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Theorem nsgconj 16558
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
StepHypRef Expression
1 nsgsubg 16557 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
213ad2ant1 1018 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  e.  (SubGrp `  G ) )
3 subgrcl 16530 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 17 . . 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
)
76subgss 16526 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
82, 7syl 17 . . . 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 )
108, 9sseldd 3443 . . 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 )
136, 11, 12grpaddsubass 16452 . . 3  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  B
)  .-  A )  =  ( A  .+  ( B  .-  A ) ) )
144, 5, 10, 5, 13syl13anc 1232 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  =  ( A  .+  ( B 
.-  A ) ) )
156, 11, 12grpnpcan 16454 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( ( B  .-  A )  .+  A
)  =  B )
164, 10, 5, 15syl3anc 1230 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  =  B )
1716, 9eqeltrd 2490 . . 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 ) )
196, 12grpsubcl 16442 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  e.  X )
204, 10, 5, 19syl3anc 1230 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( B  .-  A )  e.  X
)
216, 11nsgbi 16556 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  ( B  .-  A )  e.  X  /\  A  e.  X
)  ->  ( (
( B  .-  A
)  .+  A )  e.  S  <->  ( A  .+  ( B  .-  A ) )  e.  S ) )
2218, 20, 5, 21syl3anc 1230 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( (
( B  .-  A
)  .+  A )  e.  S  <->  ( A  .+  ( B  .-  A ) )  e.  S ) )
2317, 22mpbid 210 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( A  .+  ( B  .-  A
) )  e.  S
)
2414, 23eqeltrd 2490 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3414   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   Grpcgrp 16377   -gcsg 16379  SubGrpcsubg 16519  NrmSGrpcnsg 16520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-nsg 16523
This theorem is referenced by:  isnsg3  16559  ghmnsgima  16614  ghmnsgpreima  16615  clsnsg  20900
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