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Theorem isabld 16789
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 16040 . . . 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 16788 . 2  |-  ( ph  ->  G  e. CMnd )
8 isabl 16780 . 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 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14613   +g cplusg 14678   Mndcmnd 15897   Grpcgrp 16031  CMndccmn 16776   Abelcabl 16777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-grp 16035  df-cmn 16778  df-abl 16779
This theorem is referenced by:  subgabl  16822  gex2abl  16835  cygabl  16871  ringabl  17206  lmodabl  17535  dchrabl  23505  tgrpabl  36217  erngdvlem2N  36455  erngdvlem2-rN  36463
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