Theorem ablnsg 14115

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 2238	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
2		eqid 2238	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
3	1, 2	ablcom 14083	. . . . . 6 |- ( ( G e. Abel /\ y e. ( Base ` G ) /\ z e. ( Base ` G ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
4	3	3expb 1235	. . . . 5 |- ( ( G e. Abel /\ ( y e. ( Base ` G ) /\ z e. ( Base ` G ) ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
5	4	eleq1d 2307	. . . 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 2632	. . 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 13982	. . . 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 933	. . 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 2236	1 |- ( G e. Abel -> (NrmSGrp ` G ) = (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  SubGrpcsubg 13947  NrmSGrpcnsg 13948   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-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  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-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-subg 13950  df-nsg 13951  df-cmn 14066  df-abl 14067

This theorem is referenced by:  rngansg  14224  qus2idrng  14834  qus1  14835  qusrhm  14837