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Theorem isnsg3 19178
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 19176 . . 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 19177 . . . . 5 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝑋𝑦𝑆) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
653expb 1121 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥𝑋𝑦𝑆)) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
76ralrimivva 3202 . . 3 (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
81, 7jca 511 . 2 (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
9 simpl 482 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10 subgrcl 19149 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1110ad2antrr 726 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12 simprll 779 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧𝑋)
13 eqid 2737 . . . . . . . . . . . 12 (0g𝐺) = (0g𝐺)
14 eqid 2737 . . . . . . . . . . . 12 (invg𝐺) = (invg𝐺)
152, 3, 13, 14grplinv 19007 . . . . . . . . . . 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 7446 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
182, 14grpinvcl 19005 . . . . . . . . . . 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 780 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤𝑋)
212, 3grpass 18960 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
2211, 19, 12, 20, 21syl13anc 1374 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
232, 3, 13grplid 18985 . . . . . . . . . 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 2785 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
2625oveq1d 7446 . . . . . . 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 19030 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
2826, 27eqtrd 2777 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) = (𝑤 + 𝑧))
29 simprr 773 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30 simplr 769 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
31 oveq1 7438 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg𝐺)‘𝑧) + 𝑦))
32 id 22 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → 𝑥 = ((invg𝐺)‘𝑧))
3331, 32oveq12d 7449 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑥 + 𝑦) 𝑥) = ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)))
3433eleq1d 2826 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → (((𝑥 + 𝑦) 𝑥) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆))
35 oveq2 7439 . . . . . . . . . 10 (𝑦 = (𝑧 + 𝑤) → (((invg𝐺)‘𝑧) + 𝑦) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
3635oveq1d 7446 . . . . . . . . 9 (𝑦 = (𝑧 + 𝑤) → ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) = ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)))
3736eleq1d 2826 . . . . . . . 8 (𝑦 = (𝑧 + 𝑤) → (((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆))
3834, 37rspc2va 3634 . . . . . . 7 (((((invg𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
3919, 29, 30, 38syl21anc 838 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
4028, 39eqeltrrd 2842 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆)
4140expr 456 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ (𝑧𝑋𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
4241ralrimivva 3202 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
432, 3isnsg2 19174 . . 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 209 1 (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953  SubGrpcsubg 19138  NrmSGrpcnsg 19139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-nsg 19142
This theorem is referenced by:  nsgacs  19180  0nsg  19187  nsgid  19188  ghmnsgima  19258  ghmnsgpreima  19259  cntrsubgnsg  19361  clsnsg  24118
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