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Theorem fldax1 25531
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 2296 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2296 . . . 4  |-  ran  G  =  ran  G
4 eqid 2296 . . . 4  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4fldi 25530 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    \ cdif 3162   {csn 3653    X. cxp 4703   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   AbelOpcablo 20964   Fldcfld 21096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059  df-drngo 21089  df-com2 21094  df-fld 21097
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