Theorem isabli 13108

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 2563	. 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 13102	. 2 |- ( G e. Abel <-> ( G e. Grp /\ A. x e. B A. y e. B ( x .+ y ) = ( y .+ x ) ) )
7	1, 3, 6	mpbir2an 942	1 |- G e. Abel

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   Grpcgrp 12882   Abelcabl 13094

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-grp 12885  df-cmn 13095  df-abl 13096

This theorem is referenced by: (None)