Theorem isabld 13107

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 12894	. . 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 13106	. 2 |- ( ph -> G e. CMnd )
7		isabl 13097	. 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 978   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   Grpcgrp 12882  CMndccmn 13093   Abelcabl 13094

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-grp 12885  df-cmn 13095  df-abl 13096

This theorem is referenced by:  ringabl  13220  lmodabl  13429