  G  e.  _V )
32adantr 451 . . . 4  |-  ( ( G  e.  AbelOp  /\  S : ( CC  X.  X ) --> X )  ->  G  e.  _V )
4 cnex 8863 . . . . . . 7  |-  CC  e.  _V
5 ablogrpo 21004 . . . . . . . 8  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
6 isvc.1 . . . . . . . . 9  |-  X  =  ran  G
7 rnexg 4977 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
86, 7syl5eqel 2400 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  X  e.  _V )
95, 8syl 15 . . . . . . 7  |-  ( G  e.  AbelOp  ->  X  e.  _V )
10 xpexg 4837 . . . . . . 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 5790 . . . . . 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 21188 . 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 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822   <.cop 3677    X. cxp 4724   ran crn 4727   -->wf 5288  (class class class)co 5900   CCcc 8780   1c1 8783    + caddc 8785    x. cmul 8787   GrpOpcgr 20906   AbelOpcablo 21001   CVec
OLDcvc 21156
This theorem is referenced by:  isvci  21193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-ablo 21002  df-vc 21157
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