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Mirrors > Home > MPE Home > Th. List > isxms2 | Structured version Visualization version GIF version |
Description: Express the predicate "〈𝑋, 𝐷〉 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
isxms2 | ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isms.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
2 | isms.x | . . 3 ⊢ 𝑋 = (Base‘𝐾) | |
3 | isms.d | . . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
4 | 1, 2, 3 | isxms 22984 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
5 | 2, 1 | istps 21470 | . . . 4 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
6 | df-mopn 20469 | . . . . . . . . . 10 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
7 | 6 | dmmptss 6088 | . . . . . . . . 9 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 | toponmax 21462 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
9 | 8 | adantl 482 | . . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ 𝐽) |
10 | simpl 483 | . . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 = (MetOpen‘𝐷)) | |
11 | 9, 10 | eleqtrd 2912 | . . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ (MetOpen‘𝐷)) |
12 | elfvdm 6695 | . . . . . . . . . 10 ⊢ (𝑋 ∈ (MetOpen‘𝐷) → 𝐷 ∈ dom MetOpen) | |
13 | 11, 12 | syl 17 | . . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ dom MetOpen) |
14 | 7, 13 | sseldi 3962 | . . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ ∪ ran ∞Met) |
15 | xmetunirn 22874 | . . . . . . . 8 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
16 | 14, 15 | sylib 219 | . . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
17 | eqid 2818 | . . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
18 | 17 | mopntopon 22976 | . . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘dom dom 𝐷) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷)) |
19 | 16, 18 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (MetOpen‘𝐷) ∈ (TopOn‘dom dom 𝐷)) |
20 | 10, 19 | eqeltrd 2910 | . . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘dom dom 𝐷)) |
21 | toponuni 21450 | . . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘dom dom 𝐷) → dom dom 𝐷 = ∪ 𝐽) | |
22 | 20, 21 | syl 17 | . . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = ∪ 𝐽) |
23 | toponuni 21450 | . . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
24 | 23 | adantl 482 | . . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 = ∪ 𝐽) |
25 | 22, 24 | eqtr4d 2856 | . . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝑋) |
26 | 25 | fveq2d 6667 | . . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∞Met‘dom dom 𝐷) = (∞Met‘𝑋)) |
27 | 16, 26 | eleqtrd 2912 | . . . . . 6 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘𝑋)) |
28 | 27 | ex 413 | . . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))) |
29 | 17 | mopntopon 22976 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ (TopOn‘𝑋)) |
30 | eleq1 2897 | . . . . . 6 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ (MetOpen‘𝐷) ∈ (TopOn‘𝑋))) | |
31 | 29, 30 | syl5ibr 247 | . . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))) |
32 | 28, 31 | impbid 213 | . . . 4 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐷 ∈ (∞Met‘𝑋))) |
33 | 5, 32 | syl5bb 284 | . . 3 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐾 ∈ TopSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
34 | 33 | pm5.32ri 576 | . 2 ⊢ ((𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) |
35 | 4, 34 | bitri 276 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 × cxp 5546 dom cdm 5548 ran crn 5549 ↾ cres 5550 ‘cfv 6348 Basecbs 16471 distcds 16562 TopOpenctopn 16683 topGenctg 16699 ∞Metcxmet 20458 ballcbl 20460 MetOpencmopn 20463 TopOnctopon 21446 TopSpctps 21468 ∞MetSpcxms 22854 |
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-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-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 |
This theorem is referenced by: isms2 22987 xmsxmet 22993 setsxms 23016 tmsxms 23023 imasf1oxms 23026 ressxms 23062 prdsxms 23067 |
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