Theorem nmznsg 13078

Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-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 )
nmznsg.4	|- H = ( G ↾s N )

Assertion

Ref	Expression
nmznsg	|- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )

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

Allowed substitution hints:   H( x, y)   N( x, y)

Proof of Theorem nmznsg

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
2		elnmz.1	. . . 4 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }
3		nmzsubg.2	. . . 4 |- X = ( Base ` G )
4		nmzsubg.3	. . . 4 |- .+ = ( +g ` G )
5	2, 3, 4	ssnmz 13076	. . 3 |- ( S e. (SubGrp ` G ) -> S C_ N )
6		subgrcl 13044	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 3, 4	nmzsubg 13075	. . . . 5 |- ( G e. Grp -> N e. (SubGrp ` G ) )
8	6, 7	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> N e. (SubGrp ` G ) )
9		nmznsg.4	. . . . 5 |- H = ( G ↾s N )
10	9	subsubg 13062	. . . 4 |- ( N e. (SubGrp ` G ) -> ( S e. (SubGrp ` H ) <-> ( S e. (SubGrp ` G ) /\ S C_ N ) ) )
11	8, 10	syl 14	. . 3 |- ( S e. (SubGrp ` G ) -> ( S e. (SubGrp ` H ) <-> ( S e. (SubGrp ` G ) /\ S C_ N ) ) )
12	1, 5, 11	mpbir2and 944	. 2 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` H ) )
13	2	ssrab3 3243	. . . . . 6 |- N C_ X
14	13	sseli 3153	. . . . 5 |- ( w e. N -> w e. X )
15	2	nmzbi 13074	. . . . 5 |- ( ( z e. N /\ w e. X ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
16	14, 15	sylan2 286	. . . 4 |- ( ( z e. N /\ w e. N ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
17	16	rgen2 2563	. . 3 |- A. z e. N A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S )
18	9	subgbas 13043	. . . . 5 |- ( N e. (SubGrp ` G ) -> N = ( Base ` H ) )
19	8, 18	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> N = ( Base ` H ) )
20	19	raleqdv 2679	. . . 4 |- ( S e. (SubGrp ` G ) -> ( A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) <-> A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) )
21	19, 20	raleqbidv 2685	. . 3 |- ( S e. (SubGrp ` G ) -> ( A. z e. N A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) <-> A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) )
22	17, 21	mpbii 148	. 2 |- ( S e. (SubGrp ` G ) -> A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
23		eqid 2177	. . . 4 |- ( Base ` H ) = ( Base ` H )
24		eqid 2177	. . . 4 |- ( +g ` H ) = ( +g ` H )
25	23, 24	isnsg 13067	. . 3 |- ( S e. (NrmSGrp ` H ) <-> ( S e. (SubGrp ` H ) /\ A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z ( +g ` H ) w ) e. S <-> ( w ( +g ` H ) z ) e. S ) ) )
26	9	a1i 9	. . . . . . . . 9 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s N ) )
27	4	a1i 9	. . . . . . . . 9 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` G ) )
28	26, 27, 8, 6	ressplusgd 12589	. . . . . . . 8 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
29	28	oveqd 5894	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( z .+ w ) = ( z ( +g ` H ) w ) )
30	29	eleq1d 2246	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( z .+ w ) e. S <-> ( z ( +g ` H ) w ) e. S ) )
31	28	oveqd 5894	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( w .+ z ) = ( w ( +g ` H ) z ) )
32	31	eleq1d 2246	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( w .+ z ) e. S <-> ( w ( +g ` H ) z ) e. S ) )
33	30, 32	bibi12d 235	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) <-> ( ( z ( +g ` H ) w ) e. S <-> ( w ( +g ` H ) z ) e. S ) ) )
34	33	2ralbidv 2501	. . . 4 |- ( S e. (SubGrp ` G ) -> ( A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) <-> A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z ( +g ` H ) w ) e. S <-> ( w ( +g ` H ) z ) e. S ) ) )
35	34	anbi2d 464	. . 3 |- ( S e. (SubGrp ` G ) -> ( ( S e. (SubGrp ` H ) /\ A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) <-> ( S e. (SubGrp ` H ) /\ A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z ( +g ` H ) w ) e. S <-> ( w ( +g ` H ) z ) e. S ) ) ) )
36	25, 35	bitr4id 199	. 2 |- ( S e. (SubGrp ` G ) -> ( S e. (NrmSGrp ` H ) <-> ( S e. (SubGrp ` H ) /\ A. z e. ( Base ` H ) A. w e. ( Base ` H ) ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) ) )
37	12, 22, 36	mpbir2and 944	1 |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   {crab 2459   C_ wss 3131   ` cfv 5218  (class class class)co 5877   Basecbs 12464   ↾_s cress 12465   +g cplusg 12538   Grpcgrp 12882  SubGrpcsubg 13032  NrmSGrpcnsg 13033

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-in1 614  ax-in2 615  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-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-sbg 12887  df-subg 13035  df-nsg 13036

This theorem is referenced by: (None)