Theorem conjnmzb 17695

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		ssrab2 3687	. . . . 5 |- { y e. X | A. z e. X ( ( y .+ z ) e. S <-> ( z .+ y ) e. S ) } C_ X
3	1, 2	eqsstri 3635	. . . 4 |- N C_ X
4		simpr 477	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> A e. N )
5	3, 4	sseldi 3601	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> A e. X )
6		conjghm.x	. . . 4 |- X = ( Base ` G )
7		conjghm.p	. . . 4 |- .+ = ( +g ` G )
8		conjghm.m	. . . 4 |- .- = ( -g ` G )
9		conjsubg.f	. . . 4 |- F = ( x e. S |-> ( ( A .+ x ) .- A ) )
10	6, 7, 8, 9, 1	conjnmz 17694	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> S = ran F )
11	5, 10	jca 554	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. N ) -> ( A e. X /\ S = ran F ) )
12		simprl 794	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. X )
13		simplrr 801	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> S = ran F )
14	13	eleq2d 2687	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. S <-> ( A .+ w ) e. ran F ) )
15		subgrcl 17599	. . . . . . . . . . . . 13 |- ( S e. (SubGrp ` G ) -> G e. Grp )
16	15	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> G e. Grp )
17		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> A e. X )
18	6	subgss 17595	. . . . . . . . . . . . . 14 |- ( S e. (SubGrp ` G ) -> S C_ X )
19	18	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) -> S C_ X )
20	19	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> x e. X )
21	6, 7, 8	grpaddsubass 17505	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( A e. X /\ x e. X /\ A e. X ) ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
22	16, 17, 20, 17, 21	syl13anc 1328	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ x ) .- A ) = ( A .+ ( x .- A ) ) )
23	22	eqeq1d 2624	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( ( A .+ x ) .- A ) = ( A .+ w ) <-> ( A .+ ( x .- A ) ) = ( A .+ w ) ) )
24	6, 8	grpsubcl 17495	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ x e. X /\ A e. X ) -> ( x .- A ) e. X )
25	16, 20, 17, 24	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( x .- A ) e. X )
26		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> w e. X )
27	6, 7	grplcan 17477	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( ( x .- A ) e. X /\ w e. X /\ A e. X ) ) -> ( ( A .+ ( x .- A ) ) = ( A .+ w ) <-> ( x .- A ) = w ) )
28	16, 25, 26, 17, 27	syl13anc 1328	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ ( x .- A ) ) = ( A .+ w ) <-> ( x .- A ) = w ) )
29	6, 7, 8	grpsubadd 17503	. . . . . . . . . . 11 |- ( ( G e. Grp /\ ( x e. X /\ A e. X /\ w e. X ) ) -> ( ( x .- A ) = w <-> ( w .+ A ) = x ) )
30	16, 20, 17, 26, 29	syl13anc 1328	. . . . . . . . . 10 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( x .- A ) = w <-> ( w .+ A ) = x ) )
31	23, 28, 30	3bitrd 294	. . . . . . . . 9 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( ( A .+ x ) .- A ) = ( A .+ w ) <-> ( w .+ A ) = x ) )
32		eqcom 2629	. . . . . . . . 9 |- ( ( A .+ w ) = ( ( A .+ x ) .- A ) <-> ( ( A .+ x ) .- A ) = ( A .+ w ) )
33		eqcom 2629	. . . . . . . . 9 |- ( x = ( w .+ A ) <-> ( w .+ A ) = x )
34	31, 32, 33	3bitr4g 303	. . . . . . . 8 |- ( ( ( ( S e. (SubGrp ` G ) /\ A e. X ) /\ w e. X ) /\ x e. S ) -> ( ( A .+ w ) = ( ( A .+ x ) .- A ) <-> x = ( w .+ A ) ) )
35	34	rexbidva 3049	. . . . . . 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 ) ) )
36	35	adantlrr 757	. . . . . 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 ) ) )
37		ovex 6678	. . . . . . 7 |- ( A .+ w ) e. _V
38		eqeq1 2626	. . . . . . . 8 |- ( y = ( A .+ w ) -> ( y = ( ( A .+ x ) .- A ) <-> ( A .+ w ) = ( ( A .+ x ) .- A ) ) )
39	38	rexbidv 3052	. . . . . . 7 |- ( y = ( A .+ w ) -> ( E. x e. S y = ( ( A .+ x ) .- A ) <-> E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) ) )
40	9	rnmpt 5371	. . . . . . 7 |- ran F = { y | E. x e. S y = ( ( A .+ x ) .- A ) }
41	37, 39, 40	elab2 3354	. . . . . 6 |- ( ( A .+ w ) e. ran F <-> E. x e. S ( A .+ w ) = ( ( A .+ x ) .- A ) )
42		risset 3062	. . . . . 6 |- ( ( w .+ A ) e. S <-> E. x e. S x = ( w .+ A ) )
43	36, 41, 42	3bitr4g 303	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. ran F <-> ( w .+ A ) e. S ) )
44	14, 43	bitrd 268	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) /\ w e. X ) -> ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) )
45	44	ralrimiva 2966	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A. w e. X ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) )
46	1	elnmz 17633	. . 3 |- ( A e. N <-> ( A e. X /\ A. w e. X ( ( A .+ w ) e. S <-> ( w .+ A ) e. S ) ) )
47	12, 45, 46	sylanbrc 698	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. X /\ S = ran F ) ) -> A e. N )
48	11, 47	impbida 877	1 |- ( S e. (SubGrp ` G ) -> ( A e. N <-> ( A e. X /\ S = ran F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -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:  sylow3lem6  18047