Theorem ssnmz 14014

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 13977	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	sselda 3248	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> z e. X )
4		simpll 531	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> S e. (SubGrp ` G ) )
5		subgrcl 13982	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> z e. S )
9	7, 8	sseldd 3249	. . . . . . . . . . . 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 2238	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
12		eqid 2238	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
13	1, 10, 11, 12	grplinv 13855	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ z e. X ) -> ( ( ( invg ` G ) ` z ) .+ z ) = ( 0g ` G ) )
14	6, 9, 13	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( ( invg ` G ) ` z ) .+ z ) = ( 0g ` G ) )
15	14	oveq1d 6100	. . . . . . . . . 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 13986	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ z e. S ) -> ( ( invg ` G ) ` z ) e. S )
17	4, 8, 16	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( invg ` G ) ` z ) e. S )
18	7, 17	sseldd 3249	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( invg ` G ) ` z ) e. X )
19		simplrr 542	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. X )
20	1, 10	grpass 13814	. . . . . . . . . . 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 1280	. . . . . . . . . 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 13836	. . . . . . . . . . 11 |- ( ( G e. Grp /\ w e. X ) -> ( ( 0g ` G ) .+ w ) = w )
23	6, 19, 22	syl2anc 415	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( ( 0g ` G ) .+ w ) = w )
24	15, 21, 23	3eqtr3d 2279	. . . . . . . . 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 13987	. . . . . . . . . 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 1278	. . . . . . . . 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 2316	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. S )
29	10	subgcl 13987	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ w e. S /\ z e. S ) -> ( w .+ z ) e. S )
30	4, 28, 8, 29	syl3anc 1278	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> ( w .+ z ) e. S )
31		simpll 531	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> S e. (SubGrp ` G ) )
32		simplrl 541	. . . . . . . 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 542	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. X )
35	31, 32, 3	syl2anc 415	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> z e. X )
36		eqid 2238	. . . . . . . . . . 11 |- ( -g ` G ) = ( -g ` G )
37	1, 10, 36	grppncan 13896	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ z e. X ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
38	33, 34, 35, 37	syl3anc 1278	. . . . . . . . 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 13988	. . . . . . . . . 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 1278	. . . . . . . . 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 2316	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. S )
43	10	subgcl 13987	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ z e. S /\ w e. S ) -> ( z .+ w ) e. S )
44	31, 32, 42, 43	syl3anc 1278	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> ( z .+ w ) e. S )
45	30, 44	impbida 604	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
46	45	anassrs 404	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ z e. S ) /\ w e. X ) -> ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) )
47	46	ralrimiva 2623	. . . 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 14011	. . . 4 |- ( z e. N <-> ( z e. X /\ A. w e. X ( ( z .+ w ) e. S <-> ( w .+ z ) e. S ) ) )
50	3, 47, 49	sylanbrc 421	. . 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 3254	1 |- ( S e. (SubGrp ` G ) -> S C_ N )

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   +g cplusg 13431   0gc0g 13610   Grpcgrp 13805   invgcminusg 13806   -gcsg 13807  SubGrpcsubg 13970

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

This theorem is used by:  nmznsg  14016