Theorem isnsg3 18312

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 18310	. . 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 18311	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
6	5	3expb 1116	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
7	6	ralrimivva 3191	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
8	1, 7	jca 514	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))
9		simpl 485	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10		subgrcl 18284	. . . . . . . . . . . 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 2821	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		eqid 2821	. . . . . . . . . . . 12 ⊢ (invg‘𝐺) = (invg‘𝐺)
15	2, 3, 13, 14	grplinv 18152	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
16	11, 12, 15	syl2anc 586	. . . . . . . . . 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 7171	. . . . . . . . 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 18151	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
19	11, 12, 18	syl2anc 586	. . . . . . . . . 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 18112	. . . . . . . . . 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 1368	. . . . . . . . 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 18133	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	11, 20, 23	syl2anc 586	. . . . . . . . 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 2864	. . . . . . . 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 7171	. . . . . . 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 18172	. . . . . . 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 2856	. . . . . 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 7163	. . . . . . . . . 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 7174	. . . . . . . . 9 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → ((𝑥 + 𝑦) − 𝑥) = ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)))
34	33	eleq1d 2897	. . . . . . . 8 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (((𝑥 + 𝑦) − 𝑥) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
35		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 + 𝑤) → (((invg‘𝐺)‘𝑧) + 𝑦) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
36	35	oveq1d 7171	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 + 𝑤) → ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) = ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)))
37	36	eleq1d 2897	. . . . . . . 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 3634	. . . . . . 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 835	. . . . . 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 2914	. . . . 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 459	. . . 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 3191	. . 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 18308	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
44	9, 42, 43	sylanbrc 585	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
45	8, 44	impbii 211	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  Grpcgrp 18103  inv_gcminusg 18104  -_gcsg 18105  SubGrpcsubg 18273  NrmSGrpcnsg 18274

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-sbg 18108  df-subg 18276  df-nsg 18277

This theorem is referenced by:  nsgacs  18314  0nsg  18321  nsgid  18322  ghmnsgima  18382  ghmnsgpreima  18383  cntrsubgnsg  18471  clsnsg  22718