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Theorem isabli 15103
Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
Hypotheses
Ref Expression
isabli.g  |-  G  e. 
Grp
isabli.b  |-  B  =  ( Base `  G
)
isabli.p  |-  .+  =  ( +g  `  G )
isabli.c  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
Assertion
Ref Expression
isabli  |-  G  e. 
Abel
Distinct variable groups:    x, y, B    x, G, y
Allowed substitution hints:    .+ ( x, y)

Proof of Theorem isabli
StepHypRef Expression
1 isabli.g . 2  |-  G  e. 
Grp
2 isabli.c . . 3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  =  ( y 
.+  x ) )
32rgen2a 2609 . 2  |-  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x )
4 isabli.b . . 3  |-  B  =  ( Base `  G
)
5 isabli.p . . 3  |-  .+  =  ( +g  `  G )
64, 5isabl2 15097 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
71, 3, 6mpbir2an 886 1  |-  G  e. 
Abel
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   Abelcabel 15090
This theorem is referenced by:  cnaddablx  15158  cnaddabl  15159  zaddablx  15160
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-grp 14489  df-cmn 15091  df-abl 15092
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