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Theorem isablod 21849
Description: Properties that determine an Abelian group operation. (Changed label from isabld 15388 to isablod 21849-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (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 )
isabld.6  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
isabld.7  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
isabld.8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
Assertion
Ref Expression
isablod  |-  ( ph  ->  G  e.  AbelOp )
Distinct variable groups:    ph, x, y, z    x, G, y, z    x, X, y, z    x, U, y, z
Allowed substitution hints:    N( x, y, z)

Proof of Theorem isablod
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 isgrpda.1 . 2  |-  ( ph  ->  X  e.  _V )
2 isgrpda.2 . 2  |-  ( ph  ->  G : ( X  X.  X ) --> X )
3 isgrpda.3 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
4 isgrpda.4 . 2  |-  ( ph  ->  U  e.  X )
5 isgrpda.5 . 2  |-  ( (
ph  /\  x  e.  X )  ->  ( U G x )  =  x )
6 isabld.6 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
7 isabld.7 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
8 oveq1 6055 . . . . 5  |-  ( n  =  N  ->  (
n G x )  =  ( N G x ) )
98eqeq1d 2420 . . . 4  |-  ( n  =  N  ->  (
( n G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 3020 . . 3  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. n  e.  X  ( n G x )  =  U )
116, 7, 10syl2anc 643 . 2  |-  ( (
ph  /\  x  e.  X )  ->  E. n  e.  X  ( n G x )  =  U )
12 isabld.8 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
131, 2, 3, 4, 5, 11, 12isabloda 21848 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924    X. cxp 4843   -->wf 5417  (class class class)co 6048   AbelOpcablo 21830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-grpo 21740  df-ablo 21831
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