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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 22282 | . 2 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
6 | df-mopn 19742 | . . . . . . . 8 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
7 | 6 | dmmptss 5631 | . . . . . . 7 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 | 7 | sseli 3599 | . . . . . 6 ⊢ (𝐷 ∈ dom MetOpen → 𝐷 ∈ ∪ ran ∞Met) |
9 | simpr 477 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ ∪ ran ∞Met) | |
10 | xmetunirn 22142 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
11 | 9, 10 | sylib 208 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
12 | eqid 2622 | . . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
13 | 12 | mopnuni 22246 | . . . . . . . . . 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 5326 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐷 = dom ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
16 | dmres 5419 | . . . . . . . . . . . . . 14 ⊢ dom ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) | |
17 | 15, 16 | syl6eq 2672 | . . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐷 = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀))) |
18 | inss1 3833 | . . . . . . . . . . . . 13 ⊢ ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) ⊆ (𝑋 × 𝑋) | |
19 | 17, 18 | syl6eqss 3655 | . . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐷 ⊆ (𝑋 × 𝑋)) |
20 | dmss 5323 | . . . . . . . . . . . 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 5345 | . . . . . . . . . . 11 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
23 | 21, 22 | syl6sseq 3651 | . . . . . . . . . 10 ⊢ (𝜑 → dom dom 𝐷 ⊆ 𝑋) |
24 | 23 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → dom dom 𝐷 ⊆ 𝑋) |
25 | 14, 24 | eqsstr3d 3640 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → ∪ (MetOpen‘𝐷) ⊆ 𝑋) |
26 | sspwuni 4611 | . . . . . . . 8 ⊢ ((MetOpen‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ (MetOpen‘𝐷) ⊆ 𝑋) | |
27 | 25, 26 | sylibr 224 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
28 | 27 | ex 450 | . . . . . 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 6218 | . . . . . 6 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) = ∅) | |
31 | 0ss 3972 | . . . . . 6 ⊢ ∅ ⊆ 𝒫 𝑋 | |
32 | 30, 31 | syl6eqss 3655 | . . . . 5 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
33 | 29, 32 | pm2.61d1 171 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
34 | 1, 2, 3 | setsmsbas 22280 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
35 | 34 | pweqd 4163 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 = 𝒫 (Base‘𝐾)) |
36 | 33, 5, 35 | 3sstr3d 3647 | . . 3 ⊢ (𝜑 → (TopSet‘𝐾) ⊆ 𝒫 (Base‘𝐾)) |
37 | eqid 2622 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
38 | eqid 2622 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
39 | 37, 38 | topnid 16096 | . . 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 2656 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 〈cop 4183 ∪ cuni 4436 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 Basecbs 15857 TopSetcts 15947 distcds 15950 TopOpenctopn 16082 topGenctg 16098 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-tset 15960 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 |
This theorem is referenced by: setsxms 22284 tmslem 22287 |
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