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Theorem nsgconj 17944
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 + = (+g𝐺)
isnsg3.3 = (-g𝐺)
Assertion
Ref Expression
nsgconj ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)

Proof of Theorem nsgconj
StepHypRef Expression
1 nsgsubg 17943 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
213ad2ant1 1164 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
3 subgrcl 17916 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
42, 3syl 17 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐺 ∈ Grp)
5 simp2 1168 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐴𝑋)
6 isnsg3.1 . . . . . 6 𝑋 = (Base‘𝐺)
76subgss 17912 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
82, 7syl 17 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆𝑋)
9 simp3 1169 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑆)
108, 9sseldd 3803 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑋)
11 isnsg3.2 . . . 4 + = (+g𝐺)
12 isnsg3.3 . . . 4 = (-g𝐺)
136, 11, 12grpaddsubass 17825 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝐵𝑋𝐴𝑋)) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
144, 5, 10, 5, 13syl13anc 1492 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
156, 11, 12grpnpcan 17827 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → ((𝐵 𝐴) + 𝐴) = 𝐵)
164, 10, 5, 15syl3anc 1491 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) = 𝐵)
1716, 9eqeltrd 2882 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) ∈ 𝑆)
18 simp1 1167 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
196, 12grpsubcl 17815 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → (𝐵 𝐴) ∈ 𝑋)
204, 10, 5, 19syl3anc 1491 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐵 𝐴) ∈ 𝑋)
216, 11nsgbi 17942 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐵 𝐴) ∈ 𝑋𝐴𝑋) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2218, 20, 5, 21syl3anc 1491 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2317, 22mpbid 224 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐴 + (𝐵 𝐴)) ∈ 𝑆)
2414, 23eqeltrd 2882 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  wss 3773  cfv 6105  (class class class)co 6882  Basecbs 16188  +gcplusg 16271  Grpcgrp 17742  -gcsg 17744  SubGrpcsubg 17905  NrmSGrpcnsg 17906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-riota 6843  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-1st 7405  df-2nd 7406  df-0g 16421  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-minusg 17746  df-sbg 17747  df-subg 17908  df-nsg 17909
This theorem is referenced by:  isnsg3  17945  ghmnsgima  18001  ghmnsgpreima  18002  clsnsg  22245
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