MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnsg3 Structured version   Visualization version   GIF version

Theorem isnsg3 18301
Description: A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
isnsg3.1 𝑋 = (Base‘𝐺)
isnsg3.2 + = (+g𝐺)
isnsg3.3 = (-g𝐺)
Assertion
Ref Expression
isnsg3 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isnsg3
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgsubg 18299 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2 isnsg3.1 . . . . . 6 𝑋 = (Base‘𝐺)
3 isnsg3.2 . . . . . 6 + = (+g𝐺)
4 isnsg3.3 . . . . . 6 = (-g𝐺)
52, 3, 4nsgconj 18300 . . . . 5 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝑋𝑦𝑆) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
653expb 1117 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥𝑋𝑦𝑆)) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
76ralrimivva 3185 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
81, 7jca 515 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
9 simpl 486 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10 subgrcl 18273 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1110ad2antrr 725 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12 simprll 778 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧𝑋)
13 eqid 2824 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
14 eqid 2824 . . . . . . . . . . . 12 (invg𝐺) = (invg𝐺)
152, 3, 13, 14grplinv 18141 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1611, 12, 15syl2anc 587 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1716oveq1d 7153 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
182, 14grpinvcl 18140 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
1911, 12, 18syl2anc 587 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((invg𝐺)‘𝑧) ∈ 𝑋)
20 simprlr 779 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤𝑋)
212, 3grpass 18101 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
2211, 19, 12, 20, 21syl13anc 1369 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
232, 3, 13grplid 18122 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
2411, 20, 23syl2anc 587 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((0g𝐺) + 𝑤) = 𝑤)
2517, 22, 243eqtr3d 2867 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
2625oveq1d 7153 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) = (𝑤 ((invg𝐺)‘𝑧)))
272, 3, 4, 14, 11, 20, 12grpsubinv 18161 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
2826, 27eqtrd 2859 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
29 simprr 772 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30 simplr 768 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
31 oveq1 7145 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg𝐺)‘𝑧) + 𝑦))
32 id 22 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → 𝑥 = ((invg𝐺)‘𝑧))
3331, 32oveq12d 7156 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑥 + 𝑦) 𝑥) = ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)))
3433eleq1d 2900 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → (((𝑥 + 𝑦) 𝑥) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆))
35 oveq2 7146 . . . . . . . . . 10 (𝑦 = (𝑧 + 𝑤) → (((invg𝐺)‘𝑧) + 𝑦) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
3635oveq1d 7153 . . . . . . . . 9 (𝑦 = (𝑧 + 𝑤) → ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) = ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)))
3736eleq1d 2900 . . . . . . . 8 (𝑦 = (𝑧 + 𝑤) → (((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆))
3834, 37rspc2va 3619 . . . . . . 7 (((((invg𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
3919, 29, 30, 38syl21anc 836 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
4028, 39eqeltrrd 2917 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆)
4140expr 460 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ (𝑧𝑋𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
4241ralrimivva 3185 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
432, 3isnsg2 18297 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
449, 42, 43sylanbrc 586 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
458, 44impbii 212 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3132  cfv 6336  (class class class)co 7138  Basecbs 16472  +gcplusg 16554  0gc0g 16702  Grpcgrp 18092  invgcminusg 18093  -gcsg 18094  SubGrpcsubg 18262  NrmSGrpcnsg 18263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-0g 16704  df-mgm 17841  df-sgrp 17890  df-mnd 17901  df-grp 18095  df-minusg 18096  df-sbg 18097  df-subg 18265  df-nsg 18266
This theorem is referenced by:  nsgacs  18303  0nsg  18310  nsgid  18311  ghmnsgima  18371  ghmnsgpreima  18372  cntrsubgnsg  18460  clsnsg  22704
  Copyright terms: Public domain W3C validator