Theorem ssnmz 13662

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 13625	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
3	2	sselda 3201	. . . 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 13630	. . . . . . . . . . . . 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 3202	. . . . . . . . . . . 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 2207	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
12		eqid 2207	. . . . . . . . . . . . 13 |- ( invg ` G ) = ( invg ` G )
13	1, 10, 11, 12	grplinv 13497	. . . . . . . . . . . 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 5982	. . . . . . . . . 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 13634	. . . . . . . . . . . . 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 3202	. . . . . . . . . . 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 13456	. . . . . . . . . . 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 1252	. . . . . . . . . 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 13478	. . . . . . . . . . 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 2248	. . . . . . . . 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 13635	. . . . . . . . . 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 1250	. . . . . . . . 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 2285	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( z .+ w ) e. S ) -> w e. S )
29	10	subgcl 13635	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ w e. S /\ z e. S ) -> ( w .+ z ) e. S )
30	4, 28, 8, 29	syl3anc 1250	. . . . . . 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 2207	. . . . . . . . . . 11 |- ( -g ` G ) = ( -g ` G )
37	1, 10, 36	grppncan 13538	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ z e. X ) -> ( ( w .+ z ) ( -g ` G ) z ) = w )
38	33, 34, 35, 37	syl3anc 1250	. . . . . . . . 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 13636	. . . . . . . . . 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 1250	. . . . . . . . 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 2285	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( z e. S /\ w e. X ) ) /\ ( w .+ z ) e. S ) -> w e. S )
43	10	subgcl 13635	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ z e. S /\ w e. S ) -> ( z .+ w ) e. S )
44	31, 32, 42, 43	syl3anc 1250	. . . . . . 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 2581	. . . 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 13659	. . . 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 3207	1 |- ( S e. (SubGrp ` G ) -> S C_ N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   {crab 2490   C_ wss 3174   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448   -gcsg 13449  SubGrpcsubg 13618

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-subg 13621

This theorem is referenced by:  nmznsg  13664