Theorem isnsg4 13965

Description: A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
elnmz.1	|- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }
nmzsubg.2	|- X = ( Base ` G )
nmzsubg.3	|- .+ = ( +g ` G )

Assertion

Ref	Expression
isnsg4	|- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ N = X ) )

Distinct variable groups:   x, y, G   x, S, y   x, .+ , y   x, X, y

Allowed substitution hints:   N( x, y)

Proof of Theorem isnsg4

Step	Hyp	Ref	Expression
1		nmzsubg.2	. . 3 |- X = ( Base ` G )
2		nmzsubg.3	. . 3 |- .+ = ( +g ` G )
3	1, 2	isnsg 13955	. 2 |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) ) )
4		eqcom 2236	. . . 4 |- ( N = X <-> X = N )
5		elnmz.1	. . . . 5 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }
6	5	eqeq2i 2245	. . . 4 |- ( X = N <-> X = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) } )
7		rabid2 2723	. . . 4 |- ( X = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) } <-> A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) )
8	4, 6, 7	3bitri 206	. . 3 |- ( N = X <-> A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) )
9	8	anbi2i 457	. 2 |- ( ( S e. (SubGrp ` G ) /\ N = X ) <-> ( S e. (SubGrp ` G ) /\ A. x e. X A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) ) )
10	3, 9	bitr4i 187	1 |- ( S e. (NrmSGrp ` G ) <-> ( S e. (SubGrp ` G ) /\ N = X ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374  SubGrpcsubg 13920  NrmSGrpcnsg 13921

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-subg 13923  df-nsg 13924

This theorem is referenced by:  conjnsg  14034