Theorem nsgconj 17627

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 17626	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2	1	3ad2ant1 1082	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (SubGrp ` G ) )
3		subgrcl 17599	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
4	2, 3	syl 17	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> G e. Grp )
5		simp2 1062	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> A e. X )
6		isnsg3.1	. . . . . 6 |- X = ( Base ` G )
7	6	subgss 17595	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
8	2, 7	syl 17	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S C_ X )
9		simp3 1063	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. S )
10	8, 9	sseldd 3604	. . 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 17505	. . 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 1328	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
15	6, 11, 12	grpnpcan 17507	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( ( B .- A ) .+ A ) = B )
16	4, 10, 5, 15	syl3anc 1326	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) = B )
17	16, 9	eqeltrd 2701	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) e. S )
18		simp1 1061	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (NrmSGrp ` G ) )
19	6, 12	grpsubcl 17495	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( B .- A ) e. X )
20	4, 10, 5, 19	syl3anc 1326	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( B .- A ) e. X )
21	6, 11	nsgbi 17625	. . . 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 1326	. . 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 222	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( A .+ ( B .- A ) ) e. S )
24	14, 23	eqeltrd 2701	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588  NrmSGrpcnsg 17589

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  df-nsg 17592

This theorem is referenced by:  isnsg3  17628  ghmnsgima  17684  ghmnsgpreima  17685  clsnsg  21913