Theorem conjnmz 13409

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		conjsubg.f	. . . . . . 7 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
2		oveq2 5930	. . . . . . . 8 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( A .+ x ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
3	2	oveq1d 5937	. . . . . . 7 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( ( A .+ x ) .- A ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
4		subgrcl 13309	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> G e. Grp )
5	4	ad2antrr 488	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> G e. Grp )
6		conjghm.x	. . . . . . . . . 10 |- X = ( Base ` G )
7		eqid 2196	. . . . . . . . . 10 |- ( invg ` G ) = ( invg ` G )
8		conjnmz.1	. . . . . . . . . . . 12 |- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }
9	8	ssrab3 3269	. . . . . . . . . . 11 |- N C_ X
10		simplr 528	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A e. N )
11	9, 10	sselid 3181	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A e. X )
12	6, 7, 5, 11	grpinvcld 13181	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( invg ` G ) ` A ) e. X )
13	6	subgss 13304	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> S C_ X )
14	13	adantr 276	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S C_ X )
15	14	sselda 3183	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. X )
16		conjghm.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
17	6, 16	grpass 13141	. . . . . . . . 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 ) ) )
18	5, 12, 15, 11, 17	syl13anc 1251	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) )
19		eqid 2196	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
20	6, 16, 19, 7, 5, 11	grprinvd 13188	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( invg ` G ) ` A ) ) = ( 0g ` G ) )
21	20	oveq1d 5937	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( ( 0g ` G ) .+ w ) )
22	6, 16	grpass 13141	. . . . . . . . . . . 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	5, 11, 12, 15, 22	syl13anc 1251	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) )
24	6, 16, 19, 5, 15	grplidd 13165	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ w ) = w )
25	21, 23, 24	3eqtr3d 2237	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) = w )
26		simpr 110	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. S )
27	25, 26	eqeltrd 2273	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) e. S )
28	6, 16, 5, 12, 15	grpcld 13146	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ w ) e. X )
29	8	nmzbi 13339	. . . . . . . . . 10 |- ( ( A e. N /\ ( ( ( invg ` G ) ` A ) .+ w ) e. X ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ w ) ) e. S <-> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) e. S ) )
30	10, 28, 29	syl2anc 411	. . . . . . . . 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 ) )
31	27, 30	mpbid 147	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) e. S )
32	18, 31	eqeltrrd 2274	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. S )
33	6, 16, 5, 15, 11	grpcld 13146	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( w .+ A ) e. X )
34	6, 16, 5, 12, 33	grpcld 13146	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. X )
35	6, 16, 5, 11, 34	grpcld 13146	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. X )
36		conjghm.m	. . . . . . . . 9 |- .- = ( -g ` G )
37	6, 36	grpsubcl 13212	. . . . . . . 8 |- ( ( G e. Grp /\ ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. X /\ A e. X ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) e. X )
38	5, 35, 11, 37	syl3anc 1249	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) e. X )
39	1, 3, 32, 38	fvmptd3 5655	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
40	20	oveq1d 5937	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ ( w .+ A ) ) = ( ( 0g ` G ) .+ ( w .+ A ) ) )
41	6, 16	grpass 13141	. . . . . . . . 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 ) ) ) )
42	5, 11, 12, 33, 41	syl13anc 1251	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ ( w .+ A ) ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
43	6, 16, 19, 5, 33	grplidd 13165	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ ( w .+ A ) ) = ( w .+ A ) )
44	40, 42, 43	3eqtr3d 2237	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( w .+ A ) )
45	44	oveq1d 5937	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) = ( ( w .+ A ) .- A ) )
46	6, 16, 36	grppncan 13223	. . . . . . 7 |- ( ( G e. Grp /\ w e. X /\ A e. X ) -> ( ( w .+ A ) .- A ) = w )
47	5, 15, 11, 46	syl3anc 1249	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( w .+ A ) .- A ) = w )
48	39, 45, 47	3eqtrd 2233	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = w )
49	5	adantr 276	. . . . . . . . 9 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> G e. Grp )
50	11	adantr 276	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> A e. X )
51	14	adantr 276	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> S C_ X )
52	51	sselda 3183	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> x e. X )
53	6, 16, 49, 50, 52	grpcld 13146	. . . . . . . . 9 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> ( A .+ x ) e. X )
54	6, 36	grpsubcl 13212	. . . . . . . . 9 |- ( ( G e. Grp /\ ( A .+ x ) e. X /\ A e. X ) -> ( ( A .+ x ) .- A ) e. X )
55	49, 53, 50, 54	syl3anc 1249	. . . . . . . 8 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> ( ( A .+ x ) .- A ) e. X )
56	55	ralrimiva 2570	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A. x e. S ( ( A .+ x ) .- A ) e. X )
57	1	fnmpt 5384	. . . . . . 7 |- ( A. x e. S ( ( A .+ x ) .- A ) e. X -> F Fn S )
58	56, 57	syl 14	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> F Fn S )
59		fnfvelrn 5694	. . . . . 6 |- ( ( F Fn S /\ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. ran F )
60	58, 32, 59	syl2anc 411	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( F ` ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) e. ran F )
61	48, 60	eqeltrrd 2274	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. ran F )
62	61	ex 115	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ( w e. S -> w e. ran F ) )
63	62	ssrdv 3189	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S C_ ran F )
64	4	ad2antrr 488	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> G e. Grp )
65		simplr 528	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> A e. N )
66	9, 65	sselid 3181	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> A e. X )
67	14	sselda 3183	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> x e. X )
68	6, 16, 36	grpaddsubass 13222	. . . . . 6 |- ( ( G e. Grp /\ ( A e. X /\ x e. X /\ A e. X ) ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
69	64, 66, 67, 66, 68	syl13anc 1251	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
70	6, 16, 36	grpnpcan 13224	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( ( x .- A ) .+ A ) = x )
71	64, 67, 66, 70	syl3anc 1249	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( x .- A ) .+ A ) = x )
72		simpr 110	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> x e. S )
73	71, 72	eqeltrd 2273	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( x .- A ) .+ A ) e. S )
74	6, 36	grpsubcl 13212	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
75	64, 67, 66, 74	syl3anc 1249	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( x .- A ) e. X )
76	8	nmzbi 13339	. . . . . . 7 |- ( ( A e. N /\ ( x .- A ) e. X ) -> ( ( A .+ ( x .- A ) ) e. S <-> ( ( x .- A ) .+ A ) e. S ) )
77	65, 75, 76	syl2anc 411	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ ( x .- A ) ) e. S <-> ( ( x .- A ) .+ A ) e. S ) )
78	73, 77	mpbird 167	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( A .+ ( x .- A ) ) e. S )
79	69, 78	eqeltrd 2273	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) e. S )
80	79, 1	fmptd 5716	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> F : S --> S )
81	80	frnd 5417	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ran F C_ S )
82	63, 81	eqssd 3200	1 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   {crab 2479   C_ wss 3157   |-> cmpt 4094   ran crn 4664   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   Grpcgrp 13132   invgcminusg 13133   -gcsg 13134  SubGrpcsubg 13297

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-sbg 13137  df-subg 13300

This theorem is referenced by:  conjnmzb  13410  conjnsg  13411