Theorem nmznsg 13419

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 13417	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)
6		subgrcl 13385	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	2, 3, 4	nmzsubg 13416	. . . . 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 13403	. . . 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 946	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻))
13	2	ssrab3 3270	. . . . . 6 ⊢ 𝑁 ⊆ 𝑋
14	13	sseli 3180	. . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋)
15	2	nmzbi 13415	. . . . 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 2583	. . 3 ⊢ ∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)
18	9	subgbas 13384	. . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
19	8, 18	syl 14	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻))
20	19	raleqdv 2699	. . . 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 2709	. . 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 2196	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
24		eqid 2196	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
25	23, 24	isnsg 13408	. . 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 12831	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → + = (+g‘𝐻))
29	28	oveqd 5942	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 + 𝑤) = (𝑧(+g‘𝐻)𝑤))
30	29	eleq1d 2265	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑧(+g‘𝐻)𝑤) ∈ 𝑆))
31	28	oveqd 5942	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑤 + 𝑧) = (𝑤(+g‘𝐻)𝑧))
32	31	eleq1d 2265	. . . . . 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 2521	. . . 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 946	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  {crab 2479   ⊆ wss 3157  ‘cfv 5259  (class class class)co 5925  Basecbs 12703   ↾_s cress 12704  +_gcplusg 12780  Grpcgrp 13202  SubGrpcsubg 13373  NrmSGrpcnsg 13374

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-nsg 13377

This theorem is referenced by: (None)