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Theorem hausllycmp 22030
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 21867 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ Top)
21adantr 481 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3 eqid 2818 . . . . . 6 𝐽 = 𝐽
4 eqid 2818 . . . . . 6 {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))} = {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))}
5 simpll 763 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Haus)
6 difssd 4106 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ⊆ 𝐽)
7 simplr 765 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Comp)
81ad2antrr 722 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
9 simprl 767 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
103opncld 21569 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 584 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
12 cmpcld 21938 . . . . . . 7 ((𝐽 ∈ Comp ∧ ( 𝐽𝑥) ∈ (Clsd‘𝐽)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
137, 11, 12syl2anc 584 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
14 simprr 769 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
15 elssuni 4859 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
1615ad2antrl 724 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 𝐽)
17 dfss4 4232 . . . . . . . 8 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1816, 17sylib 219 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1914, 18eleqtrrd 2913 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ ( 𝐽 ∖ ( 𝐽𝑥)))
203, 4, 5, 6, 13, 19hauscmplem 21942 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))))
2118sseq2d 3996 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
2221anbi2d 628 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ((𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2322rexbidv 3294 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2420, 23mpbid 233 . . . 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 767 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢𝐽)
27 simprrl 777 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦𝑢)
28 opnneip 21655 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
2925, 26, 27, 28syl3anc 1363 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30 elssuni 4859 . . . . . . . . 9 (𝑢𝐽𝑢 𝐽)
3130ad2antrl 724 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 𝐽)
323sscls 21592 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
3325, 31, 32syl2anc 584 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
343clsss3 21595 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
3525, 31, 34syl2anc 584 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
363ssnei2 21652 . . . . . . 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 834 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38 simprrr 778 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39 vex 3495 . . . . . . . 8 𝑥 ∈ V
4039elpw2 5239 . . . . . . 7 (((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
4138, 40sylibr 235 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥)
4237, 41elind 4168 . . . . 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 21583 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4525, 31, 44syl2anc 584 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46 cmpcld 21938 . . . . . 6 ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
4743, 45, 46syl2anc 584 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48 oveq2 7153 . . . . . . 7 (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽t 𝑣) = (𝐽t ((cls‘𝐽)‘𝑢)))
4948eleq1d 2894 . . . . . 6 (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽t 𝑣) ∈ Comp ↔ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp))
5049rspcev 3620 . . . . 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 584 . . . 4 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5224, 51rexlimddv 3288 . . 3 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5352ralrimivva 3188 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
54 isnlly 22005 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp))
552, 53, 54sylanbrc 583 1 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  cdif 3930  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830  cfv 6348  (class class class)co 7145  t crest 16682  Topctop 21429  Clsdccld 21552  clsccl 21554  neicnei 21633  Hauscha 21844  Compccmp 21922  𝑛-Locally cnlly 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-cls 21557  df-nei 21634  df-haus 21851  df-cmp 21923  df-nlly 22003
This theorem is referenced by: (None)
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