Theorem cmppcmp 33824

Description: Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
cmppcmp	⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Proof of Theorem cmppcmp

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmptop 23298	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		cmpcref 33816	. . . . . 6 ⊢ Comp = CovHasRefFin
3	2	eleq2i 2820	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4		eqid 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscref 33810	. . . . 5 ⊢ (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
6	3, 5	bitri 275	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
7	6	simprbi 496	. . 3 ⊢ (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9		elin 3921	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
10	8, 9	sylib 218	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
11	10	simpld 494	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ 𝒫 𝐽)
12	1	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 ∈ Top)
13	10	simprd 495	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ Fin)
14		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑦)
15		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧Ref𝑦)
16		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
17		eqid 2729	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑦 = ∪ 𝑦
18	16, 17	refbas 23413	. . . . . . . . . . . . 13 ⊢ (𝑧Ref𝑦 → ∪ 𝑦 = ∪ 𝑧)
19	15, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝑦 = ∪ 𝑧)
20	14, 19	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑧)
21	4, 16	finlocfin 23423	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ ∪ 𝐽 = ∪ 𝑧) → 𝑧 ∈ (LocFin‘𝐽))
22	12, 13, 20, 21	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (LocFin‘𝐽))
23	11, 22	elind 4153	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)))
24	23, 15	jca 511	. . . . . . . 8 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦))
25	24	ex 412	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) → ((𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦)))
26	25	reximdv2 3139	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
27	26	ex 412	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) → (∪ 𝐽 = ∪ 𝑦 → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
28	27	a2d 29	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) → ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) → (∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
29	28	ralimdva 3141	. . 3 ⊢ (𝐽 ∈ Comp → (∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
30	7, 29	mpd 15	. 2 ⊢ (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
31		ispcmp 33823	. . 3 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
32	4	iscref 33810	. . 3 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
33	31, 32	bitri 275	. 2 ⊢ (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
34	1, 30, 33	sylanbrc 583	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3904  𝒫 cpw 4553  ∪ cuni 4861   class class class wbr 5095  ‘cfv 6486  Fincfn 8879  Topctop 22796  Compccmp 23289  Refcref 23405  LocFinclocfin 23407  CovHasRefccref 33808  Paracompcpcmp 33821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-reg 9503  ax-inf2 9556  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-en 8880  df-dom 8881  df-fin 8883  df-r1 9679  df-rank 9680  df-card 9854  df-ac 10029  df-top 22797  df-cmp 23290  df-ref 23408  df-locfin 23410  df-cref 33809  df-pcmp 33822

This theorem is referenced by: (None)