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Mirrors > Home > MPE Home > Th. List > setsmstopn | Structured version Visualization version GIF version |
Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsms.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
Ref | Expression |
---|---|
setsmstopn | ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.x | . . 3 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
2 | setsms.d | . . 3 ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) | |
3 | setsms.k | . . 3 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
4 | setsms.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
5 | 1, 2, 3, 4 | setsmstset 23014 | . 2 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
6 | df-mopn 20469 | . . . . . . . 8 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
7 | 6 | dmmptss 6088 | . . . . . . 7 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 | 7 | sseli 3960 | . . . . . 6 ⊢ (𝐷 ∈ dom MetOpen → 𝐷 ∈ ∪ ran ∞Met) |
9 | simpr 485 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ ∪ ran ∞Met) | |
10 | xmetunirn 22874 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
11 | 9, 10 | sylib 219 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
12 | eqid 2818 | . . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
13 | 12 | mopnuni 22978 | . . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → dom dom 𝐷 = ∪ (MetOpen‘𝐷)) |
14 | 11, 13 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → dom dom 𝐷 = ∪ (MetOpen‘𝐷)) |
15 | 2 | dmeqd 5767 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐷 = dom ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
16 | dmres 5868 | . . . . . . . . . . . . . 14 ⊢ dom ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) | |
17 | 15, 16 | syl6eq 2869 | . . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐷 = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀))) |
18 | inss1 4202 | . . . . . . . . . . . . 13 ⊢ ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) ⊆ (𝑋 × 𝑋) | |
19 | 17, 18 | eqsstrdi 4018 | . . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐷 ⊆ (𝑋 × 𝑋)) |
20 | dmss 5764 | . . . . . . . . . . . 12 ⊢ (dom 𝐷 ⊆ (𝑋 × 𝑋) → dom dom 𝐷 ⊆ dom (𝑋 × 𝑋)) | |
21 | 19, 20 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → dom dom 𝐷 ⊆ dom (𝑋 × 𝑋)) |
22 | dmxpid 5793 | . . . . . . . . . . 11 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
23 | 21, 22 | sseqtrdi 4014 | . . . . . . . . . 10 ⊢ (𝜑 → dom dom 𝐷 ⊆ 𝑋) |
24 | 23 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → dom dom 𝐷 ⊆ 𝑋) |
25 | 14, 24 | eqsstrrd 4003 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → ∪ (MetOpen‘𝐷) ⊆ 𝑋) |
26 | sspwuni 5013 | . . . . . . . 8 ⊢ ((MetOpen‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ (MetOpen‘𝐷) ⊆ 𝑋) | |
27 | 25, 26 | sylibr 235 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
28 | 27 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) ⊆ 𝒫 𝑋)) |
29 | 8, 28 | syl5 34 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) ⊆ 𝒫 𝑋)) |
30 | ndmfv 6693 | . . . . . 6 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) = ∅) | |
31 | 0ss 4347 | . . . . . 6 ⊢ ∅ ⊆ 𝒫 𝑋 | |
32 | 30, 31 | eqsstrdi 4018 | . . . . 5 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
33 | 29, 32 | pm2.61d1 181 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
34 | 1, 2, 3 | setsmsbas 23012 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
35 | 34 | pweqd 4540 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 = 𝒫 (Base‘𝐾)) |
36 | 33, 5, 35 | 3sstr3d 4010 | . . 3 ⊢ (𝜑 → (TopSet‘𝐾) ⊆ 𝒫 (Base‘𝐾)) |
37 | eqid 2818 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
38 | eqid 2818 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
39 | 37, 38 | topnid 16697 | . . 3 ⊢ ((TopSet‘𝐾) ⊆ 𝒫 (Base‘𝐾) → (TopSet‘𝐾) = (TopOpen‘𝐾)) |
40 | 36, 39 | syl 17 | . 2 ⊢ (𝜑 → (TopSet‘𝐾) = (TopOpen‘𝐾)) |
41 | 5, 40 | eqtrd 2853 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 〈cop 4563 ∪ cuni 4830 × cxp 5546 dom cdm 5548 ran crn 5549 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 sSet csts 16469 Basecbs 16471 TopSetcts 16559 distcds 16562 TopOpenctopn 16683 topGenctg 16699 ∞Metcxmet 20458 ballcbl 20460 MetOpencmopn 20463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-tset 16572 df-rest 16684 df-topn 16685 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 |
This theorem is referenced by: setsxms 23016 tmslem 23019 |
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