Theorem isabld 13429

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 13145	. . 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 13428	. 2 |- ( ph -> G e. CMnd )
7		isabl 13418	. 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 980   = wceq 1364   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   Grpcgrp 13132  CMndccmn 13414   Abelcabl 13415

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925  df-grp 13135  df-cmn 13416  df-abl 13417

This theorem is referenced by:  subgabl  13462  ablressid  13465  ringabl  13588  lmodabl  13890