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Theorem hausllycmp 21518
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 21356 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ Top)
21adantr 466 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3 eqid 2771 . . . . . 6 𝐽 = 𝐽
4 eqid 2771 . . . . . 6 {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))} = {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))}
5 simpll 750 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Haus)
6 difssd 3889 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ⊆ 𝐽)
7 simplr 752 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Comp)
81ad2antrr 705 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
9 simprl 754 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
103opncld 21058 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 573 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
12 cmpcld 21426 . . . . . . 7 ((𝐽 ∈ Comp ∧ ( 𝐽𝑥) ∈ (Clsd‘𝐽)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
137, 11, 12syl2anc 573 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
14 simprr 756 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
15 elssuni 4603 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
1615ad2antrl 707 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 𝐽)
17 dfss4 4007 . . . . . . . 8 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1816, 17sylib 208 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1914, 18eleqtrrd 2853 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ ( 𝐽 ∖ ( 𝐽𝑥)))
203, 4, 5, 6, 13, 19hauscmplem 21430 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))))
2118sseq2d 3782 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
2221anbi2d 614 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ((𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2322rexbidv 3200 . . . . 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 466 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Top)
26 simprl 754 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢𝐽)
27 simprrl 766 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦𝑢)
28 opnneip 21144 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
2925, 26, 27, 28syl3anc 1476 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30 elssuni 4603 . . . . . . . . 9 (𝑢𝐽𝑢 𝐽)
3130ad2antrl 707 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 𝐽)
323sscls 21081 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
3325, 31, 32syl2anc 573 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
343clsss3 21084 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
3525, 31, 34syl2anc 573 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
363ssnei2 21141 . . . . . . 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 1477 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38 simprrr 767 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39 vex 3354 . . . . . . . 8 𝑥 ∈ V
4039elpw2 4959 . . . . . . 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 3949 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
437adantr 466 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Comp)
443clscld 21072 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4525, 31, 44syl2anc 573 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46 cmpcld 21426 . . . . . 6 ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
4743, 45, 46syl2anc 573 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48 oveq2 6801 . . . . . . 7 (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽t 𝑣) = (𝐽t ((cls‘𝐽)‘𝑢)))
4948eleq1d 2835 . . . . . 6 (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽t 𝑣) ∈ Comp ↔ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp))
5049rspcev 3460 . . . . 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 573 . . . 4 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5224, 51rexlimddv 3183 . . 3 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5352ralrimivva 3120 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
54 isnlly 21493 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp))
552, 53, 54sylanbrc 572 1 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  cdif 3720  cin 3722  wss 3723  𝒫 cpw 4297  {csn 4316   cuni 4574  cfv 6031  (class class class)co 6793  t crest 16289  Topctop 20918  Clsdccld 21041  clsccl 21043  neicnei 21122  Hauscha 21333  Compccmp 21410  𝑛-Locally cnlly 21489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-cls 21046  df-nei 21123  df-haus 21340  df-cmp 21411  df-nlly 21491
This theorem is referenced by: (None)
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