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Theorem vcoprnelem 25669
Description: Lemma for vcoprne 25670. (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 25638 . . . . 5  |-  Rel  CVecOLD
2 df-rel 4995 . . . . 5  |-  ( Rel 
CVecOLD  <->  CVecOLD  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  CVecOLD  C_  ( _V  X.  _V )
43sseli 3485 . . 3  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  <. G ,  G >.  e.  ( _V 
X.  _V ) )
5 opelxp1 5021 . . 3  |-  ( <. G ,  G >.  e.  ( _V  X.  _V )  ->  G  e.  _V )
64, 5syl 16 . 2  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
7 eqid 2454 . . . . . 6  |-  ran  G  =  ran  G
87isvclem 25668 . . . . 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 643 . . . 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 482 . . 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 459 . . . . 5  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  G :
( CC  X.  ran  G ) --> ran  G )
12 ablogrpo 25484 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
137grpofo 25399 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
14 fofn 5779 . . . . . . . . . 10  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  G  Fn  ( ran  G  X.  ran  G ) )
1513, 14syl 16 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G  Fn  ( ran  G  X.  ran  G ) )
16 ffn 5713 . . . . . . . . 9  |-  ( G : ( CC  X.  ran  G ) --> ran  G  ->  G  Fn  ( CC 
X.  ran  G )
)
17 fndmu 5664 . . . . . . . . 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 475 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ( ran  G  X.  ran  G )  =  ( CC  X.  ran  G ) )
197grpon0 25402 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
20 xpcan2 5429 . . . . . . . . . 10  |-  ( ran 
G  =/=  (/)  ->  (
( ran  G  X.  ran  G )  =  ( CC  X.  ran  G
)  <->  ran  G  =  CC ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( ( ran  G  X.  ran  G
)  =  ( CC 
X.  ran  G )  <->  ran 
G  =  CC ) )
2221adantr 463 . . . . . . . 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 210 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ran  G  =  CC )
2412, 23sylan 469 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ran  G  =  CC )
25 xpeq2 5003 . . . . . . . 8  |-  ( ran 
G  =  CC  ->  ( CC  X.  ran  G
)  =  ( CC 
X.  CC ) )
2625feq2d 5700 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> ran 
G ) )
27 feq3 5697 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  CC ) --> ran 
G  <->  G : ( CC 
X.  CC ) --> CC ) )
2826, 27bitrd 253 . . . . . 6  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> CC ) )
2924, 28syl 16 . . . . 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 210 . . . 4  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  G :
( CC  X.  CC )
--> CC )
31303adant3 1014 . . 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 16 . 2  |-  ( ( G  e.  _V  /\  <. G ,  G >.  e. 
CVecOLD )  ->  G : ( CC  X.  CC ) --> CC )
336, 32mpancom 667 1  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    C_ wss 3461   (/)c0 3783   <.cop 4022    X. cxp 4986   ran crn 4989   Rel wrel 4993    Fn wfn 5565   -->wf 5566   -onto->wfo 5568  (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-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
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-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-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-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-grpo 25391  df-ablo 25482  df-vc 25637
This theorem is referenced by:  vcoprne  25670
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