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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 1139 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 = (metUnif‘𝐷)) | |
| 2 | simp1 1137 | . . . 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 24466 | . . . . 5 ⊢ (𝐹 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
| 6 | 5 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋)) |
| 7 | xmetpsmet 24358 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
| 8 | metuust 24573 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) | |
| 9 | 7, 8 | sylan2 593 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
| 10 | 2, 6, 9 | syl2anc 584 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
| 11 | 1, 10 | eqeltrd 2841 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 12 | xmetutop 24581 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) | |
| 13 | 2, 6, 12 | syl2anc 584 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) |
| 14 | 1 | fveq2d 6910 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘𝑈) = (unifTop‘(metUnif‘𝐷))) |
| 15 | eqid 2737 | . . . . 5 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
| 16 | 15, 3, 4 | xmstopn 24461 | . . . 4 ⊢ (𝐹 ∈ ∞MetSp → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
| 17 | 16 | 3ad2ant2 1135 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
| 18 | 13, 14, 17 | 3eqtr4rd 2788 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (unifTop‘𝑈)) |
| 19 | xmsusp.u | . . 3 ⊢ 𝑈 = (UnifSt‘𝐹) | |
| 20 | 3, 19, 15 | isusp 24270 | . 2 ⊢ (𝐹 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝐹) = (unifTop‘𝑈))) |
| 21 | 11, 18, 20 | sylanbrc 583 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 × cxp 5683 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 distcds 17306 TopOpenctopn 17466 PsMetcpsmet 21348 ∞Metcxmet 21349 MetOpencmopn 21354 metUnifcmetu 21355 UnifOncust 24208 unifTopcutop 24239 UnifStcuss 24262 UnifSpcusp 24263 ∞MetSpcxms 24327 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-metu 21363 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-fil 23854 df-ust 24209 df-utop 24240 df-usp 24266 df-xms 24330 |
| This theorem is referenced by: cmetcusp1 25387 |
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