Theorem nsgconj 13276

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 13275	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2	1	3ad2ant1 1020	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (SubGrp ` G ) )
3		subgrcl 13249	. . . 4 |- ( S e. (SubGrp ` G ) -> G e. Grp )
4	2, 3	syl 14	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> G e. Grp )
5		simp2 1000	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> A e. X )
6		isnsg3.1	. . . . . 6 |- X = ( Base ` G )
7	6	subgss 13244	. . . . 5 |- ( S e. (SubGrp ` G ) -> S C_ X )
8	2, 7	syl 14	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S C_ X )
9		simp3 1001	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. S )
10	8, 9	sseldd 3180	. . 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 13162	. . 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 1251	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
15	6, 11, 12	grpnpcan 13164	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( ( B .- A ) .+ A ) = B )
16	4, 10, 5, 15	syl3anc 1249	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) = B )
17	16, 9	eqeltrd 2270	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) e. S )
18		simp1 999	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (NrmSGrp ` G ) )
19	6, 12	grpsubcl 13152	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( B .- A ) e. X )
20	4, 10, 5, 19	syl3anc 1249	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( B .- A ) e. X )
21	6, 11	nsgbi 13274	. . . 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 1249	. . 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 147	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( A .+ ( B .- A ) ) e. S )
24	14, 23	eqeltrd 2270	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   -gcsg 13074  SubGrpcsubg 13237  NrmSGrpcnsg 13238

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-nsg 13241

This theorem is referenced by:  isnsg3  13277  ghmnsgima  13338  ghmnsgpreima  13339