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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 1134 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 = (metUnif‘𝐷)) | |
2 | simp1 1132 | . . . 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 23068 | . . . . 5 ⊢ (𝐹 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
6 | 5 | 3ad2ant2 1130 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | xmetpsmet 22960 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
8 | metuust 23172 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) | |
9 | 7, 8 | sylan2 594 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
10 | 2, 6, 9 | syl2anc 586 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
11 | 1, 10 | eqeltrd 2915 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | xmetutop 23180 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) | |
13 | 2, 6, 12 | syl2anc 586 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) |
14 | 1 | fveq2d 6676 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘𝑈) = (unifTop‘(metUnif‘𝐷))) |
15 | eqid 2823 | . . . . 5 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
16 | 15, 3, 4 | xmstopn 23063 | . . . 4 ⊢ (𝐹 ∈ ∞MetSp → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
17 | 16 | 3ad2ant2 1130 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
18 | 13, 14, 17 | 3eqtr4rd 2869 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (unifTop‘𝑈)) |
19 | xmsusp.u | . . 3 ⊢ 𝑈 = (UnifSt‘𝐹) | |
20 | 3, 19, 15 | isusp 22872 | . 2 ⊢ (𝐹 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝐹) = (unifTop‘𝑈))) |
21 | 11, 18, 20 | sylanbrc 585 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 × cxp 5555 ↾ cres 5559 ‘cfv 6357 Basecbs 16485 distcds 16576 TopOpenctopn 16697 PsMetcpsmet 20531 ∞Metcxmet 20532 MetOpencmopn 20537 metUnifcmetu 20538 UnifOncust 22810 unifTopcutop 22841 UnifStcuss 22864 UnifSpcusp 22865 ∞MetSpcxms 22929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-metu 20546 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-fil 22456 df-ust 22811 df-utop 22842 df-usp 22868 df-xms 22932 |
This theorem is referenced by: cmetcusp1 23958 |
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