Theorem conjnmzb 14060

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 3334	. . . 4 |- N C_ X
3		simpr 110	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> A e. N )
4	2, 3	sselid 3246	. . 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 14059	. . 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 535	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. X )
12		simplrr 542	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> S = ran F )
13	12	eleq2d 2308	. . . . 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 13959	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> G e. Grp )
15	14	ad3antrrr 496	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> G e. Grp )
16		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> A e. X )
17	5	subgss 13954	. . . . . . . . . . . . . 14 |- ( S e. (SubGrp ` G ) -> S C_ X )
18	17	ad2antrr 492	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) -> S C_ X )
19	18	sselda 3248	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> x e. X )
20	5, 6, 7	grpaddsubass 13872	. . . . . . . . . . . 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 1280	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
22	21	eqeq1d 2247	. . . . . . . . . 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 13862	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
24	15, 19, 16, 23	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( x .- A ) e. X )
25		simplr 533	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> w e. X )
26	5, 6	grplcan 13844	. . . . . . . . . . 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 1280	. . . . . . . . . 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 13870	. . . . . . . . . . 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 1280	. . . . . . . . . 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 2240	. . . . . . . . 9 |- ( ( A .+ w ) = ( ( A .+ x ) .- A ) <-> ( ( A .+ x ) .- A ) = ( A .+ w ) )
32		eqcom 2240	. . . . . . . . 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 2547	. . . . . . 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 487	. . . . . 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 492	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> G e. Grp )
37		simplrl 541	. . . . . . . 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 13796	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( A .+ w ) e. X )
40	8	elrnmpt 5026	. . . . . . 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 2578	. . . . . . 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 2623	. . 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 13988	. . 3 |- ( A e. N <-> ( A e. X /\ A. w e. X ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) ) )
48	11, 46, 47	sylanbrc 421	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. N )
49	10, 48	impbida 604	1 |- ( S e. (SubGrp ` G ) -> ( A e. N <-> ( A e. X /\ S = ran F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   |-> cmpt 4187   ran crn 4770   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   Grpcgrp 13782   -gcsg 13784  SubGrpcsubg 13947

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-subg 13950

This theorem is referenced by: (None)