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Mirrors > Home > MPE Home > Th. List > xmsusp | Structured version Visualization version GIF version |
Description: If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.) |
Ref | Expression |
---|---|
xmsusp.x | ⊢ 𝑋 = (Base‘𝐹) |
xmsusp.d | ⊢ 𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋)) |
xmsusp.u | ⊢ 𝑈 = (UnifSt‘𝐹) |
Ref | Expression |
---|---|
xmsusp | ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1135 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 = (metUnif‘𝐷)) | |
2 | simp1 1133 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑋 ≠ ∅) | |
3 | xmsusp.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐹) | |
4 | xmsusp.d | . . . . . 6 ⊢ 𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋)) | |
5 | 3, 4 | xmsxmet 23063 | . . . . 5 ⊢ (𝐹 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
6 | 5 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | xmetpsmet 22955 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
8 | metuust 23167 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) | |
9 | 7, 8 | sylan2 595 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
10 | 2, 6, 9 | syl2anc 587 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
11 | 1, 10 | eqeltrd 2890 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | xmetutop 23175 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) | |
13 | 2, 6, 12 | syl2anc 587 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) |
14 | 1 | fveq2d 6649 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘𝑈) = (unifTop‘(metUnif‘𝐷))) |
15 | eqid 2798 | . . . . 5 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
16 | 15, 3, 4 | xmstopn 23058 | . . . 4 ⊢ (𝐹 ∈ ∞MetSp → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
17 | 16 | 3ad2ant2 1131 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
18 | 13, 14, 17 | 3eqtr4rd 2844 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (unifTop‘𝑈)) |
19 | xmsusp.u | . . 3 ⊢ 𝑈 = (UnifSt‘𝐹) | |
20 | 3, 19, 15 | isusp 22867 | . 2 ⊢ (𝐹 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝐹) = (unifTop‘𝑈))) |
21 | 11, 18, 20 | sylanbrc 586 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 × cxp 5517 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 distcds 16566 TopOpenctopn 16687 PsMetcpsmet 20075 ∞Metcxmet 20076 MetOpencmopn 20081 metUnifcmetu 20082 UnifOncust 22805 unifTopcutop 22836 UnifStcuss 22859 UnifSpcusp 22860 ∞MetSpcxms 22924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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-ico 12732 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-metu 20090 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-fil 22451 df-ust 22806 df-utop 22837 df-usp 22863 df-xms 22927 |
This theorem is referenced by: cmetcusp1 23957 |
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