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Theorem hausllycmp 21207
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.
StepHypRef Expression
1 haustop 21045 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3 eqid 2621 . . . . . 6 𝐽 = 𝐽
4 eqid 2621 . . . . . 6 {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))} = {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))}
5 simpll 789 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Haus)
6 difssd 3716 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ⊆ 𝐽)
7 simplr 791 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Comp)
81ad2antrr 761 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
9 simprl 793 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
103opncld 20747 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 692 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
12 cmpcld 21115 . . . . . . 7 ((𝐽 ∈ Comp ∧ ( 𝐽𝑥) ∈ (Clsd‘𝐽)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
137, 11, 12syl2anc 692 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
14 simprr 795 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
15 elssuni 4433 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
1615ad2antrl 763 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 𝐽)
17 dfss4 3836 . . . . . . . 8 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1816, 17sylib 208 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1914, 18eleqtrrd 2701 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ ( 𝐽 ∖ ( 𝐽𝑥)))
203, 4, 5, 6, 13, 19hauscmplem 21119 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))))
2118sseq2d 3612 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
2221anbi2d 739 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ((𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2322rexbidv 3045 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2420, 23mpbid 222 . . . 4 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
258adantr 481 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Top)
26 simprl 793 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢𝐽)
27 simprrl 803 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦𝑢)
28 opnneip 20833 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
2925, 26, 27, 28syl3anc 1323 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30 elssuni 4433 . . . . . . . . 9 (𝑢𝐽𝑢 𝐽)
3130ad2antrl 763 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 𝐽)
323sscls 20770 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
3325, 31, 32syl2anc 692 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
343clsss3 20773 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
3525, 31, 34syl2anc 692 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
363ssnei2 20830 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑢 ∈ ((nei‘𝐽)‘{𝑦})) ∧ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝐽)) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
3725, 29, 33, 35, 36syl22anc 1324 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38 simprrr 804 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39 vex 3189 . . . . . . . 8 𝑥 ∈ V
4039elpw2 4788 . . . . . . 7 (((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
4138, 40sylibr 224 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥)
4237, 41elind 3776 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
437adantr 481 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Comp)
443clscld 20761 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4525, 31, 44syl2anc 692 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46 cmpcld 21115 . . . . . 6 ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
4743, 45, 46syl2anc 692 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48 oveq2 6612 . . . . . . 7 (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽t 𝑣) = (𝐽t ((cls‘𝐽)‘𝑢)))
4948eleq1d 2683 . . . . . 6 (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽t 𝑣) ∈ Comp ↔ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp))
5049rspcev 3295 . . . . 5 ((((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5142, 47, 50syl2anc 692 . . . 4 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5224, 51rexlimddv 3028 . . 3 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5352ralrimivva 2965 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
54 isnlly 21182 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp))
552, 53, 54sylanbrc 697 1 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  Clsdccld 20730  clsccl 20732  neicnei 20811  Hauscha 21022  Compccmp 21099  𝑛-Locally cnlly 21178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-cls 20735  df-nei 20812  df-haus 21029  df-cmp 21100  df-nlly 21180
This theorem is referenced by: (None)
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