Theorem isabli 13636

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 2592	. 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 13630	. 2 |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
7	1, 3, 6	mpbir2an 945	1 |- G e. Abel

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   Grpcgrp 13332   Abelcabl 13621

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947  df-grp 13335  df-cmn 13622  df-abl 13623

This theorem is referenced by: (None)