Theorem ablnsg 13866

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 2229	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
2		eqid 2229	. . . . . . 7 |- ( +g ` G ) = ( +g ` G )
3	1, 2	ablcom 13835	. . . . . 6 |- ( ( G e. Abel /\ y e. ( Base ` G ) /\ z e. ( Base ` G ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
4	3	3expb 1228	. . . . 5 |- ( ( G e. Abel /\ ( y e. ( Base ` G ) /\ z e. ( Base ` G ) ) ) -> ( y ( +g ` G ) z ) = ( z ( +g ` G ) y ) )
5	4	eleq1d 2298	. . . 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 2612	. . 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 13734	. . . 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 926	. . 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 2227	1 |- ( G e. Abel -> (NrmSGrp ` G ) = (SubGrp ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105  SubGrpcsubg 13699  NrmSGrpcnsg 13700   Abelcabl 13817

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-subg 13702  df-nsg 13703  df-cmn 13818  df-abl 13819

This theorem is referenced by:  rngansg  13908  qus2idrng  14483  qus1  14484  qusrhm  14486