Theorem cmppcmp 34018

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 23370	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		cmpcref 34010	. . . . . 6 ⊢ Comp = CovHasRefFin
3	2	eleq2i 2829	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4		eqid 2737	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscref 34004	. . . . 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 497	. . 3 ⊢ (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8		simprl 771	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9		elin 3906	. . . . . . . . . . . 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 731	. . . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑦)
15		simprr 773	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧Ref𝑦)
16		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
17		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑦 = ∪ 𝑦
18	16, 17	refbas 23485	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑧)
21	4, 16	finlocfin 23495	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ ∪ 𝐽 = ∪ 𝑧) → 𝑧 ∈ (LocFin‘𝐽))
22	12, 13, 20, 21	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (LocFin‘𝐽))
23	11, 22	elind 4141	. . . . . . . . 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 3148	. . . . . 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 3150	. . 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 34017	. . 3 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
32	4	iscref 34004	. . 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 584	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889  𝒫 cpw 4542  ∪ cuni 4851   class class class wbr 5086  ‘cfv 6492  Fincfn 8886  Topctop 22868  Compccmp 23361  Refcref 23477  LocFinclocfin 23479  CovHasRefccref 34002  Paracompcpcmp 34015

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553  ax-ac2 10376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-en 8887  df-dom 8888  df-fin 8890  df-r1 9679  df-rank 9680  df-card 9854  df-ac 10029  df-top 22869  df-cmp 23362  df-ref 23480  df-locfin 23482  df-cref 34003  df-pcmp 34016

This theorem is referenced by: (None)