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Theorem vcoprnelem 24101
Description: Lemma for vcoprne 24102. (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 24070 . . . . 5  |-  Rel  CVecOLD
2 df-rel 4948 . . . . 5  |-  ( Rel 
CVecOLD  <->  CVecOLD  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  CVecOLD  C_  ( _V  X.  _V )
43sseli 3453 . . 3  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  <. G ,  G >.  e.  ( _V 
X.  _V ) )
5 opelxp1 4973 . . 3  |-  ( <. G ,  G >.  e.  ( _V  X.  _V )  ->  G  e.  _V )
64, 5syl 16 . 2  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
7 eqid 2451 . . . . . 6  |-  ran  G  =  ran  G
87isvclem 24100 . . . . 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 23916 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
137grpofo 23831 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
14 fofn 5723 . . . . . . . . . 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 5660 . . . . . . . . 9  |-  ( G : ( CC  X.  ran  G ) --> ran  G  ->  G  Fn  ( CC 
X.  ran  G )
)
17 fndmu 5613 . . . . . . . . 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 23834 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
20 xpcan2 5376 . . . . . . . . . 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 4956 . . . . . . . 8  |-  ( ran 
G  =  CC  ->  ( CC  X.  ran  G
)  =  ( CC 
X.  CC ) )
2625feq2d 5648 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> ran 
G ) )
27 feq3 5645 . . . . . . 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 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   _Vcvv 3071    C_ wss 3429   (/)c0 3738   <.cop 3984    X. cxp 4939   ran crn 4942   Rel wrel 4946    Fn wfn 5514   -->wf 5515   -onto->wfo 5517  (class class class)co 6193   CCcc 9384   1c1 9387    + caddc 9389    x. cmul 9391   GrpOpcgr 23818   AbelOpcablo 23913   CVecOLDcvc 24068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-ov 6196  df-grpo 23823  df-ablo 23914  df-vc 24069
This theorem is referenced by:  vcoprne  24102
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