Theorem hausllycmp 23518

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 23355	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2	1	adantr 480	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3		eqid 2735	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2735	. . . . . 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 767	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Haus)
6		difssd 4147	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽)
7		simplr 769	. . . . . . 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 771	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽)
10	3	opncld 23057	. . . . . . . 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 23426	. . . . . . 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 773	. . . . . . 7 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
15		elssuni 4942	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
16	15	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐽)
17		dfss4 4275	. . . . . . . 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 2842	. . . . . 6 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥)))
20	3, 4, 5, 6, 13, 19	hauscmplem 23430	. . . . 5 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐽 (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝑥))))
21	18	sseq2d 4028	. . . . . . 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 3177	. . . . 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 771	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ 𝐽)
27		simprrl 781	. . . . . . . 8 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦 ∈ 𝑢)
28		opnneip 23143	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
29	25, 26, 27, 28	syl3anc 1370	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30		elssuni 4942	. . . . . . . . 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 23080	. . . . . . . 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 23083	. . . . . . . 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 23140	. . . . . . 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 839	. . . . . 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 782	. . . . . . 7 ⊢ ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑢 ∈ 𝐽 ∧ (𝑦 ∈ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39		vex 3482	. . . . . . . 8 ⊢ 𝑥 ∈ V
40	39	elpw2 5340	. . . . . . 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 4210	. . . . 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 23071	. . . . . . 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 23426	. . . . . 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 7439	. . . . . . 7 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽 ↾t 𝑣) = (𝐽 ↾t ((cls‘𝐽)‘𝑢)))
49	48	eleq1d 2824	. . . . . 6 ⊢ (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽 ↾t 𝑣) ∈ Comp ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑢)) ∈ Comp))
50	49	rspcev 3622	. . . . 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 3159	. . 3 ⊢ (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
53	52	ralrimivva 3200	. 2 ⊢ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑣) ∈ Comp)
54		isnlly 23493	. 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ∪ cuni 4912  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  Clsdccld 23040  clsccl 23042  neicnei 23121  Hauscha 23332  Compccmp 23410  𝑛-Locally cnlly 23489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cls 23045  df-nei 23122  df-haus 23339  df-cmp 23411  df-nlly 23491

This theorem is referenced by: (None)