Theorem nsgconj 13873

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 13872	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2	1	3ad2ant1 1045	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (SubGrp ` G ) )
3		subgrcl 13846	. . . 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 1025	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> A e. X )
6		isnsg3.1	. . . . . 6 |- X = ( Base ` G )
7	6	subgss 13841	. . . . 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 1026	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. S )
10	8, 9	sseldd 3229	. . 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 13753	. . 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 1276	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
15	6, 11, 12	grpnpcan 13755	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( ( B .- A ) .+ A ) = B )
16	4, 10, 5, 15	syl3anc 1274	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) = B )
17	16, 9	eqeltrd 2308	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) e. S )
18		simp1 1024	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (NrmSGrp ` G ) )
19	6, 12	grpsubcl 13743	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( B .- A ) e. X )
20	4, 10, 5, 19	syl3anc 1274	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( B .- A ) e. X )
21	6, 11	nsgbi 13871	. . . 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 1274	. . 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 2308	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   Grpcgrp 13663   -gcsg 13665  SubGrpcsubg 13834  NrmSGrpcnsg 13835

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837  df-nsg 13838

This theorem is referenced by:  isnsg3  13874  ghmnsgima  13935  ghmnsgpreima  13936