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Theorem isabloda 20966
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 20964 . 2  |-  ( ph  ->  G  e.  GrpOp )
8 grporndm 20877 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
97, 8syl 15 . . . . . . 7  |-  ( ph  ->  ran  G  =  dom  dom 
G )
10 fdm 5393 . . . . . . . . . 10  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
112, 10syl 15 . . . . . . . . 9  |-  ( ph  ->  dom  G  =  ( X  X.  X ) )
1211dmeqd 4881 . . . . . . . 8  |-  ( ph  ->  dom  dom  G  =  dom  ( X  X.  X
) )
13 dmxpid 4898 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
1412, 13syl6eq 2331 . . . . . . 7  |-  ( ph  ->  dom  dom  G  =  X )
159, 14eqtrd 2315 . . . . . 6  |-  ( ph  ->  ran  G  =  X )
1615eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  ran  G  <-> 
x  e.  X ) )
1715eleq2d 2350 . . . . 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 2634 . 2  |-  ( ph  ->  A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) )
23 eqid 2283 . . 3  |-  ran  G  =  ran  G
2423isablo 20950 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251  (class class class)co 5858   GrpOpcgr 20853   AbelOpcablo 20948
This theorem is referenced by:  isablod  20967
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-grpo 20858  df-ablo 20949
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