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Theorem isabld 16602
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 15858 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
51, 4syl 16 . . 3  |-  ( ph  ->  G  e.  Mnd )
6 isabld.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  =  ( y  .+  x ) )
72, 3, 5, 6iscmnd 16601 . 2  |-  ( ph  ->  G  e. CMnd )
8 isabl 16593 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
91, 7, 8sylanbrc 664 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   Mndcmnd 15717   Grpcgrp 15718  CMndccmn 16589   Abelcabl 16590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-grp 15853  df-cmn 16591  df-abl 16592
This theorem is referenced by:  subgabl  16632  gex2abl  16645  cygabl  16679  rngabl  17010  lmodabl  17335  dchrabl  23252  tgrpabl  35424  erngdvlem2N  35662  erngdvlem2-rN  35670
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