Theorem cmppcmp 33892

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 23311	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		cmpcref 33884	. . . . . 6 ⊢ Comp = CovHasRefFin
3	2	eleq2i 2825	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4		eqid 2733	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	iscref 33878	. . . . 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 3914	. . . . . . . . . . . 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 2733	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑧 = ∪ 𝑧
17		eqid 2733	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑦 = ∪ 𝑦
18	16, 17	refbas 23426	. . . . . . . . . . . . 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 2768	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ ∪ 𝐽 = ∪ 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → ∪ 𝐽 = ∪ 𝑧)
21	4, 16	finlocfin 23436	. . . . . . . . . . 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 4149	. . . . . . . . 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 3143	. . . . . 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 3145	. . 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 33891	. . 3 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
32	4	iscref 33878	. . 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 2113  ∀wral 3048  ∃wrex 3057   ∩ cin 3897  𝒫 cpw 4549  ∪ cuni 4858   class class class wbr 5093  ‘cfv 6486  Fincfn 8875  Topctop 22809  Compccmp 23302  Refcref 23418  LocFinclocfin 23420  CovHasRefccref 33876  Paracompcpcmp 33889

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-reg 9485  ax-inf2 9538  ax-ac2 10361

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  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 7309  df-ov 7355  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-en 8876  df-dom 8877  df-fin 8879  df-r1 9664  df-rank 9665  df-card 9839  df-ac 10014  df-top 22810  df-cmp 23303  df-ref 23421  df-locfin 23423  df-cref 33877  df-pcmp 33890

This theorem is referenced by: (None)