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Theorem nsgconj 19190
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 19189 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
213ad2ant1 1132 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
3 subgrcl 19162 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
42, 3syl 17 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐺 ∈ Grp)
5 simp2 1136 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐴𝑋)
6 isnsg3.1 . . . . . 6 𝑋 = (Base‘𝐺)
76subgss 19158 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
82, 7syl 17 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆𝑋)
9 simp3 1137 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑆)
108, 9sseldd 3996 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑋)
11 isnsg3.2 . . . 4 + = (+g𝐺)
12 isnsg3.3 . . . 4 = (-g𝐺)
136, 11, 12grpaddsubass 19061 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝐵𝑋𝐴𝑋)) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
144, 5, 10, 5, 13syl13anc 1371 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
156, 11, 12grpnpcan 19063 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → ((𝐵 𝐴) + 𝐴) = 𝐵)
164, 10, 5, 15syl3anc 1370 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) = 𝐵)
1716, 9eqeltrd 2839 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) ∈ 𝑆)
18 simp1 1135 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
196, 12grpsubcl 19051 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → (𝐵 𝐴) ∈ 𝑋)
204, 10, 5, 19syl3anc 1370 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐵 𝐴) ∈ 𝑋)
216, 11nsgbi 19188 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐵 𝐴) ∈ 𝑋𝐴𝑋) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2218, 20, 5, 21syl3anc 1370 . . 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 2839 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151  NrmSGrpcnsg 19152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155
This theorem is referenced by:  isnsg3  19191  ghmnsgima  19271  ghmnsgpreima  19272  clsnsg  24134
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