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Theorem isabli 15119
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 2622 . 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 15113 . 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 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   Abelcabel 15106
This theorem is referenced by:  cnaddablx  15174  cnaddabl  15175  zaddablx  15176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-grp 14505  df-cmn 15107  df-abl 15108
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