Theorem nmznsg 13996

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 13994	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)
6		subgrcl 13962	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	2, 3, 4	nmzsubg 13993	. . . . 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 13980	. . . 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 957	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻))
13	2	ssrab3 3334	. . . . . 6 ⊢ 𝑁 ⊆ 𝑋
14	13	sseli 3244	. . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋)
15	2	nmzbi 13992	. . . . 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 2636	. . 3 ⊢ ∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)
18	9	subgbas 13961	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
19	8, 18	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
20	19	raleqdv 2755	. . . 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 2765	. . 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 2238	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
24		eqid 2238	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
25	23, 24	isnsg 13985	. . 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 13463	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
29	28	oveqd 6095	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 + 𝑤) = (𝑧(+g‘𝐻)𝑤))
30	29	eleq1d 2307	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑧(+g‘𝐻)𝑤) ∈ 𝑆))
31	28	oveqd 6095	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑤 + 𝑧) = (𝑤(+g‘𝐻)𝑧))
32	31	eleq1d 2307	. . . . . 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 2574	. . . 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 468	. . 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 957	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  ‘cfv 5375  (class class class)co 6078  Basecbs 13333   ↾_s cress 13334  +_gcplusg 13411  Grpcgrp 13785  SubGrpcsubg 13950  NrmSGrpcnsg 13951

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-i2m1 8277  ax-0lt1 8278  ax-0id 8280  ax-rnegex 8281  ax-pre-ltirr 8284  ax-pre-ltadd 8288

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-inn 9287  df-2 9345  df-ndx 13336  df-slot 13337  df-base 13339  df-sets 13340  df-iress 13341  df-plusg 13424  df-0g 13592  df-mgm 13656  df-sgrp 13697  df-mnd 13710  df-grp 13788  df-minusg 13789  df-sbg 13790  df-subg 13953  df-nsg 13954

This theorem is referenced by: (None)