Theorem nmznsg 13880

Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypotheses

Ref	Expression
elnmz.1	⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2	⊢ 𝑋 = (Base‘𝐺)
nmzsubg.3	⊢ + = (+g‘𝐺)
nmznsg.4	⊢ 𝐻 = (𝐺 ↾s 𝑁)

Assertion

Ref	Expression
nmznsg	⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐻(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem nmznsg

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
2		elnmz.1	. . . 4 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
3		nmzsubg.2	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
4		nmzsubg.3	. . . 4 ⊢ + = (+g‘𝐺)
5	2, 3, 4	ssnmz 13878	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)
6		subgrcl 13846	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	2, 3, 4	nmzsubg 13877	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺))
8	6, 7	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
9		nmznsg.4	. . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑁)
10	9	subsubg 13864	. . . 4 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁)))
11	8, 10	syl 14	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁)))
12	1, 5, 11	mpbir2and 953	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻))
13	2	ssrab3 3314	. . . . . 6 ⊢ 𝑁 ⊆ 𝑋
14	13	sseli 3224	. . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋)
15	2	nmzbi 13876	. . . . 5 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
16	14, 15	sylan2 286	. . . 4 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
17	16	rgen2 2619	. . 3 ⊢ ∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)
18	9	subgbas 13845	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
19	8, 18	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
20	19	raleqdv 2737	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
21	19, 20	raleqbidv 2747	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
22	17, 21	mpbii 148	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
23		eqid 2231	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
24		eqid 2231	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
25	23, 24	isnsg 13869	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g‘𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g‘𝐻)𝑧) ∈ 𝑆)))
26	9	a1i 9	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑁))
27	4	a1i 9	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → + = (+g‘𝐺))
28	26, 27, 8, 6	ressplusgd 13292	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
29	28	oveqd 6045	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 + 𝑤) = (𝑧(+g‘𝐻)𝑤))
30	29	eleq1d 2300	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑧(+g‘𝐻)𝑤) ∈ 𝑆))
31	28	oveqd 6045	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑤 + 𝑧) = (𝑤(+g‘𝐻)𝑧))
32	31	eleq1d 2300	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝑤 + 𝑧) ∈ 𝑆 ↔ (𝑤(+g‘𝐻)𝑧) ∈ 𝑆))
33	30, 32	bibi12d 235	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ((𝑧(+g‘𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g‘𝐻)𝑧) ∈ 𝑆)))
34	33	2ralbidv 2557	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g‘𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g‘𝐻)𝑧) ∈ 𝑆)))
35	34	anbi2d 464	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧(+g‘𝐻)𝑤) ∈ 𝑆 ↔ (𝑤(+g‘𝐻)𝑧) ∈ 𝑆))))
36	25, 35	bitr4id 199	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))))
37	12, 22, 36	mpbir2and 953	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  {crab 2515   ⊆ wss 3201  ‘cfv 5333  (class class class)co 6028  Basecbs 13162   ↾_s cress 13163  +_gcplusg 13240  Grpcgrp 13663  SubGrpcsubg 13834  NrmSGrpcnsg 13835

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-subg 13837  df-nsg 13838

This theorem is referenced by: (None)