MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isabld Structured version   Unicode version

Theorem isabld 15427
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 14819 . . . 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 15426 . 2  |-  ( ph  ->  G  e. CMnd )
8 isabl 15418 . 2  |-  ( G  e.  Abel  <->  ( G  e. 
Grp  /\  G  e. CMnd ) )
91, 7, 8sylanbrc 647 1  |-  ( ph  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Mndcmnd 14686   Grpcgrp 14687  CMndccmn 15414   Abelcabel 15415
This theorem is referenced by:  subgabl  15457  gex2abl  15468  cygabl  15502  rngabl  15695  lmodabl  15993  dchrabl  21040  tgrpabl  31610  erngdvlem2N  31848  erngdvlem2-rN  31856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-grp 14814  df-cmn 15416  df-abl 15417
  Copyright terms: Public domain W3C validator