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Theorem nsgconj 19177
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 19176 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
213ad2ant1 1134 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
3 subgrcl 19149 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
42, 3syl 17 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐺 ∈ Grp)
5 simp2 1138 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐴𝑋)
6 isnsg3.1 . . . . . 6 𝑋 = (Base‘𝐺)
76subgss 19145 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
82, 7syl 17 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆𝑋)
9 simp3 1139 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑆)
108, 9sseldd 3984 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑋)
11 isnsg3.2 . . . 4 + = (+g𝐺)
12 isnsg3.3 . . . 4 = (-g𝐺)
136, 11, 12grpaddsubass 19048 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝐵𝑋𝐴𝑋)) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
144, 5, 10, 5, 13syl13anc 1374 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
156, 11, 12grpnpcan 19050 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → ((𝐵 𝐴) + 𝐴) = 𝐵)
164, 10, 5, 15syl3anc 1373 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) = 𝐵)
1716, 9eqeltrd 2841 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) ∈ 𝑆)
18 simp1 1137 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
196, 12grpsubcl 19038 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → (𝐵 𝐴) ∈ 𝑋)
204, 10, 5, 19syl3anc 1373 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐵 𝐴) ∈ 𝑋)
216, 11nsgbi 19175 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐵 𝐴) ∈ 𝑋𝐴𝑋) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2218, 20, 5, 21syl3anc 1373 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2317, 22mpbid 232 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐴 + (𝐵 𝐴)) ∈ 𝑆)
2414, 23eqeltrd 2841 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  -gcsg 18953  SubGrpcsubg 19138  NrmSGrpcnsg 19139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-nsg 19142
This theorem is referenced by:  isnsg3  19178  ghmnsgima  19258  ghmnsgpreima  19259  clsnsg  24118
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