Theorem ssnmz 13076

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 13039	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	sselda 3157	. . . 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 13044	. . . . . . . . . . . . 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 3158	. . . . . . . . . . . 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 2177	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
12		eqid 2177	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
13	1, 10, 11, 12	grplinv 12927	. . . . . . . . . . . 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 5892	. . . . . . . . . 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 13048	. . . . . . . . . . . . 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 3158	. . . . . . . . . . 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 12891	. . . . . . . . . . 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 1240	. . . . . . . . . 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 12911	. . . . . . . . . . 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 2218	. . . . . . . . 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 13049	. . . . . . . . . 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 1238	. . . . . . . . 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 2255	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. S )
29	10	subgcl 13049	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ w e. S /\ z e. S ) -> ( w .+ z ) e. S )
30	4, 28, 8, 29	syl3anc 1238	. . . . . . 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 2177	. . . . . . . . . . 11 |- ( -g ` G ) = ( -g ` G )
37	1, 10, 36	grppncan 12966	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ z e. X ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
38	33, 34, 35, 37	syl3anc 1238	. . . . . . . . 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 13050	. . . . . . . . . 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 1238	. . . . . . . . 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 2255	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. S )
43	10	subgcl 13049	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ z e. S /\ w e. S ) -> ( z .+ w ) e. S )
44	31, 32, 42, 43	syl3anc 1238	. . . . . . 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 2550	. . . 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 13073	. . . 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 3163	1 |- ( S e. (SubGrp ` G ) -> S C_ N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   {crab 2459   C_ wss 3131   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710   Grpcgrp 12882   invgcminusg 12883   -gcsg 12884  SubGrpcsubg 13032

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-sbg 12887  df-subg 13035

This theorem is referenced by:  nmznsg  13078