Theorem ssnmz 13878

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 13841	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	sselda 3228	. . . 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 13846	. . . . . . . . . . . . 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 537	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> z e. S )
9	7, 8	sseldd 3229	. . . . . . . . . . . 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 2231	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
12		eqid 2231	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
13	1, 10, 11, 12	grplinv 13713	. . . . . . . . . . . 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 6043	. . . . . . . . . 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 13850	. . . . . . . . . . . . 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 3229	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( invg ` G ) ` z ) e. X )
19		simplrr 538	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. X )
20	1, 10	grpass 13672	. . . . . . . . . . 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 1276	. . . . . . . . . 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 13694	. . . . . . . . . . 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 2272	. . . . . . . . 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 13851	. . . . . . . . . 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 1274	. . . . . . . . 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 2309	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. S )
29	10	subgcl 13851	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ w e. S /\ z e. S ) -> ( w .+ z ) e. S )
30	4, 28, 8, 29	syl3anc 1274	. . . . . . 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 537	. . . . . . . 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 538	. . . . . . . . . 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 2231	. . . . . . . . . . 11 |- ( -g ` G ) = ( -g ` G )
37	1, 10, 36	grppncan 13754	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ z e. X ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
38	33, 34, 35, 37	syl3anc 1274	. . . . . . . . 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 13852	. . . . . . . . . 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 1274	. . . . . . . . 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 2309	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. S )
43	10	subgcl 13851	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ z e. S /\ w e. S ) -> ( z .+ w ) e. S )
44	31, 32, 42, 43	syl3anc 1274	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( z .+ w ) e. S )
45	30, 44	impbida 600	. . . . . 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 2606	. . . 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 13875	. . . 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 3234	1 |- ( S e. (SubGrp ` G ) -> S C_ N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664   -gcsg 13665  SubGrpcsubg 13834

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837

This theorem is referenced by:  nmznsg  13880