Theorem conjnmz 17694

Description: A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	|- X = ( Base ` G )
conjghm.p	|- .+ = ( +g ` G )
conjghm.m	|- .- = ( -g ` G )
conjsubg.f	|- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
conjnmz.1	|- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }

Assertion

Ref	Expression
conjnmz	|- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )

Distinct variable groups:   x, y, .-   x, z, .+ , y   x, A, y, z   y, F, z   x, N   x, G, y, z   x, S, y, z   x, X, y, z

Allowed substitution hints:   F( x)   .- ( z)   N( y, z)

Proof of Theorem conjnmz

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgrcl 17599	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> G e. Grp )
2	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> G e. Grp )
3		conjnmz.1	. . . . . . . . . . . 12 |- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }
4		ssrab2 3687	. . . . . . . . . . . 12 |- { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } C_ X
5	3, 4	eqsstri 3635	. . . . . . . . . . 11 |- N C_ X
6		simplr 792	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A e. N )
7	5, 6	sseldi 3601	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A e. X )
8		conjghm.x	. . . . . . . . . . 11 |- X = ( Base ` G )
9		eqid 2622	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
10	8, 9	grpinvcl 17467	. . . . . . . . . 10 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
11	2, 7, 10	syl2anc 693	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( invg ` G ) ` A ) e. X )
12	8	subgss 17595	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> S C_ X )
13	12	adantr 481	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S C_ X )
14	13	sselda 3603	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. X )
15		conjghm.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
16	8, 15	grpass 17431	. . . . . . . . 9 |- ( ( G e. Grp /\ ( ( ( invg ` G ) ` A ) e. X /\ w e. X /\ A e. X ) ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) )
17	2, 11, 14, 7, 16	syl13anc 1328	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) )
18		eqid 2622	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
19	8, 15, 18, 9	grprinv 17469	. . . . . . . . . . . . 13 |- ( ( G e. Grp /\ A e. X ) -> ( A .+ ( ( invg ` G ) ` A ) ) = ( 0g ` G ) )
20	2, 7, 19	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( invg ` G ) ` A ) ) = ( 0g ` G ) )
21	20	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( ( 0g ` G ) .+ w ) )
22	8, 15	grpass 17431	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( A e. X /\ ( ( invg ` G ) ` A ) e. X /\ w e. X ) ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) )
23	2, 7, 11, 14, 22	syl13anc 1328	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) )
24	8, 15, 18	grplid 17452	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ w e. X ) -> ( ( 0g ` G ) .+ w ) = w )
25	2, 14, 24	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ w ) = w )
26	21, 23, 25	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) = w )
27		simpr 477	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. S )
28	26, 27	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) e. S )
29	8, 15	grpcl 17430	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( ( invg ` G ) ` A ) e. X /\ w e. X ) -> ( ( ( invg ` G ) ` A ) .+ w ) e. X )
30	2, 11, 14, 29	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ w ) e. X )
31	3	nmzbi 17634	. . . . . . . . . 10 |- ( ( A e. N /\ ( ( ( invg ` G ) ` A ) .+ w ) e. X ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) e. S <-> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) e. S ) )
32	6, 30, 31	syl2anc 693	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) e. S <-> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) e. S ) )
33	28, 32	mpbid 222	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) e. S )
34	17, 33	eqeltrrd 2702	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. S )
35		oveq2 6658	. . . . . . . . 9 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( A .+ x ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
36	35	oveq1d 6665	. . . . . . . 8 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( ( A .+ x ) .- A ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
37		conjsubg.f	. . . . . . . 8 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
38		ovex 6678	. . . . . . . 8 |- ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) e. _V
39	36, 37, 38	fvmpt 6282	. . . . . . 7 |- ( ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. S -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
40	34, 39	syl 17	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
41	20	oveq1d 6665	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ ( w .+ A ) ) = ( ( 0g ` G ) .+ ( w .+ A ) ) )
42	8, 15	grpcl 17430	. . . . . . . . . 10 |- ( ( G e. Grp /\ w e. X /\ A e. X ) -> ( w .+ A ) e. X )
43	2, 14, 7, 42	syl3anc 1326	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( w .+ A ) e. X )
44	8, 15	grpass 17431	. . . . . . . . 9 |- ( ( G e. Grp /\ ( A e. X /\ ( ( invg ` G ) ` A ) e. X /\ ( w .+ A ) e. X ) ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ ( w .+ A ) ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
45	2, 7, 11, 43, 44	syl13anc 1328	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ ( w .+ A ) ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
46	8, 15, 18	grplid 17452	. . . . . . . . 9 |- ( ( G e. Grp /\ ( w .+ A ) e. X ) -> ( ( 0g ` G ) .+ ( w .+ A ) ) = ( w .+ A ) )
47	2, 43, 46	syl2anc 693	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ ( w .+ A ) ) = ( w .+ A ) )
48	41, 45, 47	3eqtr3d 2664	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( w .+ A ) )
49	48	oveq1d 6665	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) = ( ( w .+ A ) .- A ) )
50		conjghm.m	. . . . . . . 8 |- .- = ( -g ` G )
51	8, 15, 50	grppncan 17506	. . . . . . 7 |- ( ( G e. Grp /\ w e. X /\ A e. X ) -> ( ( w .+ A ) .- A ) = w )
52	2, 14, 7, 51	syl3anc 1326	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( w .+ A ) .- A ) = w )
53	40, 49, 52	3eqtrd 2660	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = w )
54		ovex 6678	. . . . . . 7 |- ( ( A .+ x ) .- A ) e. _V
55	54, 37	fnmpti 6022	. . . . . 6 |- F Fn S
56		fnfvelrn 6356	. . . . . 6 |- ( ( F Fn S /\ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. ran F )
57	55, 34, 56	sylancr 695	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. ran F )
58	53, 57	eqeltrrd 2702	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. ran F )
59	58	ex 450	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ( w e. S -> w e. ran F ) )
60	59	ssrdv 3609	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S C_ ran F )
61	1	ad2antrr 762	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> G e. Grp )
62		simplr 792	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> A e. N )
63	5, 62	sseldi 3601	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> A e. X )
64	13	sselda 3603	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> x e. X )
65	8, 15, 50	grpaddsubass 17505	. . . . . 6 |- ( ( G e. Grp /\ ( A e. X /\ x e. X /\ A e. X ) ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
66	61, 63, 64, 63, 65	syl13anc 1328	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
67	8, 15, 50	grpnpcan 17507	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( ( x .- A ) .+ A ) = x )
68	61, 64, 63, 67	syl3anc 1326	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( x .- A ) .+ A ) = x )
69		simpr 477	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> x e. S )
70	68, 69	eqeltrd 2701	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( x .- A ) .+ A ) e. S )
71	8, 50	grpsubcl 17495	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
72	61, 64, 63, 71	syl3anc 1326	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( x .- A ) e. X )
73	3	nmzbi 17634	. . . . . . 7 |- ( ( A e. N /\ ( x .- A ) e. X ) -> ( ( A .+ ( x .- A ) ) e. S <-> ( ( x .- A ) .+ A ) e. S ) )
74	62, 72, 73	syl2anc 693	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ ( x .- A ) ) e. S <-> ( ( x .- A ) .+ A ) e. S ) )
75	70, 74	mpbird 247	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( A .+ ( x .- A ) ) e. S )
76	66, 75	eqeltrd 2701	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) e. S )
77	76, 37	fmptd 6385	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> F : S --> S )
78		frn 6053	. . 3 |- ( F : S --> S -> ran F C_ S )
79	77, 78	syl 17	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ran F C_ S )
80	60, 79	eqssd 3620	1 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424  SubGrpcsubg 17588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591

This theorem is referenced by:  conjnmzb  17695  conjnsg  17696  sylow3lem2  18043