Theorem hausllycmp 21107

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 20945	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3		eqid 2610	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2610	. . . . . 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 786	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Haus)
6		difssd 3700	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽)
7		simplr 788	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Comp)
8	1	ad2antrr 758	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top)
9		simprl 790	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
10	3	opncld 20647	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
11	8, 9, 10	syl2anc 691	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
12		cmpcld 21015	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (∪ 𝐽 ∖ 𝑥)) ∈ Comp)
13	7, 11, 12	syl2anc 691	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (𝐽 ↾t (∪ 𝐽 ∖ 𝑥)) ∈ Comp)
14		simprr 792	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
15		elssuni 4403	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
16	15	ad2antrl 760	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
17		dfss4 3820	. . . . . . . 8 ⊢ (𝑥 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
18	16, 17	sylib 207	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) = 𝑥)
19	14, 18	eleqtrrd 2691	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
20	3, 4, 5, 6, 13, 19	hauscmplem 21019	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
21	18	sseq2d 3596	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
22	21	anbi2d 736	. . . . . 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 3034	. . . . 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 221	. . . 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 790	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ 𝐽)
27		simprrl 800	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦 ∈ 𝑢)
28		opnneip 20733	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
29	25, 26, 27, 28	syl3anc 1318	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30		elssuni 4403	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽)
31	30	ad2antrl 760	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ∪ 𝐽)
32	3	sscls 20670	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
33	25, 31, 32	syl2anc 691	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
34	3	clsss3 20673	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
35	25, 31, 34	syl2anc 691	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽)
36	3	ssnei2 20730	. . . . . . 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 1319	. . . . . 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 801	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
40	39	elpw2 4755	. . . . . . 7 ⊢ (((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
41	38, 40	sylibr 223	. . . . . 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 3760	. . . . 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 20661	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
45	25, 31, 44	syl2anc 691	. . . . . 6 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46		cmpcld 21015	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp)
47	43, 45, 46	syl2anc 691	. . . . 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 6557	. . . . . . 7 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t ((cls‘𝐽)‘𝑢)))
49	48	eleq1d 2672	. . . . . 6 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽 ↾t 𝑣) ∈ Comp ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp))
50	49	rspcev 3282	. . . . 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 691	. . . 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 3017	. . 3 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
53	52	ralrimivva 2954	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
54		isnlly 21082	. 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 695	1 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  Clsdccld 20630  clsccl 20632  neicnei 20711  Hauscha 20922  Compccmp 20999  𝑛-Locally cnlly 21078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cls 20635  df-nei 20712  df-haus 20929  df-cmp 21000  df-nlly 21080

This theorem is referenced by: (None)