Theorem ssnmz 13341

Description: A subgroup is a subset of its normalizer. (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
ssnmz	|- ( S e. (SubGrp ` G ) -> S C_ N )

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

Allowed substitution hints:   N( x, y)

Proof of Theorem ssnmz

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

Step	Hyp	Ref	Expression
1		nmzsubg.2	. . . . . 6 |- X = ( Base ` G )
2	1	subgss 13304	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	sselda 3183	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> z e. X )
4		simpll 527	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> S e. (SubGrp ` G ) )
5		subgrcl 13309	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> G e. Grp )
6	4, 5	syl 14	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> G e. Grp )
7	4, 2	syl 14	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> S C_ X )
8		simplrl 535	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> z e. S )
9	7, 8	sseldd 3184	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> z e. X )
10		nmzsubg.3	. . . . . . . . . . . . 13 |- .+ = ( +g ` G )
11		eqid 2196	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
12		eqid 2196	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
13	1, 10, 11, 12	grplinv 13182	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ z e. X ) -> ( ( ( invg ` G ) ` z ) .+ z ) = ( 0g ` G ) )
14	6, 9, 13	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( invg ` G ) ` z ) .+ z ) = ( 0g ` G ) )
15	14	oveq1d 5937	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( ( invg ` G ) ` z ) .+ z ) .+ w ) = ( ( 0g ` G ) .+ w ) )
16	12	subginvcl 13313	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> ( ( invg ` G ) ` z ) e. S )
17	4, 8, 16	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( invg ` G ) ` z ) e. S )
18	7, 17	sseldd 3184	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( invg ` G ) ` z ) e. X )
19		simplrr 536	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. X )
20	1, 10	grpass 13141	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( ( ( invg ` G ) ` z ) e. X /\ z e. X /\ w e. X ) ) -> ( ( ( ( invg ` G ) ` z ) .+ z ) .+ w ) = ( ( ( invg ` G ) ` z ) .+ ( z .+ w ) ) )
21	6, 18, 9, 19, 20	syl13anc 1251	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( ( invg ` G ) ` z ) .+ z ) .+ w ) = ( ( ( invg ` G ) ` z ) .+ ( z .+ w ) ) )
22	1, 10, 11	grplid 13163	. . . . . . . . . . 11 |- ( ( G e. Grp /\ w e. X ) -> ( ( 0g ` G ) .+ w ) = w )
23	6, 19, 22	syl2anc 411	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( 0g ` G ) .+ w ) = w )
24	15, 21, 23	3eqtr3d 2237	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( invg ` G ) ` z ) .+ ( z .+ w ) ) = w )
25		simpr 110	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( z .+ w ) e. S )
26	10	subgcl 13314	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ ( ( invg ` G ) ` z ) e. S /\ ( z .+ w ) e. S ) -> ( ( ( invg ` G ) ` z ) .+ ( z .+ w ) ) e. S )
27	4, 17, 25, 26	syl3anc 1249	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( invg ` G ) ` z ) .+ ( z .+ w ) ) e. S )
28	24, 27	eqeltrrd 2274	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. S )
29	10	subgcl 13314	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ w e. S /\ z e. S ) -> ( w .+ z ) e. S )
30	4, 28, 8, 29	syl3anc 1249	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( w .+ z ) e. S )
31		simpll 527	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> S e. (SubGrp ` G ) )
32		simplrl 535	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> z e. S )
33	31, 5	syl 14	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> G e. Grp )
34		simplrr 536	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. X )
35	31, 32, 3	syl2anc 411	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> z e. X )
36		eqid 2196	. . . . . . . . . . 11 |- ( -g ` G ) = ( -g ` G )
37	1, 10, 36	grppncan 13223	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ z e. X ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
38	33, 34, 35, 37	syl3anc 1249	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
39		simpr 110	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( w .+ z ) e. S )
40	36	subgsubcl 13315	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ ( w .+ z ) e. S /\ z e. S ) -> ( ( w .+ z ) ( -g ` G ) z ) e. S )
41	31, 39, 32, 40	syl3anc 1249	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( ( w .+ z ) ( -g ` G ) z ) e. S )
42	38, 41	eqeltrrd 2274	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. S )
43	10	subgcl 13314	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ z e. S /\ w e. S ) -> ( z .+ w ) e. S )
44	31, 32, 42, 43	syl3anc 1249	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( z .+ w ) e. S )
45	30, 44	impbida 596	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
46	45	anassrs 400	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ z e. S ) /\ w e. X ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
47	46	ralrimiva 2570	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> A. w e. X ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
48		elnmz.1	. . . . 5 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. S <-> ( y .+ x ) e. S ) }
49	48	elnmz 13338	. . . 4 |- ( z e. N <-> ( z e. X /\ A. w e. X ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) )
50	3, 47, 49	sylanbrc 417	. . 3 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> z e. N )
51	50	ex 115	. 2 |- ( S e. (SubGrp ` G ) -> ( z e. S -> z e. N ) )
52	51	ssrdv 3189	1 |- ( S e. (SubGrp ` G ) -> S C_ N )

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 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133   -gcsg 13134  SubGrpcsubg 13297

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 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

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 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137  df-subg 13300

This theorem is referenced by:  nmznsg  13343