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Theorem vcoprnelem 23907
Description: Lemma for vcoprne 23908. (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 23876 . . . . 5  |-  Rel  CVecOLD
2 df-rel 4842 . . . . 5  |-  ( Rel 
CVecOLD  <->  CVecOLD  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  CVecOLD  C_  ( _V  X.  _V )
43sseli 3347 . . 3  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  <. G ,  G >.  e.  ( _V 
X.  _V ) )
5 opelxp1 4867 . . 3  |-  ( <. G ,  G >.  e.  ( _V  X.  _V )  ->  G  e.  _V )
64, 5syl 16 . 2  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
7 eqid 2438 . . . . . 6  |-  ran  G  =  ran  G
87isvclem 23906 . . . . 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 645 . . . 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 484 . . 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 461 . . . . 5  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  G :
( CC  X.  ran  G ) --> ran  G )
12 ablogrpo 23722 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
137grpofo 23637 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
14 fofn 5617 . . . . . . . . . 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 5554 . . . . . . . . 9  |-  ( G : ( CC  X.  ran  G ) --> ran  G  ->  G  Fn  ( CC 
X.  ran  G )
)
17 fndmu 5507 . . . . . . . . 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 477 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ( ran  G  X.  ran  G )  =  ( CC  X.  ran  G ) )
197grpon0 23640 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
20 xpcan2 5270 . . . . . . . . . 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 465 . . . . . . . 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 471 . . . . . 6  |-  ( ( G  e.  AbelOp  /\  G : ( CC  X.  ran  G ) --> ran  G
)  ->  ran  G  =  CC )
25 xpeq2 4850 . . . . . . . 8  |-  ( ran 
G  =  CC  ->  ( CC  X.  ran  G
)  =  ( CC 
X.  CC ) )
2625feq2d 5542 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> ran 
G ) )
27 feq3 5539 . . . . . . 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 1008 . . 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 669 1  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G :
( CC  X.  CC )
--> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967    C_ wss 3323   (/)c0 3632   <.cop 3878    X. cxp 4833   ran crn 4836   Rel wrel 4840    Fn wfn 5408   -->wf 5409   -onto->wfo 5411  (class class class)co 6086   CCcc 9272   1c1 9275    + caddc 9277    x. cmul 9279   GrpOpcgr 23624   AbelOpcablo 23719   CVecOLDcvc 23874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6089  df-grpo 23629  df-ablo 23720  df-vc 23875
This theorem is referenced by:  vcoprne  23908
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