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Theorem isabl2 12893
Description: The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
iscmn.b  |-  B  =  ( Base `  G
)
iscmn.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
isabl2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
Distinct variable groups:    x, y, B   
x, G, y
Allowed substitution hints:    .+ ( x, y)

Proof of Theorem isabl2
StepHypRef Expression
1 isabl 12888 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
2 grpmnd 12745 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3 iscmn.b . . . . . 6  |-  B  =  ( Base `  G
)
4 iscmn.p . . . . . 6  |-  .+  =  ( +g  `  G )
53, 4iscmn 12892 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
65baib 919 . . . 4  |-  ( G  e.  Mnd  ->  ( G  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
72, 6syl 14 . . 3  |-  ( G  e.  Grp  ->  ( G  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
87pm5.32i 454 . 2  |-  ( ( G  e.  Grp  /\  G  e. CMnd )  <->  ( G  e.  Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
91, 8bitri 184 1  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   Mndcmnd 12682   Grpcgrp 12738  CMndccmn 12884   Abelcabl 12885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-in 3133  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868  df-grp 12741  df-cmn 12886  df-abl 12887
This theorem is referenced by:  isabli  12899
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