Theorem nsgconj 15818

Description: The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypotheses

Ref	Expression
isnsg3.1	|- X = ( Base ` G )
isnsg3.2	|- .+ = ( +g ` G )
isnsg3.3	|- .- = ( -g ` G )

Assertion

Ref	Expression
nsgconj	|- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Proof of Theorem nsgconj

Step	Hyp	Ref	Expression
1		nsgsubg 15817	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2	1	3ad2ant1 1009	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (SubGrp ` G ) )
3		subgrcl 15790	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
4	2, 3	syl 16	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> G e. Grp )
5		simp2 989	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> A e. X )
6		isnsg3.1	. . . . . 6 |- X = ( Base ` G )
7	6	subgss 15786	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
8	2, 7	syl 16	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S C_ X )
9		simp3 990	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. S )
10	8, 9	sseldd 3457	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. X )
11		isnsg3.2	. . . 4 |- .+ = ( +g ` G )
12		isnsg3.3	. . . 4 |- .- = ( -g ` G )
13	6, 11, 12	grpaddsubass 15719	. . 3 |- ( ( G e. Grp /\ ( A e. X /\ B e. X /\ A e. X ) ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
14	4, 5, 10, 5, 13	syl13anc 1221	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
15	6, 11, 12	grpnpcan 15721	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( ( B .- A ) .+ A ) = B )
16	4, 10, 5, 15	syl3anc 1219	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) = B )
17	16, 9	eqeltrd 2539	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) e. S )
18		simp1 988	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (NrmSGrp ` G ) )
19	6, 12	grpsubcl 15710	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( B .- A ) e. X )
20	4, 10, 5, 19	syl3anc 1219	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( B .- A ) e. X )
21	6, 11	nsgbi 15816	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ ( B .- A ) e. X /\ A e. X ) -> ( ( ( B .- A ) .+ A ) e. S <-> ( A .+ ( B .- A ) ) e. S ) )
22	18, 20, 5, 21	syl3anc 1219	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( ( B .- A ) .+ A ) e. S <-> ( A .+ ( B .- A ) ) e. S ) )
23	17, 22	mpbid 210	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( A .+ ( B .- A ) ) e. S )
24	14, 23	eqeltrd 2539	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Syntax hints:   -> wi 4   <-> wb 184   /\ w3a 965   = wceq 1370   e. wcel 1758   C_ wss 3428   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   Grpcgrp 15514   -gcsg 15517  SubGrpcsubg 15779  NrmSGrpcnsg 15780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-0g 14484  df-mnd 15519  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-nsg 15783

This theorem is referenced by:  isnsg3  15819  ghmnsgima  15874  ghmnsgpreima  15875  clsnsg  19798