Theorem isabli 13370

Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)

Hypotheses

Ref	Expression
isabli.g	|- G e. Grp
isabli.b	|- B = ( Base ` G )
isabli.p	|- .+ = ( +g ` G )
isabli.c	|- ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )

Assertion

Ref	Expression
isabli	|- G e. Abel

Distinct variable groups:   x, y, B   x, G, y

Allowed substitution hints:   .+ ( x, y)

Proof of Theorem isabli

Step	Hyp	Ref	Expression
1		isabli.g	. 2 |- G e. Grp
2		isabli.c	. . 3 |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )
3	2	rgen2 2580	. 2 |- A. x e. B A. y e. B ( x .+ y ) = ( y .+ x )
4		isabli.b	. . 3 |- B = ( Base ` G )
5		isabli.p	. . 3 |- .+ = ( +g ` G )
6	4, 5	isabl2 13364	. 2 |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
7	1, 3, 6	mpbir2an 944	1 |- G e. Abel

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   Abelcabl 13355

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-grp 13075  df-cmn 13356  df-abl 13357

This theorem is referenced by: (None)