Theorem nmznsg 13319

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 13317	. . 3 |- ( S e. (SubGrp ` G ) -> S C_ N )
6		subgrcl 13285	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
7	2, 3, 4	nmzsubg 13316	. . . . 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 13303	. . . 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 946	. 2 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` H ) )
13	2	ssrab3 3269	. . . . . 6 |- N C_ X
14	13	sseli 3179	. . . . 5 |- ( w e. N -> w e. X )
15	2	nmzbi 13315	. . . . 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 2583	. . 3 |- A. z e. N A. w e. N ( ( z .+ w ) e. S <-> ( w .+ z ) e. S )
18	9	subgbas 13284	. . . . 5 |- ( N e. (SubGrp ` G ) -> N = ( Base ` H ) )
19	8, 18	syl 14	. . . 4 |- ( S e. (SubGrp ` G ) -> N = ( Base ` H ) )
20	19	raleqdv 2699	. . . 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 2709	. . 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 2196	. . . 4 |- ( Base ` H ) = ( Base ` H )
24		eqid 2196	. . . 4 |- ( +g ` H ) = ( +g ` H )
25	23, 24	isnsg 13308	. . 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 12782	. . . . . . . 8 |- ( S e. (SubGrp ` G ) -> .+ = ( +g ` H ) )
29	28	oveqd 5939	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( z .+ w ) = ( z ( +g ` H ) w ) )
30	29	eleq1d 2265	. . . . . 6 |- ( S e. (SubGrp ` G ) -> ( ( z .+ w ) e. S <-> ( z ( +g ` H ) w ) e. S ) )
31	28	oveqd 5939	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> ( w .+ z ) = ( w ( +g ` H ) z ) )
32	31	eleq1d 2265	. . . . . 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 2521	. . . 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 946	1 |- ( S e. (SubGrp ` G ) -> S e. (NrmSGrp ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   {crab 2479   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12654   ↾_s cress 12655   +g cplusg 12731   Grpcgrp 13108  SubGrpcsubg 13273  NrmSGrpcnsg 13274

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-addcom 7977  ax-addass 7979  ax-i2m1 7982  ax-0lt1 7983  ax-0id 7985  ax-rnegex 7986  ax-pre-ltirr 7989  ax-pre-ltadd 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8061  df-mnf 8062  df-ltxr 8064  df-inn 8988  df-2 9046  df-ndx 12657  df-slot 12658  df-base 12660  df-sets 12661  df-iress 12662  df-plusg 12744  df-0g 12905  df-mgm 12975  df-sgrp 13021  df-mnd 13034  df-grp 13111  df-minusg 13112  df-sbg 13113  df-subg 13276  df-nsg 13277

This theorem is referenced by: (None)