Theorem nmznsg 14016

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 14014	. . 3 |- ( S e. (SubGrp ` G ) -> S C_ N )
6		subgrcl 13982	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 3, 4	nmzsubg 14013	. . . . 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 14000	. . . 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 957	. 2 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` H ) )
13	2	ssrab3 3334	. . . . . 6 |- N C_ X
14	13	sseli 3244	. . . . 5 |- ( w e. N -> w e. X )
15	2	nmzbi 14012	. . . . 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 2636	. . 3 |- A. z e. N A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S )
18	9	subgbas 13981	. . . . 5 |- ( N e. (SubGrp ` G ) -> N = ( Base ` H ) )
19	8, 18	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> N = ( Base ` H ) )
20	19	raleqdv 2755	. . . 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 2765	. . 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 2238	. . . 4 |- ( Base ` H ) = ( Base ` H )
24		eqid 2238	. . . 4 |- ( +g ` H ) = ( +g ` H )
25	23, 24	isnsg 14005	. . 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 13483	. . . . . . . 8 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
29	28	oveqd 6102	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( z .+ w ) = ( z ( +g ` H ) w ) )
30	29	eleq1d 2307	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( z .+ w ) e. S <-> ( z ( +g ` H ) w ) e. S ) )
31	28	oveqd 6102	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( w .+ z ) = ( w ( +g ` H ) z ) )
32	31	eleq1d 2307	. . . . . 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 2574	. . . 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 468	. . 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 957	1 |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   C_ wss 3220   ` cfv 5377  (class class class)co 6085   Basecbs 13352   ↾_s cress 13353   +g cplusg 13431   Grpcgrp 13805  SubGrpcsubg 13970  NrmSGrpcnsg 13971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-sbg 13810  df-subg 13973  df-nsg 13974

This theorem is used by: (None)