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Theorem cmppcmp 32496
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 22762 . 2 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 cmpcref 32488 . . . . . 6 Comp = CovHasRefFin
32eleq2i 2826 . . . . 5 (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4 eqid 2733 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
54iscref 32482 . . . . 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 498 . . 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 3927 . . . . . . . . . . . 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 496 . . . . . . . . . 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 497 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 βˆͺ 𝑧 = βˆͺ 𝑧
17 eqid 2733 . . . . . . . . . . . . . 14 βˆͺ 𝑦 = βˆͺ 𝑦
1816, 17refbas 22877 . . . . . . . . . . . . 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 2773 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ βˆͺ 𝐽 = βˆͺ 𝑧)
214, 16finlocfin 22887 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ βˆͺ 𝐽 = βˆͺ 𝑧) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2212, 13, 20, 21syl3anc 1372 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (LocFinβ€˜π½))
2311, 22elind 4155 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ 𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)))
2423, 15jca 513 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ βˆͺ 𝐽 = βˆͺ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) β†’ (𝑧 ∈ (𝒫 𝐽 ∩ (LocFinβ€˜π½)) ∧ 𝑧Ref𝑦))
2524ex 414 . . . . . . 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 3158 . . . . . 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 414 . . . . 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 3161 . . 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 32495 . . 3 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFinβ€˜π½))
324iscref 32482 . . 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 584 1 (𝐽 ∈ Comp β†’ 𝐽 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3910  π’« cpw 4561  βˆͺ cuni 4866   class class class wbr 5106  β€˜cfv 6497  Fincfn 8886  Topctop 22258  Compccmp 22753  Refcref 22869  LocFinclocfin 22871  CovHasRefccref 32480  Paracompcpcmp 32493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582  ax-ac2 10404
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-fin 8890  df-r1 9705  df-rank 9706  df-card 9880  df-ac 10057  df-top 22259  df-cmp 22754  df-ref 22872  df-locfin 22874  df-cref 32481  df-pcmp 32494
This theorem is referenced by: (None)
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