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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 24473 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
5 | 2, 1 | istps 22956 | . . . 4 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
6 | df-mopn 21378 | . . . . . . . . . 10 ⊢ MetOpen = (𝑥 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑥))) | |
7 | 6 | dmmptss 6263 | . . . . . . . . 9 ⊢ dom MetOpen ⊆ ∪ ran ∞Met |
8 | toponmax 22948 | . . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
9 | 8 | adantl 481 | . . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ 𝐽) |
10 | simpl 482 | . . . . . . . . . . 11 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 = (MetOpen‘𝐷)) | |
11 | 9, 10 | eleqtrd 2841 | . . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 ∈ (MetOpen‘𝐷)) |
12 | elfvdm 6944 | . . . . . . . . . 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 | sselid 3993 | . . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ ∪ ran ∞Met) |
15 | xmetunirn 24363 | . . . . . . . 8 ⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷)) | |
16 | 14, 15 | sylib 218 | . . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘dom dom 𝐷)) |
17 | eqid 2735 | . . . . . . . . . . . . 13 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
18 | 17 | mopntopon 24465 | . . . . . . . . . . . 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 2839 | . . . . . . . . . 10 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘dom dom 𝐷)) |
21 | toponuni 22936 | . . . . . . . . . 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 22936 | . . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
24 | 23 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝑋 = ∪ 𝐽) |
25 | 22, 24 | eqtr4d 2778 | . . . . . . . 8 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → dom dom 𝐷 = 𝑋) |
26 | 25 | fveq2d 6911 | . . . . . . 7 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∞Met‘dom dom 𝐷) = (∞Met‘𝑋)) |
27 | 16, 26 | eleqtrd 2841 | . . . . . 6 ⊢ ((𝐽 = (MetOpen‘𝐷) ∧ 𝐽 ∈ (TopOn‘𝑋)) → 𝐷 ∈ (∞Met‘𝑋)) |
28 | 27 | ex 412 | . . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))) |
29 | 17 | mopntopon 24465 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (MetOpen‘𝐷) ∈ (TopOn‘𝑋)) |
30 | eleq1 2827 | . . . . . 6 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ (MetOpen‘𝐷) ∈ (TopOn‘𝑋))) | |
31 | 29, 30 | imbitrrid 246 | . . . . 5 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))) |
32 | 28, 31 | impbid 212 | . . . 4 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐷 ∈ (∞Met‘𝑋))) |
33 | 5, 32 | bitrid 283 | . . 3 ⊢ (𝐽 = (MetOpen‘𝐷) → (𝐾 ∈ TopSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
34 | 33 | pm5.32ri 575 | . 2 ⊢ ((𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) |
35 | 4, 34 | bitri 275 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 × cxp 5687 dom cdm 5689 ran crn 5690 ↾ cres 5691 ‘cfv 6563 Basecbs 17245 distcds 17307 TopOpenctopn 17468 topGenctg 17484 ∞Metcxmet 21367 ballcbl 21369 MetOpencmopn 21372 TopOnctopon 22932 TopSpctps 22954 ∞MetSpcxms 24343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-xms 24346 |
This theorem is referenced by: isms2 24476 xmsxmet 24482 setsxms 24507 tmsxms 24515 imasf1oxms 24518 ressxms 24554 prdsxms 24559 |
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