Theorem ablnsg 13268

Description: Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)

Assertion

Ref	Expression
ablnsg	|- ( G e. Abel -> (NrmSGrp ` G ) = (SubGrp ` G ) )

Proof of Theorem ablnsg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2189	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
2		eqid 2189	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
3	1, 2	ablcom 13239	. . . . . 6 |- ( ( G e. Abel /\ y e. ( Base ` G ) /\ z e. ( Base ` G ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
4	3	3expb 1206	. . . . 5 |- ( ( G e. Abel /\ ( y e. ( Base ` G ) /\ z e. ( Base ` G ) ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
5	4	eleq1d 2258	. . . 4 |- ( ( G e. Abel /\ ( y e. ( Base ` G ) /\ z e. ( Base ` G ) ) ) -> ( ( y ( +g ` G ) z ) e. x <-> ( z ( +g ` G ) y ) e. x ) )
6	5	ralrimivva 2572	. . 3 |- ( G e. Abel -> A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) e. x <-> ( z ( +g ` G ) y ) e. x ) )
7	1, 2	isnsg 13138	. . . 4 |- ( x e. (NrmSGrp ` G ) <-> ( x e. (SubGrp ` G ) /\ A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) e. x <-> ( z ( +g ` G ) y ) e. x ) ) )
8	7	rbaib 922	. . 3 |- ( A. y e. ( Base ` G ) A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) e. x <-> ( z ( +g ` G ) y ) e. x ) -> ( x e. (NrmSGrp ` G ) <-> x e. (SubGrp ` G ) ) )
9	6, 8	syl 14	. 2 |- ( G e. Abel -> ( x e. (NrmSGrp ` G ) <-> x e. (SubGrp ` G ) ) )
10	9	eqrdv 2187	1 |- ( G e. Abel -> (NrmSGrp ` G ) = (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586  SubGrpcsubg 13103  NrmSGrpcnsg 13104   Abelcabl 13221

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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898  df-inn 8949  df-2 9007  df-ndx 12514  df-slot 12515  df-base 12517  df-plusg 12599  df-subg 13106  df-nsg 13107  df-cmn 13222  df-abl 13223

This theorem is referenced by:  rngansg  13301  qus2idrng  13837  qus1  13838