Theorem nmznsg 13947

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 13945	. . 3 |- ( S e. (SubGrp ` G ) -> S C_ N )
6		subgrcl 13913	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 3, 4	nmzsubg 13944	. . . . 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 13931	. . . 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 953	. 2 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` H ) )
13	2	ssrab3 3326	. . . . . 6 |- N C_ X
14	13	sseli 3236	. . . . 5 |- ( w e. N -> w e. X )
15	2	nmzbi 13943	. . . . 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 2630	. . 3 |- A. z e. N A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S )
18	9	subgbas 13912	. . . . 5 |- ( N e. (SubGrp ` G ) -> N = ( Base ` H ) )
19	8, 18	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> N = ( Base ` H ) )
20	19	raleqdv 2749	. . . 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 2759	. . 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 2234	. . . 4 |- ( Base ` H ) = ( Base ` H )
24		eqid 2234	. . . 4 |- ( +g ` H ) = ( +g ` H )
25	23, 24	isnsg 13936	. . 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 13359	. . . . . . . 8 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
29	28	oveqd 6069	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( z .+ w ) = ( z ( +g ` H ) w ) )
30	29	eleq1d 2303	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( z .+ w ) e. S <-> ( z ( +g ` H ) w ) e. S ) )
31	28	oveqd 6069	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( w .+ z ) = ( w ( +g ` H ) z ) )
32	31	eleq1d 2303	. . . . . 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 2568	. . . 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 953	1 |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   C_ wss 3213   ` cfv 5354  (class class class)co 6052   Basecbs 13229   ↾_s cress 13230   +g cplusg 13307   Grpcgrp 13730  SubGrpcsubg 13901  NrmSGrpcnsg 13902

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 619  ax-in2 620  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-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-subg 13904  df-nsg 13905

This theorem is referenced by: (None)