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Theorem isnsg3 18884
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 18882 . . 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 18883 . . . . 5 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝑋𝑦𝑆) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
653expb 1119 . . . 4 ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥𝑋𝑦𝑆)) → ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
76ralrimivva 3193 . . 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 18856 . . . . . . . . . . . 12 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1110ad2antrr 723 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12 simprll 776 . . . . . . . . . . 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 18724 . . . . . . . . . . 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 7352 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
182, 14grpinvcl 18723 . . . . . . . . . . 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 777 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤𝑋)
212, 3grpass 18682 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
2211, 19, 12, 20, 21syl13anc 1371 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
232, 3, 13grplid 18705 . . . . . . . . . 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 7352 . . . . . . 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 18744 . . . . . . 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 770 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30 simplr 766 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆)
31 oveq1 7344 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg𝐺)‘𝑧) + 𝑦))
32 id 22 . . . . . . . . . 10 (𝑥 = ((invg𝐺)‘𝑧) → 𝑥 = ((invg𝐺)‘𝑧))
3331, 32oveq12d 7355 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑥 + 𝑦) 𝑥) = ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)))
3433eleq1d 2821 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → (((𝑥 + 𝑦) 𝑥) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆))
35 oveq2 7345 . . . . . . . . . 10 (𝑦 = (𝑧 + 𝑤) → (((invg𝐺)‘𝑧) + 𝑦) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
3635oveq1d 7352 . . . . . . . . 9 (𝑦 = (𝑧 + 𝑤) → ((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) = ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)))
3736eleq1d 2821 . . . . . . . 8 (𝑦 = (𝑧 + 𝑤) → (((((invg𝐺)‘𝑧) + 𝑦) ((invg𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆))
3834, 37rspc2va 3580 . . . . . . 7 (((((invg𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
3919, 29, 30, 38syl21anc 835 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) ∧ ((𝑧𝑋𝑤𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ((invg𝐺)‘𝑧)) ∈ 𝑆)
4028, 39eqeltrrd 2838 . . . . 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 3193 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥𝑋𝑦𝑆 ((𝑥 + 𝑦) 𝑥) ∈ 𝑆) → ∀𝑧𝑋𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
432, 3isnsg2 18880 . . 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 1540  wcel 2105  wral 3061  cfv 6479  (class class class)co 7337  Basecbs 17009  +gcplusg 17059  0gc0g 17247  Grpcgrp 18673  invgcminusg 18674  -gcsg 18675  SubGrpcsubg 18845  NrmSGrpcnsg 18846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-minusg 18677  df-sbg 18678  df-subg 18848  df-nsg 18849
This theorem is referenced by:  nsgacs  18886  0nsg  18893  nsgid  18894  ghmnsgima  18954  ghmnsgpreima  18955  cntrsubgnsg  19043  clsnsg  23367
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