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Theorem fldax1 25428
Description: 1st "axiom" of a field. The addition is an abelian group. (Contributed by FL, 11-Jul-2010.)
Hypothesis
Ref Expression
fldax1.1  |-  G  =  ( 1st `  R
)
Assertion
Ref Expression
fldax1  |-  ( R  e.  Fld  ->  G  e.  AbelOp )

Proof of Theorem fldax1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldax1.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2283 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2283 . . . 4  |-  ran  G  =  ran  G
4 eqid 2283 . . . 4  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4fldi 25427 . . 3  |-  ( R  e.  Fld  ->  (
( G  e.  AbelOp  /\  ( 2nd `  R
) : ( ran 
G  X.  ran  G
) --> ran  G )  /\  ( A. x  e. 
ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( ( x ( 2nd `  R ) y ) ( 2nd `  R
) z )  =  ( x ( 2nd `  R ) ( y ( 2nd `  R
) z ) )  /\  ( x ( 2nd `  R ) ( y G z ) )  =  ( ( x ( 2nd `  R ) y ) G ( x ( 2nd `  R ) z ) )  /\  ( ( x G y ) ( 2nd `  R ) z )  =  ( ( x ( 2nd `  R
) z ) G ( y ( 2nd `  R ) z ) ) )  /\  E. x  e.  ran  G A. y  e.  ran  G ( ( x ( 2nd `  R ) y )  =  y  /\  (
y ( 2nd `  R
) x )  =  y ) )  /\  ( ( ( 2nd `  R )  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp  /\  A. x  e. 
ran  G A. y  e.  ran  G ( x ( 2nd `  R
) y )  =  ( y ( 2nd `  R ) x ) ) ) )
65simp1d 967 . 2  |-  ( R  e.  Fld  ->  ( G  e.  AbelOp  /\  ( 2nd `  R ) : ( ran  G  X.  ran  G ) --> ran  G
) )
76simpld 445 1  |-  ( R  e.  Fld  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    \ cdif 3149   {csn 3640    X. cxp 4687   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   AbelOpcablo 20948   Fldcfld 21080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043  df-drngo 21073  df-com2 21078  df-fld 21081
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