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Theorem isablod 21919
Description: Properties that determine an Abelian group operation. (Changed label from isabld 15456 to isablod 21919-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 6117 . . . . 5  |-  ( n  =  N  ->  (
n G x )  =  ( N G x ) )
98eqeq1d 2450 . . . 4  |-  ( n  =  N  ->  (
( n G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 3058 . . 3  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. n  e.  X  ( n G x )  =  U )
116, 7, 10syl2anc 644 . 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 21918 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   E.wrex 2712   _Vcvv 2962    X. cxp 4905   -->wf 5479  (class class class)co 6110   AbelOpcablo 21900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-grpo 21810  df-ablo 21901
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