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Theorem isnsg3 18962
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 18960 . . 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 18961 . . . . 5 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝑋𝑦𝑆) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
653expb 1120 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥𝑋𝑦𝑆)) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
76ralrimivva 3197 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
81, 7jca 512 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
9 simpl 483 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10 subgrcl 18933 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1110ad2antrr 724 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12 simprll 777 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧𝑋)
13 eqid 2736 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
14 eqid 2736 . . . . . . . . . . . 12 (invg𝐺) = (invg𝐺)
152, 3, 13, 14grplinv 18800 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1611, 12, 15syl2anc 584 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1716oveq1d 7372 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
182, 14grpinvcl 18798 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → ((invg𝐺)‘𝑧) ∈ 𝑋)
1911, 12, 18syl2anc 584 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((invg𝐺)‘𝑧) ∈ 𝑋)
20 simprlr 778 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤𝑋)
212, 3grpass 18757 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
2211, 19, 12, 20, 21syl13anc 1372 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
232, 3, 13grplid 18780 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
2411, 20, 23syl2anc 584 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((0g𝐺) + 𝑤) = 𝑤)
2517, 22, 243eqtr3d 2784 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
2625oveq1d 7372 . . . . . . 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 18820 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
2826, 27eqtrd 2776 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
29 simprr 771 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30 simplr 767 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
31 oveq1 7364 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg𝐺)‘𝑧) + 𝑦))
32 id 22 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → 𝑥 = ((invg𝐺)‘𝑧))
3331, 32oveq12d 7375 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑥 + 𝑦) 𝑥) = ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)))
3433eleq1d 2822 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → (((𝑥 + 𝑦) 𝑥) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆))
35 oveq2 7365 . . . . . . . . . 10 (𝑦 = (𝑧 + 𝑤) → (((invg𝐺)‘𝑧) + 𝑦) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
3635oveq1d 7372 . . . . . . . . 9 (𝑦 = (𝑧 + 𝑤) → ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) = ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)))
3736eleq1d 2822 . . . . . . . 8 (𝑦 = (𝑧 + 𝑤) → (((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆))
3834, 37rspc2va 3591 . . . . . . 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 2839 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆)
4140expr 457 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ (𝑧𝑋𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
4241ralrimivva 3197 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
432, 3isnsg2 18958 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
449, 42, 43sylanbrc 583 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
458, 44impbii 208 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749  -gcsg 18750  SubGrpcsubg 18922  NrmSGrpcnsg 18923
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-nsg 18926
This theorem is referenced by:  nsgacs  18964  0nsg  18971  nsgid  18972  ghmnsgima  19032  ghmnsgpreima  19033  cntrsubgnsg  19121  clsnsg  23461
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