Theorem conjnmzb 13219

Description: Alternative condition for elementhood in the 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
conjnmzb	|- ( S e. (SubGrp ` G ) -> ( A e. N <-> ( A e. X /\ 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 conjnmzb

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		conjnmz.1	. . . . 5 |- N = { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) }
2	1	ssrab3 3256	. . . 4 |- N C_ X
3		simpr 110	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> A e. N )
4	2, 3	sselid 3168	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> A e. X )
5		conjghm.x	. . . 4 |- X = ( Base ` G )
6		conjghm.p	. . . 4 |- .+ = ( +g ` G )
7		conjghm.m	. . . 4 |- .- = ( -g ` G )
8		conjsubg.f	. . . 4 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
9	5, 6, 7, 8, 1	conjnmz 13218	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )
10	4, 9	jca 306	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ( A e. X /\ S = ran F ) )
11		simprl 529	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. X )
12		simplrr 536	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> S = ran F )
13	12	eleq2d 2259	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. S <-> ( A .+ w ) e. ran F ) )
14		subgrcl 13118	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> G e. Grp )
15	14	ad3antrrr 492	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> G e. Grp )
16		simpllr 534	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> A e. X )
17	5	subgss 13113	. . . . . . . . . . . . . 14 |- ( S e. (SubGrp ` G ) -> S C_ X )
18	17	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) -> S C_ X )
19	18	sselda 3170	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> x e. X )
20	5, 6, 7	grpaddsubass 13034	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( A e. X /\ x e. X /\ A e. X ) ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
21	15, 16, 19, 16, 20	syl13anc 1251	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
22	21	eqeq1d 2198	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( ( A .+ x ) .- A ) = ( A .+ w ) <-> ( A .+ ( x .- A ) ) = ( A .+ w ) ) )
23	5, 7	grpsubcl 13024	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
24	15, 19, 16, 23	syl3anc 1249	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( x .- A ) e. X )
25		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> w e. X )
26	5, 6	grplcan 13006	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( ( x .- A ) e. X /\ w e. X /\ A e. X ) ) -> ( ( A .+ ( x .- A ) ) = ( A .+ w ) <-> ( x .- A ) = w ) )
27	15, 24, 25, 16, 26	syl13anc 1251	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ ( x .- A ) ) = ( A .+ w ) <-> ( x .- A ) = w ) )
28	5, 6, 7	grpsubadd 13032	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( x e. X /\ A e. X /\ w e. X ) ) -> ( ( x .- A ) = w <-> ( w .+ A ) = x ) )
29	15, 19, 16, 25, 28	syl13anc 1251	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( x .- A ) = w <-> ( w .+ A ) = x ) )
30	22, 27, 29	3bitrd 214	. . . . . . . . 9 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( ( A .+ x ) .- A ) = ( A .+ w ) <-> ( w .+ A ) = x ) )
31		eqcom 2191	. . . . . . . . 9 |- ( ( A .+ w ) = ( ( A .+ x ) .- A ) <-> ( ( A .+ x ) .- A ) = ( A .+ w ) )
32		eqcom 2191	. . . . . . . . 9 |- ( x = ( w .+ A ) <-> ( w .+ A ) = x )
33	30, 31, 32	3bitr4g 223	. . . . . . . 8 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ w ) = ( ( A .+ x ) .- A ) <-> x = ( w .+ A ) ) )
34	33	rexbidva 2487	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) -> ( E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) <-> E. x e. S x = ( w .+ A ) ) )
35	34	adantlrr 483	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) <-> E. x e. S x = ( w .+ A ) ) )
36	14	ad2antrr 488	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> G e. Grp )
37		simplrl 535	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> A e. X )
38		simpr 110	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> w e. X )
39	5, 6, 36, 37, 38	grpcld 12959	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( A .+ w ) e. X )
40	8	elrnmpt 4894	. . . . . . 7 |- ( ( A .+ w ) e. X -> ( ( A .+ w ) e. ran F <-> E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) ) )
41	39, 40	syl 14	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. ran F <-> E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) ) )
42		risset 2518	. . . . . . 7 |- ( ( w .+ A ) e. S <-> E. x e. S x = ( w .+ A ) )
43	42	a1i 9	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( w .+ A ) e. S <-> E. x e. S x = ( w .+ A ) ) )
44	35, 41, 43	3bitr4d 220	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. ran F <-> ( w .+ A ) e. S ) )
45	13, 44	bitrd 188	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) )
46	45	ralrimiva 2563	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A. w e. X ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) )
47	1	elnmz 13147	. . 3 |- ( A e. N <-> ( A e. X /\ A. w e. X ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) ) )
48	11, 46, 47	sylanbrc 417	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. N )
49	10, 48	impbida 596	1 |- ( S e. (SubGrp ` G ) -> ( A e. N <-> ( A e. X /\ S = ran F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   C_ wss 3144   |-> cmpt 4079   ran crn 4645   ` cfv 5235  (class class class)co 5896   Basecbs 12512   +g cplusg 12589   Grpcgrp 12945   -gcsg 12947  SubGrpcsubg 13106

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-inn 8950  df-2 9008  df-ndx 12515  df-slot 12516  df-base 12518  df-plusg 12602  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-grp 12948  df-minusg 12949  df-sbg 12950  df-subg 13109

This theorem is referenced by: (None)