| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 24489 | . 2 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾)) |
| 6 | df-mopn 21360 | . . . . . . . 8 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
| 7 | 6 | dmmptss 6261 | . . . . . . 7 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
| 8 | 7 | sseli 3979 | . . . . . 6 ⊢ (𝐷 ∈ dom MetOpen → 𝐷 ∈ ∪ ran ∞Met) |
| 9 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ ∪ ran ∞Met) | |
| 10 | xmetunirn 24347 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
| 11 | 9, 10 | sylib 218 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
| 12 | eqid 2737 | . . . . . . . . . . 11 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
| 13 | 12 | mopnuni 24451 | . . . . . . . . . 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 5916 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐷 = dom ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
| 16 | dmres 6030 | . . . . . . . . . . . . . 14 ⊢ dom ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) | |
| 17 | 15, 16 | eqtrdi 2793 | . . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐷 = ((𝑋 × 𝑋) ∩ dom (dist‘𝑀))) |
| 18 | inss1 4237 | . . . . . . . . . . . . 13 ⊢ ((𝑋 × 𝑋) ∩ dom (dist‘𝑀)) ⊆ (𝑋 × 𝑋) | |
| 19 | 17, 18 | eqsstrdi 4028 | . . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐷 ⊆ (𝑋 × 𝑋)) |
| 20 | dmss 5913 | . . . . . . . . . . . 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 5941 | . . . . . . . . . . 11 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
| 23 | 21, 22 | sseqtrdi 4024 | . . . . . . . . . 10 ⊢ (𝜑 → dom dom 𝐷 ⊆ 𝑋) |
| 24 | 23 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → dom dom 𝐷 ⊆ 𝑋) |
| 25 | 14, 24 | eqsstrrd 4019 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → ∪ (MetOpen‘𝐷) ⊆ 𝑋) |
| 26 | sspwuni 5100 | . . . . . . . 8 ⊢ ((MetOpen‘𝐷) ⊆ 𝒫 𝑋 ↔ ∪ (MetOpen‘𝐷) ⊆ 𝑋) | |
| 27 | 25, 26 | sylibr 234 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐷 ∈ ∪ ran ∞Met) → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
| 28 | 27 | ex 412 | . . . . . 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 6941 | . . . . . 6 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) = ∅) | |
| 31 | 0ss 4400 | . . . . . 6 ⊢ ∅ ⊆ 𝒫 𝑋 | |
| 32 | 30, 31 | eqsstrdi 4028 | . . . . 5 ⊢ (¬ 𝐷 ∈ dom MetOpen → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
| 33 | 29, 32 | pm2.61d1 180 | . . . 4 ⊢ (𝜑 → (MetOpen‘𝐷) ⊆ 𝒫 𝑋) |
| 34 | 1, 2, 3 | setsmsbas 24485 | . . . . 5 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
| 35 | 34 | pweqd 4617 | . . . 4 ⊢ (𝜑 → 𝒫 𝑋 = 𝒫 (Base‘𝐾)) |
| 36 | 33, 5, 35 | 3sstr3d 4038 | . . 3 ⊢ (𝜑 → (TopSet‘𝐾) ⊆ 𝒫 (Base‘𝐾)) |
| 37 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 38 | eqid 2737 | . . . 4 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
| 39 | 37, 38 | topnid 17480 | . . 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 2777 | 1 ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 〈cop 4632 ∪ cuni 4907 × cxp 5683 dom cdm 5685 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 Basecbs 17247 TopSetcts 17303 distcds 17306 TopOpenctopn 17466 topGenctg 17482 ∞Metcxmet 21349 ballcbl 21351 MetOpencmopn 21354 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-tset 17316 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 |
| This theorem is referenced by: setsxms 24491 tmslem 24494 tmslemOLD 24495 |
| Copyright terms: Public domain | W3C validator |