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Theorem nsgconj 17834
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 17833 . . . . 5 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
213ad2ant1 1126 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
3 subgrcl 17806 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
42, 3syl 17 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐺 ∈ Grp)
5 simp2 1130 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐴𝑋)
6 isnsg3.1 . . . . . 6 𝑋 = (Base‘𝐺)
76subgss 17802 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
82, 7syl 17 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆𝑋)
9 simp3 1131 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑆)
108, 9sseldd 3751 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝐵𝑋)
11 isnsg3.2 . . . 4 + = (+g𝐺)
12 isnsg3.3 . . . 4 = (-g𝐺)
136, 11, 12grpaddsubass 17712 . . 3 ((𝐺 ∈ Grp ∧ (𝐴𝑋𝐵𝑋𝐴𝑋)) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
144, 5, 10, 5, 13syl13anc 1477 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) = (𝐴 + (𝐵 𝐴)))
156, 11, 12grpnpcan 17714 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → ((𝐵 𝐴) + 𝐴) = 𝐵)
164, 10, 5, 15syl3anc 1475 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) = 𝐵)
1716, 9eqeltrd 2849 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐵 𝐴) + 𝐴) ∈ 𝑆)
18 simp1 1129 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
196, 12grpsubcl 17702 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵𝑋𝐴𝑋) → (𝐵 𝐴) ∈ 𝑋)
204, 10, 5, 19syl3anc 1475 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐵 𝐴) ∈ 𝑋)
216, 11nsgbi 17832 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝐵 𝐴) ∈ 𝑋𝐴𝑋) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2218, 20, 5, 21syl3anc 1475 . . 3 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (((𝐵 𝐴) + 𝐴) ∈ 𝑆 ↔ (𝐴 + (𝐵 𝐴)) ∈ 𝑆))
2317, 22mpbid 222 . 2 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → (𝐴 + (𝐵 𝐴)) ∈ 𝑆)
2414, 23eqeltrd 2849 1 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴𝑋𝐵𝑆) → ((𝐴 + 𝐵) 𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1070   = wceq 1630  wcel 2144  wss 3721  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Grpcgrp 17629  -gcsg 17631  SubGrpcsubg 17795  NrmSGrpcnsg 17796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-sbg 17634  df-subg 17798  df-nsg 17799
This theorem is referenced by:  isnsg3  17835  ghmnsgima  17891  ghmnsgpreima  17892  clsnsg  22132
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