Theorem isnsg3 13739

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 13737	. . 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 13738	. . . . 5 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
6	5	3expb 1228	. . . 4 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
7	6	ralrimivva 2612	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)
8	1, 7	jca 306	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))
9		simpl 109	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
10		subgrcl 13711	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
11	10	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp)
12		simprll 537	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧 ∈ 𝑋)
13		eqid 2229	. . . . . . . . . . . 12 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		eqid 2229	. . . . . . . . . . . 12 ⊢ (invg‘𝐺) = (invg‘𝐺)
15	2, 3, 13, 14	grplinv 13578	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
16	11, 12, 15	syl2anc 411	. . . . . . . . . 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 6015	. . . . . . . . 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 13576	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
19	11, 12, 18	syl2anc 411	. . . . . . . . . 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 538	. . . . . . . . . 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 13537	. . . . . . . . . 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 1273	. . . . . . . . 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 13559	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	11, 20, 23	syl2anc 411	. . . . . . . . 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 2270	. . . . . . . 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 6015	. . . . . . 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 13601	. . . . . . 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 2262	. . . . . 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 531	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆)
30		simplr 528	. . . . . . 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 6007	. . . . . . . . . 10 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg‘𝐺)‘𝑧) + 𝑦))
32		id 19	. . . . . . . . . 10 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → 𝑥 = ((invg‘𝐺)‘𝑧))
33	31, 32	oveq12d 6018	. . . . . . . . 9 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → ((𝑥 + 𝑦) − 𝑥) = ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)))
34	33	eleq1d 2298	. . . . . . . 8 ⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (((𝑥 + 𝑦) − 𝑥) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) ∈ 𝑆))
35		oveq2 6008	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 + 𝑤) → (((invg‘𝐺)‘𝑧) + 𝑦) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
36	35	oveq1d 6015	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 + 𝑤) → ((((invg‘𝐺)‘𝑧) + 𝑦) − ((invg‘𝐺)‘𝑧)) = ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) − ((invg‘𝐺)‘𝑧)))
37	36	eleq1d 2298	. . . . . . . 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 2921	. . . . . . 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 1270	. . . . . 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 2307	. . . . 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 375	. . . 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 2612	. . 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 13735	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)))
44	9, 42, 43	sylanbrc 417	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺))
45	8, 44	impbii 126	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  +_gcplusg 13105  0_gc0g 13284  Grpcgrp 13528  inv_gcminusg 13529  -_gcsg 13530  SubGrpcsubg 13699  NrmSGrpcnsg 13700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-subg 13702  df-nsg 13703

This theorem is referenced by:  0nsg  13746  nsgid  13747  ghmnsgima  13800  ghmnsgpreima  13801