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Mirrors > Home > MPE Home > Th. List > alexsub | Structured version Visualization version GIF version |
Description: The Alexander Subbase Theorem: If 𝐵 is a subbase for the topology 𝐽, and any cover taken from 𝐵 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 24075 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
alexsub.1 | ⊢ (𝜑 → 𝑋 ∈ UFL) |
alexsub.2 | ⊢ (𝜑 → 𝑋 = ∪ 𝐵) |
alexsub.3 | ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) |
alexsub.4 | ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
Ref | Expression |
---|---|
alexsub | ⊢ (𝜑 → 𝐽 ∈ Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alexsub.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ UFL) | |
2 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 ∈ UFL) |
3 | alexsub.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 = ∪ 𝐵) | |
4 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑋 = ∪ 𝐵) |
5 | alexsub.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵))) | |
6 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝐽 = (topGen‘(fi‘𝐵))) |
7 | alexsub.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) | |
8 | 7 | adantlr 715 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) |
9 | simprl 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → 𝑓 ∈ (UFil‘𝑋)) | |
10 | simprr 773 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → (𝐽 fLim 𝑓) = ∅) | |
11 | 2, 4, 6, 8, 9, 10 | alexsublem 24068 | . . . . . . 7 ⊢ ¬ (𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) |
12 | 11 | pm2.21i 119 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ (UFil‘𝑋) ∧ (𝐽 fLim 𝑓) = ∅)) → ¬ (𝐽 fLim 𝑓) = ∅) |
13 | 12 | expr 456 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ((𝐽 fLim 𝑓) = ∅ → ¬ (𝐽 fLim 𝑓) = ∅)) |
14 | 13 | pm2.01d 190 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → ¬ (𝐽 fLim 𝑓) = ∅) |
15 | 14 | neqned 2945 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (UFil‘𝑋)) → (𝐽 fLim 𝑓) ≠ ∅) |
16 | 15 | ralrimiva 3144 | . 2 ⊢ (𝜑 → ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅) |
17 | fibas 23000 | . . . . . 6 ⊢ (fi‘𝐵) ∈ TopBases | |
18 | tgtopon 22994 | . . . . . 6 ⊢ ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵))) | |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ (topGen‘(fi‘𝐵)) ∈ (TopOn‘∪ (fi‘𝐵)) |
20 | 5, 19 | eqeltrdi 2847 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ (fi‘𝐵))) |
21 | 1 | elexd 3502 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V) |
22 | 3, 21 | eqeltrrd 2840 | . . . . . . . 8 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
23 | uniexb 7783 | . . . . . . . 8 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
24 | 22, 23 | sylibr 234 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
25 | fiuni 9466 | . . . . . . 7 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵)) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → ∪ 𝐵 = ∪ (fi‘𝐵)) |
27 | 3, 26 | eqtrd 2775 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ (fi‘𝐵)) |
28 | 27 | fveq2d 6911 | . . . 4 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ (fi‘𝐵))) |
29 | 20, 28 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
30 | ufilcmp 24056 | . . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) | |
31 | 1, 29, 30 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅)) |
32 | 16, 31 | mpbird 257 | 1 ⊢ (𝜑 → 𝐽 ∈ Comp) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ficfi 9448 topGenctg 17484 TopOnctopon 22932 TopBasesctb 22968 Compccmp 23410 UFilcufil 23923 UFLcufl 23924 fLim cflim 23958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-en 8985 df-dom 8986 df-fin 8988 df-fi 9449 df-topgen 17490 df-fbas 21379 df-fg 21380 df-top 22916 df-topon 22933 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-cmp 23411 df-fil 23870 df-ufil 23925 df-ufl 23926 df-flim 23963 df-fcls 23965 |
This theorem is referenced by: alexsubb 24070 ptcmplem5 24080 |
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