Theorem nsgconj 13076

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 13075	. . . . 5 |- ( S e. (NrmSGrp ` G ) -> S e. (SubGrp ` G ) )
2	1	3ad2ant1 1018	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (SubGrp ` G ) )
3		subgrcl 13049	. . . 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 998	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> A e. X )
6		isnsg3.1	. . . . . 6 |- X = ( Base ` G )
7	6	subgss 13044	. . . . 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 999	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> B e. S )
10	8, 9	sseldd 3158	. . 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 12967	. . 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 1240	. 2 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) = ( A .+ ( B .- A ) ) )
15	6, 11, 12	grpnpcan 12969	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( ( B .- A ) .+ A ) = B )
16	4, 10, 5, 15	syl3anc 1238	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) = B )
17	16, 9	eqeltrd 2254	. . 3 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( B .- A ) .+ A ) e. S )
18		simp1 997	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> S e. (NrmSGrp ` G ) )
19	6, 12	grpsubcl 12957	. . . . 5 |- ( ( G e. Grp /\ B e. X /\ A e. X ) -> ( B .- A ) e. X )
20	4, 10, 5, 19	syl3anc 1238	. . . 4 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( B .- A ) e. X )
21	6, 11	nsgbi 13074	. . . 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 1238	. . 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 2254	1 |- ( ( S e. (NrmSGrp ` G ) /\ A e. X /\ B e. S ) -> ( ( A .+ B ) .- A ) e. S )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3131   ` cfv 5218  (class class class)co 5878   Basecbs 12465   +g cplusg 12539   Grpcgrp 12884   -gcsg 12886  SubGrpcsubg 13037  NrmSGrpcnsg 13038

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-inn 8923  df-2 8981  df-ndx 12468  df-slot 12469  df-base 12471  df-plusg 12552  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12825  df-grp 12887  df-minusg 12888  df-sbg 12889  df-subg 13040  df-nsg 13041

This theorem is referenced by:  isnsg3  13077