Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fldax5 Unicode version

Theorem fldax5 25535
Description: 5th "axiom" of a field. Existence of a neutral element. (Contributed by FL, 11-Jul-2010.)
Hypotheses
Ref Expression
fldax.1  |-  G  =  ( 1st `  R
)
fldax.2  |-  H  =  ( 2nd `  R
)
fldax.3  |-  X  =  ran  G
Assertion
Ref Expression
fldax5  |-  ( R  e.  Fld  ->  E. x  e.  X  A. y  e.  X  ( y H x )  =  y )
Distinct variable groups:    x, G, y    x, H, y    x, R, y    x, X, y

Proof of Theorem fldax5
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fldax.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 fldax.2 . . . . 5  |-  H  =  ( 2nd `  R
)
3 fldax.3 . . . . 5  |-  X  =  ran  G
4 eqid 2296 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4fldi 25530 . . . 4  |-  ( R  e.  Fld  ->  (
( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X )  /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  (
( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )  /\  ( ( H  |`  ( ( X  \  { (GId `  G ) } )  X.  ( X  \  { (GId `  G ) } ) ) )  e.  GrpOp  /\ 
A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) ) )
65simp2d 968 . . 3  |-  ( R  e.  Fld  ->  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) ) )
76simprd 449 . 2  |-  ( R  e.  Fld  ->  E. x  e.  X  A. y  e.  X  ( (
x H y )  =  y  /\  (
y H x )  =  y ) )
8 simpr 447 . . . 4  |-  ( ( ( x H y )  =  y  /\  ( y H x )  =  y )  ->  ( y H x )  =  y )
98ralimi 2631 . . 3  |-  ( A. y  e.  X  (
( x H y )  =  y  /\  ( y H x )  =  y )  ->  A. y  e.  X  ( y H x )  =  y )
109reximi 2663 . 2  |-  ( E. x  e.  X  A. y  e.  X  (
( x H y )  =  y  /\  ( y H x )  =  y )  ->  E. x  e.  X  A. y  e.  X  ( y H x )  =  y )
117, 10syl 15 1  |-  ( R  e.  Fld  ->  E. x  e.  X  A. y  e.  X  ( y H x )  =  y )
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
  Copyright terms: Public domain W3C validator