Theorem hausllycmp 23357

Description: A compact Hausdorff space is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
hausllycmp	⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)

Proof of Theorem hausllycmp

Dummy variables 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 23194	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2729	. . . . . 6 ⊢ {𝑧 ∈ 𝐽 ∣ ∃𝑣 ∈ 𝐽 (𝑦 ∈ 𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ (∪ 𝐽 ∖ 𝑧))} = {𝑧 ∈ 𝐽 ∣ ∃𝑣 ∈ 𝐽 (𝑦 ∈ 𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ (∪ 𝐽 ∖ 𝑧))}
5		simpll 766	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Haus)
6		difssd 4096	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽)
7		simplr 768	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Comp)
8	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
9		simprl 770	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
10	3	opncld 22896	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
11	8, 9, 10	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
12		cmpcld 23265	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (∪ 𝐽 ∖ 𝑥)) ∈ Comp)
13	7, 11, 12	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (𝐽 ↾t (∪ 𝐽 ∖ 𝑥)) ∈ Comp)
14		simprr 772	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
15		elssuni 4897	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
16	15	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
17		dfss4 4228	. . . . . . . 8 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
18	16, 17	sylib 218	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
19	14, 18	eleqtrrd 2831	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
20	3, 4, 5, 6, 13, 19	hauscmplem 23269	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
21	18	sseq2d 3976	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
22	21	anbi2d 630	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ((𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))) ↔ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
23	22	rexbidv 3157	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))) ↔ ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
24	20, 23	mpbid 232	. . . 4 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
25	8	adantr 480	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Top)
26		simprl 770	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ 𝐽)
27		simprrl 780	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦 ∈ 𝑢)
28		opnneip 22982	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
29	25, 26, 27, 28	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30		elssuni 4897	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
31	30	ad2antrl 728	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ∪ 𝐽)
32	3	sscls 22919	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
33	25, 31, 32	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
34	3	clsss3 22922	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
35	25, 31, 34	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
36	3	ssnei2 22979	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑦})) ∧ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) ∧ ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
37	25, 29, 33, 35, 36	syl22anc 838	. . . . . 6 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38		simprrr 781	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
40	39	elpw2 5284	. . . . . . 7 ⊢ (((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
41	38, 40	sylibr 234	. . . . . 6 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥)
42	37, 41	elind 4159	. . . . 5 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
43	7	adantr 480	. . . . . 6 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Comp)
44	3	clscld 22910	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
45	25, 31, 44	syl2anc 584	. . . . . 6 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46		cmpcld 23265	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp)
47	43, 45, 46	syl2anc 584	. . . . 5 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48		oveq2 7377	. . . . . . 7 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t ((cls‘𝐽)‘𝑢)))
49	48	eleq1d 2813	. . . . . 6 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽 ↾t 𝑣) ∈ Comp ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp))
50	49	rspcev 3585	. . . . 5 ⊢ ((((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
51	42, 47, 50	syl2anc 584	. . . 4 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
52	24, 51	rexlimddv 3140	. . 3 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
53	52	ralrimivva 3178	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
54		isnlly 23332	. 2 ⊢ (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp))
55	2, 53, 54	sylanbrc 583	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3402   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867  ‘cfv 6499  (class class class)co 7369   ↾_t crest 17359  Topctop 22756  Clsdccld 22879  clsccl 22881  neicnei 22960  Hauscha 23171  Compccmp 23249  𝑛-Locally cnlly 23328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-1o 8411  df-2o 8412  df-en 8896  df-dom 8897  df-fin 8899  df-fi 9338  df-rest 17361  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-cld 22882  df-cls 22884  df-nei 22961  df-haus 23178  df-cmp 23250  df-nlly 23330

This theorem is referenced by: (None)