Theorem isnsg3 19143

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 19141	. . 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 19142	. . . . 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 3187	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
8	1, 7	jca 511	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))
9		simpl 482	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10		subgrcl 19114	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	10	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12		simprll 778	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
13		eqid 2735	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		eqid 2735	. . . . . . . . . . . 12 ⊢ (invg‘𝐺) = (invg‘𝐺)
15	2, 3, 13, 14	grplinv 18972	. . . . . . . . . . 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 7420	. . . . . . . . 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 18970	. . . . . . . . . . 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 779	. . . . . . . . . 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 18925	. . . . . . . . . 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 1374	. . . . . . . . 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 18950	. . . . . . . . . 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 2778	. . . . . . . 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 7420	. . . . . . 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 18995	. . . . . . 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 2770	. . . . . 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 772	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30		simplr 768	. . . . . . 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 7412	. . . . . . . . . 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 7423	. . . . . . . . 9 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → ((𝑥 + 𝑦) − 𝑥) = ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)))
34	33	eleq1d 2819	. . . . . . . 8 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (((𝑥 + 𝑦) − 𝑥) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
35		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 + 𝑤) → (((invg‘𝐺)‘𝑧) + 𝑦) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
36	35	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 + 𝑤) → ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) = ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)))
37	36	eleq1d 2819	. . . . . . . 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 3613	. . . . . . 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 837	. . . . . 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 2835	. . . . 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 456	. . . 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 3187	. . 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 19139	. . 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 209	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453  Grpcgrp 18916  inv_gcminusg 18917  -_gcsg 18918  SubGrpcsubg 19103  NrmSGrpcnsg 19104

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-nsg 19107

This theorem is referenced by:  nsgacs  19145  0nsg  19152  nsgid  19153  ghmnsgima  19223  ghmnsgpreima  19224  cntrsubgnsg  19326  clsnsg  24048