Theorem isabld 13885

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 13595	. . 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 13884	. 2 |- ( ph -> G e. CMnd )
7		isabl 13874	. 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 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   Grpcgrp 13582  CMndccmn 13870   Abelcabl 13871

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-grp 13585  df-cmn 13872  df-abl 13873

This theorem is referenced by:  subgabl  13918  ablressid  13921  ringabl  14044  lmodabl  14347