Theorem cmppcmp 31211

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 22000	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		cmpcref 31203	. . . . . 6 ⊢ Comp = CovHasRefFin
3	2	eleq2i 2881	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4		eqid 2798	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscref 31197	. . . . 5 ⊢ (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
6	3, 5	bitri 278	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
7	6	simprbi 500	. . 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 3897	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
10	8, 9	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ Fin))
11	10	simpld 498	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ 𝒫 𝐽)
12	1	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 ∈ Top)
13	10	simprd 499	. . . . . . . . . . 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 2798	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
17		eqid 2798	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑦 = ∪ 𝑦
18	16, 17	refbas 22115	. . . . . . . . . . . . 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 2833	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑧)
21	4, 16	finlocfin 22125	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ ∪ 𝐽 = ∪ 𝑧) → 𝑧 ∈ (LocFin‘𝐽))
22	12, 13, 20, 21	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (LocFin‘𝐽))
23	11, 22	elind 4121	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)))
24	23, 15	jca 515	. . . . . . . 8 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦))
25	24	ex 416	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) → ((𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦)))
26	25	reximdv2 3230	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
27	26	ex 416	. . . . 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 31210	. . 3 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
32	4	iscref 31197	. . 3 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
33	31, 32	bitri 278	. 2 ⊢ (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
34	1, 30, 33	sylanbrc 586	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ‘cfv 6324  Fincfn 8492  Topctop 21498  Compccmp 21991  Refcref 22107  LocFinclocfin 22109  CovHasRefccref 31195  Paracompcpcmp 31208

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088  ax-ac2 9874

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-fin 8496  df-r1 9177  df-rank 9178  df-card 9352  df-ac 9527  df-top 21499  df-cmp 21992  df-ref 22110  df-locfin 22112  df-cref 31196  df-pcmp 31209

This theorem is referenced by: (None)