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Theorem isabld 15102
Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
Hypotheses
Ref Expression
isabld.b  |-  ( ph  ->  B  =  ( Base `  G ) )
isabld.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
isabld.g  |-  ( ph  ->  G  e.  Grp )
isabld.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
Assertion
Ref Expression
isabld  |-  ( ph  ->  G  e.  Abel )
Distinct variable groups:    x, y, B    x, G, y    ph, x, y
Allowed substitution hints:    .+ ( x, y)

Proof of Theorem isabld
StepHypRef Expression
1 isabld.g . 2  |-  ( ph  ->  G  e.  Grp )
2 isabld.b . . 3  |-  ( ph  ->  B  =  ( Base `  G ) )
3 isabld.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  G ) )
4 grpmnd 14494 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
51, 4syl 15 . . 3  |-  ( ph  ->  G  e.  Mnd )
6 isabld.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
72, 3, 5, 6iscmnd 15101 . 2  |-  ( ph  ->  G  e. CMnd )
8 isabl 15093 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
91, 7, 8sylanbrc 645 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Mndcmnd 14361   Grpcgrp 14362  CMndccmn 15089   Abelcabel 15090
This theorem is referenced by:  subgabl  15132  gex2abl  15143  cygabl  15177  rngabl  15370  lmodabl  15672  dchrabl  20493  tgrpabl  30940  erngdvlem2N  31178  erngdvlem2-rN  31186
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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