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Theorem cmppcmp 33395
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 23286 . 2 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 cmpcref 33387 . . . . . 6 Comp = CovHasRefFin
32eleq2i 2820 . . . . 5 (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4 eqid 2727 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
54iscref 33381 . . . . 5 (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
63, 5bitri 275 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
76simprbi 496 . . 3 (𝐽 ∈ Comp β†’ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8 simprl 770 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9 elin 3960 . . . . . . . . . . . 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 494 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ 𝒫 𝐽)
121ad3antrrr 729 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝐽 ∈ Top)
1310simprd 495 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ Fin)
14 simplr 768 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝐽 = βˆͺ 𝑦)
15 simprr 772 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧Ref𝑦)
16 eqid 2727 . . . . . . . . . . . . . 14 βˆͺ 𝑧 = βˆͺ 𝑧
17 eqid 2727 . . . . . . . . . . . . . 14 βˆͺ 𝑦 = βˆͺ 𝑦
1816, 17refbas 23401 . . . . . . . . . . . . 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 2767 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝐽 = βˆͺ 𝑧)
214, 16finlocfin 23411 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ βˆͺ 𝐽 = βˆͺ 𝑧) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2212, 13, 20, 21syl3anc 1369 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2311, 22elind 4190 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)))
2423, 15jca 511 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ (𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)) ∧ 𝑧Ref𝑦))
2524ex 412 . . . . . . 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 3159 . . . . . 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 412 . . . . 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 3162 . . 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 33394 . . 3 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
324iscref 33381 . . 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 275 . 2 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ βˆ€π‘¦ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ 𝑦 β†’ βˆƒπ‘§ ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½))𝑧Ref𝑦)))
341, 30, 33sylanbrc 582 1 (𝐽 ∈ Comp β†’ 𝐽 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065   ∩ cin 3943  π’« cpw 4598  βˆͺ cuni 4903   class class class wbr 5142  β€˜cfv 6542  Fincfn 8955  Topctop 22782  Compccmp 23277  Refcref 23393  LocFinclocfin 23395  CovHasRefccref 33379  Paracompcpcmp 33392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656  ax-ac2 10478
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-fin 8959  df-r1 9779  df-rank 9780  df-card 9954  df-ac 10131  df-top 22783  df-cmp 23278  df-ref 23396  df-locfin 23398  df-cref 33380  df-pcmp 33393
This theorem is referenced by: (None)
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