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Theorem hausllycmp 21519
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 21357 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ Top)
21adantr 472 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3 eqid 2760 . . . . . 6 𝐽 = 𝐽
4 eqid 2760 . . . . . 6 {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))} = {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))}
5 simpll 807 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Haus)
6 difssd 3881 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ⊆ 𝐽)
7 simplr 809 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Comp)
81ad2antrr 764 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
9 simprl 811 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
103opncld 21059 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 696 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
12 cmpcld 21427 . . . . . . 7 ((𝐽 ∈ Comp ∧ ( 𝐽𝑥) ∈ (Clsd‘𝐽)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
137, 11, 12syl2anc 696 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
14 simprr 813 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
15 elssuni 4619 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
1615ad2antrl 766 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 𝐽)
17 dfss4 4001 . . . . . . . 8 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1816, 17sylib 208 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1914, 18eleqtrrd 2842 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ ( 𝐽 ∖ ( 𝐽𝑥)))
203, 4, 5, 6, 13, 19hauscmplem 21431 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))))
2118sseq2d 3774 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
2221anbi2d 742 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ((𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2322rexbidv 3190 . . . . 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 472 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Top)
26 simprl 811 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢𝐽)
27 simprrl 823 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦𝑢)
28 opnneip 21145 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
2925, 26, 27, 28syl3anc 1477 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30 elssuni 4619 . . . . . . . . 9 (𝑢𝐽𝑢 𝐽)
3130ad2antrl 766 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 𝐽)
323sscls 21082 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
3325, 31, 32syl2anc 696 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
343clsss3 21085 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
3525, 31, 34syl2anc 696 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
363ssnei2 21142 . . . . . . 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 1478 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38 simprrr 824 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39 vex 3343 . . . . . . . 8 𝑥 ∈ V
4039elpw2 4977 . . . . . . 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 3941 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
437adantr 472 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝐽 ∈ Comp)
443clscld 21073 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4525, 31, 44syl2anc 696 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46 cmpcld 21427 . . . . . 6 ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
4743, 45, 46syl2anc 696 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48 oveq2 6822 . . . . . . 7 (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽t 𝑣) = (𝐽t ((cls‘𝐽)‘𝑢)))
4948eleq1d 2824 . . . . . 6 (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽t 𝑣) ∈ Comp ↔ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp))
5049rspcev 3449 . . . . 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 696 . . . 4 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5224, 51rexlimddv 3173 . . 3 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5352ralrimivva 3109 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
54 isnlly 21494 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp))
552, 53, 54sylanbrc 701 1 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588  cfv 6049  (class class class)co 6814  t crest 16303  Topctop 20920  Clsdccld 21042  clsccl 21044  neicnei 21123  Hauscha 21334  Compccmp 21411  𝑛-Locally cnlly 21490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-fin 8127  df-fi 8484  df-rest 16305  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972  df-cld 21045  df-cls 21047  df-nei 21124  df-haus 21341  df-cmp 21412  df-nlly 21492
This theorem is referenced by: (None)
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