MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vcoprnelem Structured version   Unicode version

Theorem vcoprnelem 25147
Description: Lemma for vcoprne 25148. (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 25116 . . . . 5  |-  Rel  CVecOLD
2 df-rel 5006 . . . . 5  |-  ( Rel 
CVecOLD  <->  CVecOLD  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  CVecOLD  C_  ( _V  X.  _V )
43sseli 3500 . . 3  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  <. G ,  G >.  e.  ( _V 
X.  _V ) )
5 opelxp1 5031 . . 3  |-  ( <. G ,  G >.  e.  ( _V  X.  _V )  ->  G  e.  _V )
64, 5syl 16 . 2  |-  ( <. G ,  G >.  e. 
CVecOLD  ->  G  e.  _V )
7 eqid 2467 . . . . . 6  |-  ran  G  =  ran  G
87isvclem 25146 . . . . 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 24962 . . . . . . 7  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
137grpofo 24877 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
14 fofn 5795 . . . . . . . . . 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 5729 . . . . . . . . 9  |-  ( G : ( CC  X.  ran  G ) --> ran  G  ->  G  Fn  ( CC 
X.  ran  G )
)
17 fndmu 5680 . . . . . . . . 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 24880 . . . . . . . . . 10  |-  ( G  e.  GrpOp  ->  ran  G  =/=  (/) )
20 xpcan2 5442 . . . . . . . . . 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 5014 . . . . . . . 8  |-  ( ran 
G  =  CC  ->  ( CC  X.  ran  G
)  =  ( CC 
X.  CC ) )
2625feq2d 5716 . . . . . . 7  |-  ( ran 
G  =  CC  ->  ( G : ( CC 
X.  ran  G ) --> ran  G  <->  G : ( CC 
X.  CC ) --> ran 
G ) )
27 feq3 5713 . . . . . . 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 1016 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   <.cop 4033    X. cxp 4997   ran crn 5000   Rel wrel 5004    Fn wfn 5581   -->wf 5582   -onto->wfo 5584  (class class class)co 6282   CCcc 9486   1c1 9489    + caddc 9491    x. cmul 9493   GrpOpcgr 24864   AbelOpcablo 24959   CVecOLDcvc 25114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-ov 6285  df-grpo 24869  df-ablo 24960  df-vc 25115
This theorem is referenced by:  vcoprne  25148
  Copyright terms: Public domain W3C validator