Theorem conjnmz 13776

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 5977	. . . . . . . 8 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( A .+ x ) = ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) )
3	2	oveq1d 5984	. . . . . . 7 |- ( x = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) -> ( ( A .+ x ) .- A ) = ( ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) .- A ) )
4		subgrcl 13676	. . . . . . . . . 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 2207	. . . . . . . . . 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 3288	. . . . . . . . . . 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 3200	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A e. X )
12	6, 7, 5, 11	grpinvcld 13542	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( invg ` G ) ` A ) e. X )
13	6	subgss 13671	. . . . . . . . . . 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 3202	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> w e. X )
16		conjghm.p	. . . . . . . . . 10 |- .+ = ( +g ` G )
17	6, 16	grpass 13502	. . . . . . . . 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 1252	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( ( invg ` G ) ` A ) .+ w ) .+ A ) = ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) )
19		eqid 2207	. . . . . . . . . . . . 13 |- ( 0g ` G ) = ( 0g ` G )
20	6, 16, 19, 7, 5, 11	grprinvd 13549	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( invg ` G ) ` A ) ) = ( 0g ` G ) )
21	20	oveq1d 5984	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( A .+ ( ( invg ` G ) ` A ) ) .+ w ) = ( ( 0g ` G ) .+ w ) )
22	6, 16	grpass 13502	. . . . . . . . . . . 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 1252	. . . . . . . . . . 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 13526	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ w ) = w )
25	21, 23, 24	3eqtr3d 2248	. . . . . . . . . 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 2284	. . . . . . . . 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 13507	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( ( invg ` G ) ` A ) .+ w ) e. X )
29	8	nmzbi 13706	. . . . . . . . . 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 2285	. . . . . . 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 13507	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( w .+ A ) e. X )
34	6, 16, 5, 12, 33	grpcld 13507	. . . . . . . . 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 13507	. . . . . . . 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 13573	. . . . . . . 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 1250	. . . . . . 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 5698	. . . . . 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 5984	. . . . . . . 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 13502	. . . . . . . . 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 1252	. . . . . . . 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 13526	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( 0g ` G ) .+ ( w .+ A ) ) = ( w .+ A ) )
44	40, 42, 43	3eqtr3d 2248	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( A .+ ( ( ( invg ` G ) ` A ) .+ ( w .+ A ) ) ) = ( w .+ A ) )
45	44	oveq1d 5984	. . . . . 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 13584	. . . . . . 7 |- ( ( G e. Grp /\ w e. X /\ A e. X ) -> ( ( w .+ A ) .- A ) = w )
47	5, 15, 11, 46	syl3anc 1250	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> ( ( w .+ A ) .- A ) = w )
48	39, 45, 47	3eqtrd 2244	. . . . 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 3202	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> x e. X )
53	6, 16, 49, 50, 52	grpcld 13507	. . . . . . . . 9 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> ( A .+ x ) e. X )
54	6, 36	grpsubcl 13573	. . . . . . . . 9 |- ( ( G e. Grp /\ ( A .+ x ) e. X /\ A e. X ) -> ( ( A .+ x ) .- A ) e. X )
55	49, 53, 50, 54	syl3anc 1250	. . . . . . . 8 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) /\ x e. S ) -> ( ( A .+ x ) .- A ) e. X )
56	55	ralrimiva 2581	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ w e. S ) -> A. x e. S ( ( A .+ x ) .- A ) e. X )
57	1	fnmpt 5423	. . . . . . 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 5737	. . . . . 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 2285	. . . 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 3208	. 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 3200	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> A e. X )
67	14	sselda 3202	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> x e. X )
68	6, 16, 36	grpaddsubass 13583	. . . . . 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 1252	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
70	6, 16, 36	grpnpcan 13585	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( ( x .- A ) .+ A ) = x )
71	64, 67, 66, 70	syl3anc 1250	. . . . . . 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 2284	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( x .- A ) .+ A ) e. S )
74	6, 36	grpsubcl 13573	. . . . . . . 8 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
75	64, 67, 66, 74	syl3anc 1250	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( x .- A ) e. X )
76	8	nmzbi 13706	. . . . . . 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 2284	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ A e. N ) /\ x e. S ) -> ( ( A .+ x ) .- A ) e. S )
80	79, 1	fmptd 5759	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> F : S --> S )
81	80	frnd 5456	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ran F C_ S )
82	63, 81	eqssd 3219	1 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   {crab 2490   C_ wss 3175   |-> cmpt 4122   ran crn 4695   Fn wfn 5286   ` cfv 5291  (class class class)co 5969   Basecbs 12993   +g cplusg 13070   0gc0g 13249   Grpcgrp 13493   invgcminusg 13494   -gcsg 13495  SubGrpcsubg 13664

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 4176  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1re 8056  ax-addrcl 8059

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-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-inn 9074  df-2 9132  df-ndx 12996  df-slot 12997  df-base 12999  df-plusg 13083  df-0g 13251  df-mgm 13349  df-sgrp 13395  df-mnd 13410  df-grp 13496  df-minusg 13497  df-sbg 13498  df-subg 13667

This theorem is referenced by:  conjnmzb  13777  conjnsg  13778