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Theorem isabloda 20982
Description: Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpda.1  |-  ( ph  ->  X  e.  _V )
isgrpda.2  |-  ( ph  ->  G : ( X  X.  X ) --> X )
isgrpda.3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
isgrpda.4  |-  ( ph  ->  U  e.  X )
isgrpda.5  |-  ( (
ph  /\  x  e.  X )  ->  ( U G x )  =  x )
isablda.6  |-  ( (
ph  /\  x  e.  X )  ->  E. n  e.  X  ( n G x )  =  U )
isablda.7  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
Assertion
Ref Expression
isabloda  |-  ( ph  ->  G  e.  AbelOp )
Distinct variable groups:    ph, x, y, z    n, G, x, y, z    n, X, x, y, z    U, n, x, y, z
Allowed substitution hint:    ph( n)

Proof of Theorem isabloda
StepHypRef Expression
1 isgrpda.1 . . 3  |-  ( ph  ->  X  e.  _V )
2 isgrpda.2 . . 3  |-  ( ph  ->  G : ( X  X.  X ) --> X )
3 isgrpda.3 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
4 isgrpda.4 . . 3  |-  ( ph  ->  U  e.  X )
5 isgrpda.5 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( U G x )  =  x )
6 isablda.6 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  E. n  e.  X  ( n G x )  =  U )
71, 2, 3, 4, 5, 6isgrpda 20980 . 2  |-  ( ph  ->  G  e.  GrpOp )
8 grporndm 20893 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  ran  G  =  dom  dom 
G )
10 fdm 5409 . . . . . . . . . 10  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
112, 10syl 15 . . . . . . . . 9  |-  ( ph  ->  dom  G  =  ( X  X.  X ) )
1211dmeqd 4897 . . . . . . . 8  |-  ( ph  ->  dom  dom  G  =  dom  ( X  X.  X
) )
13 dmxpid 4914 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
1412, 13syl6eq 2344 . . . . . . 7  |-  ( ph  ->  dom  dom  G  =  X )
159, 14eqtrd 2328 . . . . . 6  |-  ( ph  ->  ran  G  =  X )
1615eleq2d 2363 . . . . 5  |-  ( ph  ->  ( x  e.  ran  G  <-> 
x  e.  X ) )
1715eleq2d 2363 . . . . 5  |-  ( ph  ->  ( y  e.  ran  G  <-> 
y  e.  X ) )
1816, 17anbi12d 691 . . . 4  |-  ( ph  ->  ( ( x  e. 
ran  G  /\  y  e.  ran  G )  <->  ( x  e.  X  /\  y  e.  X ) ) )
19 isablda.7 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
2019ex 423 . . . 4  |-  ( ph  ->  ( ( x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) ) )
2118, 20sylbid 206 . . 3  |-  ( ph  ->  ( ( x  e. 
ran  G  /\  y  e.  ran  G )  -> 
( x G y )  =  ( y G x ) ) )
2221ralrimivv 2647 . 2  |-  ( ph  ->  A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) )
23 eqid 2296 . . 3  |-  ran  G  =  ran  G
2423isablo 20966 . 2  |-  ( G  e.  AbelOp 
<->  ( G  e.  GrpOp  /\ 
A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) ) )
257, 22, 24sylanbrc 645 1  |-  ( ph  ->  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   _Vcvv 2801    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964
This theorem is referenced by:  isablod  20983
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-grpo 20874  df-ablo 20965
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