Theorem isabld 13136

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

Step	Hyp	Ref	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	1	grpmndd 12911	. . 3 |- ( ph -> G e. Mnd )
5		isabld.c	. . 3 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )
6	2, 3, 4, 5	iscmnd 13135	. 2 |- ( ph -> G e. CMnd )
7		isabl 13125	. 2 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
8	1, 6, 7	sylanbrc 417	1 |- ( ph -> G e. Abel )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12476   +g cplusg 12551   Grpcgrp 12899  CMndccmn 13121   Abelcabl 13122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-in 3147  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891  df-grp 12902  df-cmn 13123  df-abl 13124

This theorem is referenced by:  subgabl  13167  ablressid  13170  ringabl  13284  lmodabl  13523