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Theorem nsgconj 12997
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 12996 . . . . 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 12970 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
42, 3syl 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
)
76subgss 12965 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
82, 7syl 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 )
108, 9sseldd 3156 . . 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 12892 . . 3  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  B  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  B
)  .-  A )  =  ( A  .+  ( B  .-  A ) ) )
144, 5, 10, 5, 13syl13anc 1240 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( A  .+  B )  .-  A )  =  ( A  .+  ( B 
.-  A ) ) )
156, 11, 12grpnpcan 12894 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( ( B  .-  A )  .+  A
)  =  B )
164, 10, 5, 15syl3anc 1238 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( ( B  .-  A )  .+  A )  =  B )
1716, 9eqeltrd 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 ) )
196, 12grpsubcl 12882 . . . . 5  |-  ( ( G  e.  Grp  /\  B  e.  X  /\  A  e.  X )  ->  ( B  .-  A
)  e.  X )
204, 10, 5, 19syl3anc 1238 . . . 4  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( B  .-  A )  e.  X
)
216, 11nsgbi 12995 . . . 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 1238 . . 3  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( (
( B  .-  A
)  .+  A )  e.  S  <->  ( A  .+  ( B  .-  A ) )  e.  S ) )
2317, 22mpbid 147 . 2  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  A  e.  X  /\  B  e.  S
)  ->  ( A  .+  ( B  .-  A
) )  e.  S
)
2414, 23eqeltrd 2254 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 105    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   Grpcgrp 12809   -gcsg 12811  SubGrpcsubg 12958  NrmSGrpcnsg 12959
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 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-sbg 12814  df-subg 12961  df-nsg 12962
This theorem is referenced by:  isnsg3  12998
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