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Theorem hausllycmp 21577
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 21415 . . 3 (𝐽 ∈ Haus → 𝐽 ∈ Top)
21adantr 472 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ Top)
3 eqid 2765 . . . . . 6 𝐽 = 𝐽
4 eqid 2765 . . . . . 6 {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))} = {𝑧𝐽 ∣ ∃𝑣𝐽 (𝑦𝑣 ∧ ((cls‘𝐽)‘𝑣) ⊆ ( 𝐽𝑧))}
5 simpll 783 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Haus)
6 difssd 3900 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ⊆ 𝐽)
7 simplr 785 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Comp)
81ad2antrr 717 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝐽 ∈ Top)
9 simprl 787 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥𝐽)
103opncld 21117 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑥𝐽) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
118, 9, 10syl2anc 579 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
12 cmpcld 21485 . . . . . . 7 ((𝐽 ∈ Comp ∧ ( 𝐽𝑥) ∈ (Clsd‘𝐽)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
137, 11, 12syl2anc 579 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (𝐽t ( 𝐽𝑥)) ∈ Comp)
14 simprr 789 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦𝑥)
15 elssuni 4625 . . . . . . . . 9 (𝑥𝐽𝑥 𝐽)
1615ad2antrl 719 . . . . . . . 8 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑥 𝐽)
17 dfss4 4023 . . . . . . . 8 (𝑥 𝐽 ↔ ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1816, 17sylib 209 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ( 𝐽 ∖ ( 𝐽𝑥)) = 𝑥)
1914, 18eleqtrrd 2847 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → 𝑦 ∈ ( 𝐽 ∖ ( 𝐽𝑥)))
203, 4, 5, 6, 13, 19hauscmplem 21489 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))))
2118sseq2d 3793 . . . . . . 7 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥)) ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))
2221anbi2d 622 . . . . . 6 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ((𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2322rexbidv 3199 . . . . 5 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → (∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ ( 𝐽 ∖ ( 𝐽𝑥))) ↔ ∃𝑢𝐽 (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)))
2420, 23mpbid 223 . . . 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 787 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢𝐽)
27 simprrl 799 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑦𝑢)
28 opnneip 21203 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
2925, 26, 27, 28syl3anc 1490 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
30 elssuni 4625 . . . . . . . . 9 (𝑢𝐽𝑢 𝐽)
3130ad2antrl 719 . . . . . . . 8 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 𝐽)
323sscls 21140 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
3325, 31, 32syl2anc 579 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
343clsss3 21143 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
3525, 31, 34syl2anc 579 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
363ssnei2 21200 . . . . . . 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 867 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ ((nei‘𝐽)‘{𝑦}))
38 simprrr 800 . . . . . . 7 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
39 vex 3353 . . . . . . . 8 𝑥 ∈ V
4039elpw2 4986 . . . . . . 7 (((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥 ↔ ((cls‘𝐽)‘𝑢) ⊆ 𝑥)
4138, 40sylibr 225 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ 𝒫 𝑥)
4237, 41elind 3960 . . . . 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 21131 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4525, 31, 44syl2anc 579 . . . . . 6 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
46 cmpcld 21485 . . . . . 6 ((𝐽 ∈ Comp ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
4743, 45, 46syl2anc 579 . . . . 5 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp)
48 oveq2 6850 . . . . . . 7 (𝑣 = ((cls‘𝐽)‘𝑢) → (𝐽t 𝑣) = (𝐽t ((cls‘𝐽)‘𝑢)))
4948eleq1d 2829 . . . . . 6 (𝑣 = ((cls‘𝐽)‘𝑢) → ((𝐽t 𝑣) ∈ Comp ↔ (𝐽t ((cls‘𝐽)‘𝑢)) ∈ Comp))
5049rspcev 3461 . . . . 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 579 . . . 4 ((((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) ∧ (𝑢𝐽 ∧ (𝑦𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑥))) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5224, 51rexlimddv 3182 . . 3 (((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
5352ralrimivva 3118 . 2 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp)
54 isnlly 21552 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑣 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑣) ∈ Comp))
552, 53, 54sylanbrc 578 1 ((𝐽 ∈ Haus ∧ 𝐽 ∈ Comp) → 𝐽 ∈ 𝑛-Locally Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  {crab 3059  cdif 3729  cin 3731  wss 3732  𝒫 cpw 4315  {csn 4334   cuni 4594  cfv 6068  (class class class)co 6842  t crest 16347  Topctop 20977  Clsdccld 21100  clsccl 21102  neicnei 21181  Hauscha 21392  Compccmp 21469  𝑛-Locally cnlly 21548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-rest 16349  df-topgen 16370  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-cls 21105  df-nei 21182  df-haus 21399  df-cmp 21470  df-nlly 21550
This theorem is referenced by: (None)
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