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Theorem nsgconj 14650
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 14649 . . . . 5  |-  ( S  e.  (NrmSGrp `  G
)  ->  S  e.  (SubGrp `  G ) )
213ad2ant1 976 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  e.  (SubGrp `  G ) )
3 subgrcl 14626 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 15 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  G  e.  Grp )
5 simp2 956 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  A  e.  X )
6 isnsg3.1 . . . . . 6  |-  X  =  ( Base `  G
)
76subgss 14622 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
82, 7syl 15 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  C_  X
)
9 simp3 957 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  B  e.  S )
108, 9sseldd 3181 . . 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 14555 . . 3  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  B
)  .-  A )  =  ( A  .+  ( B  .-  A ) ) )
144, 5, 10, 5, 13syl13anc 1184 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  =  ( A  .+  ( B 
.-  A ) ) )
156, 11, 12grpnpcan 14557 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( ( B  .-  A )  .+  A
)  =  B )
164, 10, 5, 15syl3anc 1182 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  =  B )
1716, 9eqeltrd 2357 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  e.  S
)
18 simp1 955 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  S  e.  (NrmSGrp `  G ) )
196, 12grpsubcl 14546 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  e.  X )
204, 10, 5, 19syl3anc 1182 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( B  .-  A )  e.  X
)
216, 11nsgbi 14648 . . . 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 1182 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( (
( B  .-  A
)  .+  A )  e.  S  <->  ( A  .+  ( B  .-  A ) )  e.  S ) )
2317, 22mpbid 201 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( A  .+  ( B  .-  A
) )  e.  S
)
2414, 23eqeltrd 2357 1  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615  NrmSGrpcnsg 14616
This theorem is referenced by:  isnsg3  14651  ghmnsgima  14706  ghmnsgpreima  14707  clsnsg  17792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-nsg 14619
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