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Theorem vcoprnelem 26278
Description: Lemma for vcoprne 26279. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
vcoprnelem  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )

Proof of Theorem vcoprnelem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vcrel 26247 . . . . 5  |-  Rel  CVecOLD
2 df-rel 4846 . . . . 5  |-  ( Rel 
CVecOLD  <->  CVecOLD  C_  ( _V  X.  _V ) )
31, 2mpbi 213 . . . 4  |-  CVecOLD  C_  ( _V  X.  _V )
43sseli 3414 . . 3  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  <. G ,  G >.  e.  ( _V 
X.  _V ) )
5 opelxp1 4872 . . 3  |-  ( <. G ,  G >.  e.  ( _V  X.  _V )  ->  G  e.  _V )
64, 5syl 17 . 2  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
7 eqid 2471 . . . . . 6  |-  ran  G  =  ran  G
87isvclem 26277 . . . . 5  |-  ( ( G  e.  _V  /\  G  e.  _V )  ->  ( <. G ,  G >.  e.  CVecOLD  <->  ( G  e.  AbelOp  /\  G :
( CC  X.  ran  G ) --> ran  G  /\  A. x  e.  ran  G
( ( 1 G x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  G ( y G ( x G z ) )  =  ( ( y G x ) G ( y G z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) G x )  =  ( ( y G x ) G ( z G x ) )  /\  ( ( y  x.  z ) G x )  =  ( y G ( z G x ) ) ) ) ) ) ) )
98anidms 657 . . . 4  |-  ( G  e.  _V  ->  ( <. G ,  G >.  e. 
CVecOLD  <->  ( G  e. 
AbelOp  /\  G : ( CC  X.  ran  G
) --> ran  G  /\  A. x  e.  ran  G
( ( 1 G x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  G ( y G ( x G z ) )  =  ( ( y G x ) G ( y G z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) G x )  =  ( ( y G x ) G ( z G x ) )  /\  ( ( y  x.  z ) G x )  =  ( y G ( z G x ) ) ) ) ) ) ) )
109biimpa 492 . . 3  |-  ( ( G  e.  _V  /\  <. G ,  G >.  e. 
CVecOLD )  ->  ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G  /\  A. x  e.  ran  G ( ( 1 G x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  G ( y G ( x G z ) )  =  ( ( y G x ) G ( y G z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) G x )  =  ( ( y G x ) G ( z G x ) )  /\  ( ( y  x.  z ) G x )  =  ( y G ( z G x ) ) ) ) ) ) )
11 simpr 468 . . . . 5  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  G :
( CC  X.  ran  G ) --> ran  G )
12 ablogrpo 26093 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
137grpofo 26008 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
14 fofn 5808 . . . . . . . . . 10  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G  Fn  ( ran  G  X.  ran  G ) )
1513, 14syl 17 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G  Fn  ( ran  G  X.  ran  G ) )
16 ffn 5739 . . . . . . . . 9  |-  ( G : ( CC  X.  ran  G ) --> ran  G  ->  G  Fn  ( CC 
X.  ran  G )
)
17 fndmu 5687 . . . . . . . . 9  |-  ( ( G  Fn  ( ran 
G  X.  ran  G
)  /\  G  Fn  ( CC  X.  ran  G
) )  ->  ( ran  G  X.  ran  G
)  =  ( CC 
X.  ran  G )
)
1815, 16, 17syl2an 485 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ( ran  G  X.  ran  G )  =  ( CC  X.  ran  G ) )
197grpon0 26011 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
20 xpcan2 5280 . . . . . . . . . 10  |-  ( ran 
G  =/=  (/)  ->  (
( ran  G  X.  ran  G )  =  ( CC  X.  ran  G
)  <->  ran  G  =  CC ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( ( ran  G  X.  ran  G
)  =  ( CC 
X.  ran  G )  <->  ran 
G  =  CC ) )
2221adantr 472 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ( ( ran  G  X.  ran  G
)  =  ( CC 
X.  ran  G )  <->  ran 
G  =  CC ) )
2318, 22mpbid 215 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ran  G  =  CC )
2412, 23sylan 479 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ran  G  =  CC )
25 xpeq2 4854 . . . . . . . 8  |-  ( ran 
G  =  CC  ->  ( CC  X.  ran  G
)  =  ( CC 
X.  CC ) )
2625feq2d 5725 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> ran 
G ) )
27 feq3 5722 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  CC ) --> ran 
G  <->  G : ( CC 
X.  CC ) --> CC ) )
2826, 27bitrd 261 . . . . . 6  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> CC ) )
2924, 28syl 17 . . . . 5  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ( G : ( CC  X.  ran  G ) --> ran  G  <->  G : ( CC  X.  CC ) --> CC ) )
3011, 29mpbid 215 . . . 4  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  G :
( CC  X.  CC )
--> CC )
31303adant3 1050 . . 3  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G  /\  A. x  e.  ran  G ( ( 1 G x )  =  x  /\  A. y  e.  CC  ( A. z  e.  ran  G ( y G ( x G z ) )  =  ( ( y G x ) G ( y G z ) )  /\  A. z  e.  CC  ( ( ( y  +  z ) G x )  =  ( ( y G x ) G ( z G x ) )  /\  ( ( y  x.  z ) G x )  =  ( y G ( z G x ) ) ) ) ) )  ->  G :
( CC  X.  CC )
--> CC )
3210, 31syl 17 . 2  |-  ( ( G  e.  _V  /\  <. G ,  G >.  e. 
CVecOLD )  ->  G : ( CC  X.  CC ) --> CC )
336, 32mpancom 682 1  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    C_ wss 3390   (/)c0 3722   <.cop 3965    X. cxp 4837   ran crn 4840   Rel wrel 4844    Fn wfn 5584   -->wf 5585   -onto->wfo 5587  (class class class)co 6308   CCcc 9555   1c1 9558    + caddc 9560    x. cmul 9562   GrpOpcgr 25995   AbelOpcablo 26090   CVecOLDcvc 26245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-ov 6311  df-grpo 26000  df-ablo 26091  df-vc 26246
This theorem is referenced by:  vcoprne  26279
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