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Theorem isnsg3 17675
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 17673 . . 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 17674 . . . . 5 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝑋𝑦𝑆) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
653expb 1285 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥𝑋𝑦𝑆)) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
76ralrimivva 3000 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
81, 7jca 553 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
9 simpl 472 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10 subgrcl 17646 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1110ad2antrr 762 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12 simprll 819 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧𝑋)
13 eqid 2651 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
14 eqid 2651 . . . . . . . . . . . 12 (invg𝐺) = (invg𝐺)
152, 3, 13, 14grplinv 17515 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1611, 12, 15syl2anc 694 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1716oveq1d 6705 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
182, 14grpinvcl 17514 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
1911, 12, 18syl2anc 694 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((invg𝐺)‘𝑧) ∈ 𝑋)
20 simprlr 820 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤𝑋)
212, 3grpass 17478 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
2211, 19, 12, 20, 21syl13anc 1368 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
232, 3, 13grplid 17499 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
2411, 20, 23syl2anc 694 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((0g𝐺) + 𝑤) = 𝑤)
2517, 22, 243eqtr3d 2693 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
2625oveq1d 6705 . . . . . . 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 17535 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
2826, 27eqtrd 2685 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
29 simprr 811 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30 simplr 807 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
31 oveq1 6697 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg𝐺)‘𝑧) + 𝑦))
32 id 22 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → 𝑥 = ((invg𝐺)‘𝑧))
3331, 32oveq12d 6708 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑥 + 𝑦) 𝑥) = ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)))
3433eleq1d 2715 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → (((𝑥 + 𝑦) 𝑥) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆))
35 oveq2 6698 . . . . . . . . . 10 (𝑦 = (𝑧 + 𝑤) → (((invg𝐺)‘𝑧) + 𝑦) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
3635oveq1d 6705 . . . . . . . . 9 (𝑦 = (𝑧 + 𝑤) → ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) = ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)))
3736eleq1d 2715 . . . . . . . 8 (𝑦 = (𝑧 + 𝑤) → (((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆))
3834, 37rspc2va 3354 . . . . . . 7 (((((invg𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
3919, 29, 30, 38syl21anc 1365 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
4028, 39eqeltrrd 2731 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆)
4140expr 642 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ (𝑧𝑋𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
4241ralrimivva 3000 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
432, 3isnsg2 17671 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
449, 42, 43sylanbrc 699 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
458, 44impbii 199 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Grpcgrp 17469  invgcminusg 17470  -gcsg 17471  SubGrpcsubg 17635  NrmSGrpcnsg 17636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-nsg 17639
This theorem is referenced by:  nsgacs  17677  0nsg  17686  nsgid  17687  ghmnsgima  17731  ghmnsgpreima  17732  cntrsubgnsg  17819  clsnsg  21960
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