Theorem cmppcmp 33866

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 23308	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		cmpcref 33858	. . . . . 6 ⊢ Comp = CovHasRefFin
3	2	eleq2i 2823	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4		eqid 2731	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscref 33852	. . . . 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 3918	. . . . . . . . . . . 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 2731	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
17		eqid 2731	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑦 = ∪ 𝑦
18	16, 17	refbas 23423	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑧)
21	4, 16	finlocfin 23433	. . . . . . . . . . 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 4150	. . . . . . . . 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 3142	. . . . . 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 3144	. . 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 33865	. . 3 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
32	4	iscref 33852	. . 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3901  𝒫 cpw 4550  ∪ cuni 4859   class class class wbr 5091  ‘cfv 6481  Fincfn 8869  Topctop 22806  Compccmp 23299  Refcref 23415  LocFinclocfin 23417  CovHasRefccref 33850  Paracompcpcmp 33863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-reg 9478  ax-inf2 9531  ax-ac2 10351

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-en 8870  df-dom 8871  df-fin 8873  df-r1 9654  df-rank 9655  df-card 9829  df-ac 10004  df-top 22807  df-cmp 23300  df-ref 23418  df-locfin 23420  df-cref 33851  df-pcmp 33864

This theorem is referenced by: (None)