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Theorem nsgconj 14749
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 14748 . . . . 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 14725 . . . 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 14721 . . . . 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 3257 . . 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 14654 . . 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 14656 . . . . 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 2432 . . 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 14645 . . . . 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 14747 . . . 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 2432 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 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337  (class class class)co 5945   Basecbs 13245   +g cplusg 13305   Grpcgrp 14461   -gcsg 14464  SubGrpcsubg 14714  NrmSGrpcnsg 14715
This theorem is referenced by:  isnsg3  14750  ghmnsgima  14805  ghmnsgpreima  14806  clsnsg  17894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-nsg 14718
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