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Theorem isabloda 21889
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 21887 . 2  |-  ( ph  ->  G  e.  GrpOp )
8 grporndm 21800 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
97, 8syl 16 . . . . . . 7  |-  ( ph  ->  ran  G  =  dom  dom 
G )
10 fdm 5597 . . . . . . . . . 10  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
112, 10syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  G  =  ( X  X.  X ) )
1211dmeqd 5074 . . . . . . . 8  |-  ( ph  ->  dom  dom  G  =  dom  ( X  X.  X
) )
13 dmxpid 5091 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
1412, 13syl6eq 2486 . . . . . . 7  |-  ( ph  ->  dom  dom  G  =  X )
159, 14eqtrd 2470 . . . . . 6  |-  ( ph  ->  ran  G  =  X )
1615eleq2d 2505 . . . . 5  |-  ( ph  ->  ( x  e.  ran  G  <-> 
x  e.  X ) )
1715eleq2d 2505 . . . . 5  |-  ( ph  ->  ( y  e.  ran  G  <-> 
y  e.  X ) )
1816, 17anbi12d 693 . . . 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 425 . . . 4  |-  ( ph  ->  ( ( x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) ) )
2118, 20sylbid 208 . . 3  |-  ( ph  ->  ( ( x  e. 
ran  G  /\  y  e.  ran  G )  -> 
( x G y )  =  ( y G x ) ) )
2221ralrimivv 2799 . 2  |-  ( ph  ->  A. x  e.  ran  G A. y  e.  ran  G ( x G y )  =  ( y G x ) )
23 eqid 2438 . . 3  |-  ran  G  =  ran  G
2423isablo 21873 . 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 647 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    X. cxp 4878   dom cdm 4880   ran crn 4881   -->wf 5452  (class class class)co 6083   GrpOpcgr 21776   AbelOpcablo 21871
This theorem is referenced by:  isablod  21890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-grpo 21781  df-ablo 21872
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