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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 23232 | . 2 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
6 | df-mopn 20215 | . . . . . . . 8 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
7 | 6 | dmmptss 6073 | . . . . . . 7 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 | 7 | sseli 3873 | . . . . . 6 ⊢ (𝐷 ∈ dom MetOpen → 𝐷 ∈ ∪ ran ∞Met) |
9 | simpr 488 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ ∪ ran ∞Met) | |
10 | xmetunirn 23092 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
11 | 9, 10 | sylib 221 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
12 | eqid 2738 | . . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
13 | 12 | mopnuni 23196 | . . . . . . . . . 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 5748 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐷 = dom ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
16 | dmres 5847 | . . . . . . . . . . . . . 14 ⊢ dom ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) | |
17 | 15, 16 | eqtrdi 2789 | . . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐷 = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀))) |
18 | inss1 4119 | . . . . . . . . . . . . 13 ⊢ ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) ⊆ (𝑋 × 𝑋) | |
19 | 17, 18 | eqsstrdi 3931 | . . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐷 ⊆ (𝑋 × 𝑋)) |
20 | dmss 5745 | . . . . . . . . . . . 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 5773 | . . . . . . . . . . 11 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
23 | 21, 22 | sseqtrdi 3927 | . . . . . . . . . 10 ⊢ (𝜑 → dom dom 𝐷 ⊆ 𝑋) |
24 | 23 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → dom dom 𝐷 ⊆ 𝑋) |
25 | 14, 24 | eqsstrrd 3916 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → ∪ (MetOpen‘𝐷) ⊆ 𝑋) |
26 | sspwuni 4985 | . . . . . . . 8 ⊢ ((MetOpen‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ (MetOpen‘𝐷) ⊆ 𝑋) | |
27 | 25, 26 | sylibr 237 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
28 | 27 | ex 416 | . . . . . 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 6706 | . . . . . 6 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) = ∅) | |
31 | 0ss 4285 | . . . . . 6 ⊢ ∅ ⊆ 𝒫 𝑋 | |
32 | 30, 31 | eqsstrdi 3931 | . . . . 5 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
33 | 29, 32 | pm2.61d1 183 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
34 | 1, 2, 3 | setsmsbas 23230 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
35 | 34 | pweqd 4507 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 = 𝒫 (Base‘𝐾)) |
36 | 33, 5, 35 | 3sstr3d 3923 | . . 3 ⊢ (𝜑 → (TopSet‘𝐾) ⊆ 𝒫 (Base‘𝐾)) |
37 | eqid 2738 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
38 | eqid 2738 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
39 | 37, 38 | topnid 16814 | . . 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 2773 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 〈cop 4522 ∪ cuni 4796 × cxp 5523 dom cdm 5525 ran crn 5526 ↾ cres 5527 ‘cfv 6339 (class class class)co 7172 ndxcnx 16585 sSet csts 16586 Basecbs 16588 TopSetcts 16676 distcds 16679 TopOpenctopn 16800 topGenctg 16816 ∞Metcxmet 20204 ballcbl 20206 MetOpencmopn 20209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-sup 8981 df-inf 8982 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-tset 16689 df-rest 16801 df-topn 16802 df-topgen 16822 df-psmet 20211 df-xmet 20212 df-bl 20214 df-mopn 20215 df-top 21647 df-topon 21664 df-bases 21699 |
This theorem is referenced by: setsxms 23234 tmslem 23237 |
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