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Theorem cmppcmp 33124
Description: Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
cmppcmp (𝐽 ∈ Comp β†’ 𝐽 ∈ Paracomp)
Proof of Theorem cmppcmp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmptop 23119 . 2 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 cmpcref 33116 . . . . . 6 Comp = CovHasRefFin
32eleq2i 2825 . . . . 5 (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4 eqid 2732 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
54iscref 33110 . . . . 5 (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
63, 5bitri 274 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
76simprbi 497 . . 3 (𝐽 ∈ Comp β†’ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8 simprl 769 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9 elin 3964 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
108, 9sylib 217 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
1110simpld 495 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ 𝒫 𝐽)
121ad3antrrr 728 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝐽 ∈ Top)
1310simprd 496 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ Fin)
14 simplr 767 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝐽 = βˆͺ 𝑦)
15 simprr 771 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧Ref𝑦)
16 eqid 2732 . . . . . . . . . . . . . 14 βˆͺ 𝑧 = βˆͺ 𝑧
17 eqid 2732 . . . . . . . . . . . . . 14 βˆͺ 𝑦 = βˆͺ 𝑦
1816, 17refbas 23234 . . . . . . . . . . . . 13 (𝑧Ref𝑦 β†’ βˆͺ 𝑦 = βˆͺ 𝑧)
1915, 18syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝑦 = βˆͺ 𝑧)
2014, 19eqtrd 2772 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝐽 = βˆͺ 𝑧)
214, 16finlocfin 23244 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ βˆͺ 𝐽 = βˆͺ 𝑧) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2212, 13, 20, 21syl3anc 1371 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2311, 22elind 4194 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)))
2423, 15jca 512 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ (𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)) ∧ 𝑧Ref𝑦))
2524ex 413 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) β†’ ((𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦) β†’ (𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)) ∧ 𝑧Ref𝑦)))
2625reximdv2 3164 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) β†’ (βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦))
2726ex 413 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) β†’ (βˆͺ 𝐽 = βˆͺ 𝑦 β†’ (βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
2827a2d 29 . . . 4 ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) β†’ ((βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) β†’ (βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
2928ralimdva 3167 . . 3 (𝐽 ∈ Comp β†’ (βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) β†’ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
307, 29mpd 15 . 2 (𝐽 ∈ Comp β†’ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦))
31 ispcmp 33123 . . 3 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
324iscref 33110 . . 3 (𝐽 ∈ CovHasRef(LocFinβ€˜π½) ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
3331, 32bitri 274 . 2 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
341, 30, 33sylanbrc 583 1 (𝐽 ∈ Comp β†’ 𝐽 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148  β€˜cfv 6543  Fincfn 8941  Topctop 22615  Compccmp 23110  Refcref 23226  LocFinclocfin 23228  CovHasRefccref 33108  Paracompcpcmp 33121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-fin 8945  df-r1 9761  df-rank 9762  df-card 9936  df-ac 10113  df-top 22616  df-cmp 23111  df-ref 23229  df-locfin 23231  df-cref 33109  df-pcmp 33122
This theorem is referenced by: (None)
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