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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 24313 | . . . . 5 ⊢ (𝐹 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
6 | 5 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | xmetpsmet 24205 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
8 | metuust 24420 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) | |
9 | 7, 8 | sylan2 592 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
10 | 2, 6, 9 | syl2anc 583 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
11 | 1, 10 | eqeltrd 2827 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | xmetutop 24428 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) | |
13 | 2, 6, 12 | syl2anc 583 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) |
14 | 1 | fveq2d 6888 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘𝑈) = (unifTop‘(metUnif‘𝐷))) |
15 | eqid 2726 | . . . . 5 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
16 | 15, 3, 4 | xmstopn 24308 | . . . 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 2777 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (unifTop‘𝑈)) |
19 | xmsusp.u | . . 3 ⊢ 𝑈 = (UnifSt‘𝐹) | |
20 | 3, 19, 15 | isusp 24117 | . 2 ⊢ (𝐹 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝐹) = (unifTop‘𝑈))) |
21 | 11, 18, 20 | sylanbrc 582 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 × cxp 5667 ↾ cres 5671 ‘cfv 6536 Basecbs 17151 distcds 17213 TopOpenctopn 17374 PsMetcpsmet 21220 ∞Metcxmet 21221 MetOpencmopn 21226 metUnifcmetu 21227 UnifOncust 24055 unifTopcutop 24086 UnifStcuss 24109 UnifSpcusp 24110 ∞MetSpcxms 24174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-metu 21235 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-fil 23701 df-ust 24056 df-utop 24087 df-usp 24113 df-xms 24177 |
This theorem is referenced by: cmetcusp1 25232 |
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