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Theorem isabl2 15190
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 15186 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
2 grpmnd 14587 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3 iscmn.b . . . . . 6  |-  B  =  ( Base `  G
)
4 iscmn.p . . . . . 6  |-  .+  =  ( +g  `  G )
53, 4iscmn 15189 . . . . 5  |-  ( G  e. CMnd 
<->  ( G  e.  Mnd  /\ 
A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
65baib 871 . . . 4  |-  ( G  e.  Mnd  ->  ( G  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
72, 6syl 15 . . 3  |-  ( G  e.  Grp  ->  ( G  e. CMnd  <->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
87pm5.32i 618 . 2  |-  ( ( G  e.  Grp  /\  G  e. CMnd )  <->  ( G  e.  Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
91, 8bitri 240 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:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299   Mndcmnd 14454   Grpcgrp 14455  CMndccmn 15182   Abelcabel 15183
This theorem is referenced by:  isabli  15196  invghm  15223  divsabl  15250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945  df-grp 14582  df-cmn 15184  df-abl 15185
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