Theorem isabli 14080

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 2636	. 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 14074	. 2 |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
7	1, 3, 6	mpbir2an 955	1 |- G e. Abel

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   Grpcgrp 13782   Abelcabl 14065

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-grp 13785  df-cmn 14066  df-abl 14067

This theorem is referenced by: (None)