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	⊢ + = (+g‘𝐺)
isnsg3.3	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
isnsg3	⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((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.

Step	Hyp	Ref	Expression
1		nsgsubg 18960	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		isnsg3.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
3		isnsg3.2	. . . . . 6 ⊢ + = (+g‘𝐺)
4		isnsg3.3	. . . . . 6 ⊢ − = (-g‘𝐺)
5	2, 3, 4	nsgconj 18961	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
6	5	3expb 1120	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
7	6	ralrimivva 3197	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
8	1, 7	jca 512	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))
9		simpl 483	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10		subgrcl 18933	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	10	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12		simprll 777	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
13		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		eqid 2736	. . . . . . . . . . . 12 ⊢ (invg‘𝐺) = (invg‘𝐺)
15	2, 3, 13, 14	grplinv 18800	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
16	11, 12, 15	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
17	16	oveq1d 7372	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g‘𝐺) + 𝑤))
18	2, 14	grpinvcl 18798	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
19	11, 12, 18	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
20		simprlr 778	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤 ∈ 𝑋)
21	2, 3	grpass 18757	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
22	11, 19, 12, 20, 21	syl13anc 1372	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
23	2, 3, 13	grplid 18780	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	11, 20, 23	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((0g‘𝐺) + 𝑤) = 𝑤)
25	17, 22, 24	3eqtr3d 2784	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
26	25	oveq1d 7372	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)) = (𝑤 − ((invg‘𝐺)‘𝑧)))
27	2, 3, 4, 14, 11, 20, 12	grpsubinv 18820	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 − ((invg‘𝐺)‘𝑧)) = (𝑤 + 𝑧))
28	26, 27	eqtrd 2776	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)) = (𝑤 + 𝑧))
29		simprr 771	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30		simplr 767	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
31		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg‘𝐺)‘𝑧) + 𝑦))
32		id 22	. . . . . . . . . 10 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → 𝑥 = ((invg‘𝐺)‘𝑧))
33	31, 32	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → ((𝑥 + 𝑦) − 𝑥) = ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)))
34	33	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (((𝑥 + 𝑦) − 𝑥) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
35		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 + 𝑤) → (((invg‘𝐺)‘𝑧) + 𝑦) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
36	35	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 + 𝑤) → ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) = ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)))
37	36	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = (𝑧 + 𝑤) → (((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
38	34, 37	rspc2va 3591	. . . . . . 7 ⊢ (((((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆)
39	19, 29, 30, 38	syl21anc 836	. . . . . 6 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆)
40	28, 39	eqeltrrd 2839	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆)
41	40	expr 457	. . . 4 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
42	41	ralrimivva 3197	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))
43	2, 3	isnsg2 18958	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
44	9, 42, 43	sylanbrc 583	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
45	8, 44	impbii 208	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))

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  0_gc0g 17321  Grpcgrp 18748  inv_gcminusg 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