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Theorem nsgconj 15818
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 15817 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
213ad2ant1 1009 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  e.  (SubGrp `  G ) )
3 subgrcl 15790 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 16 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  G  e.  Grp )
5 simp2 989 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  A  e.  X )
6 isnsg3.1 . . . . . 6  |-  X  =  ( Base `  G
)
76subgss 15786 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
82, 7syl 16 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  C_  X
)
9 simp3 990 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  B  e.  S )
108, 9sseldd 3457 . . 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 15719 . . 3  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  B
)  .-  A )  =  ( A  .+  ( B  .-  A ) ) )
144, 5, 10, 5, 13syl13anc 1221 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  =  ( A  .+  ( B 
.-  A ) ) )
156, 11, 12grpnpcan 15721 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( ( B  .-  A )  .+  A
)  =  B )
164, 10, 5, 15syl3anc 1219 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  =  B )
1716, 9eqeltrd 2539 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  e.  S
)
18 simp1 988 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  e.  (NrmSGrp `  G ) )
196, 12grpsubcl 15710 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  e.  X )
204, 10, 5, 19syl3anc 1219 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( B  .-  A )  e.  X
)
216, 11nsgbi 15816 . . . 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 1219 . . 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 2539 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 965    = wceq 1370    e. wcel 1758    C_ wss 3428   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   Grpcgrp 15514   -gcsg 15517  SubGrpcsubg 15779  NrmSGrpcnsg 15780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-0g 14484  df-mnd 15519  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-nsg 15783
This theorem is referenced by:  isnsg3  15819  ghmnsgima  15874  ghmnsgpreima  15875  clsnsg  19798
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