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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 1056 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 = (metUnif‘𝐷)) | |
2 | simp1 1054 | . . . 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 22071 | . . . . 5 ⊢ (𝐹 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
6 | 5 | 3ad2ant2 1076 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | xmetpsmet 21963 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
8 | metuust 22175 | . . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) | |
9 | 7, 8 | sylan2 490 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
10 | 2, 6, 9 | syl2anc 691 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋)) |
11 | 1, 10 | eqeltrd 2688 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝑈 ∈ (UnifOn‘𝑋)) |
12 | xmetutop 22183 | . . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) | |
13 | 2, 6, 12 | syl2anc 691 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷)) |
14 | 1 | fveq2d 6107 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (unifTop‘𝑈) = (unifTop‘(metUnif‘𝐷))) |
15 | eqid 2610 | . . . . 5 ⊢ (TopOpen‘𝐹) = (TopOpen‘𝐹) | |
16 | 15, 3, 4 | xmstopn 22066 | . . . 4 ⊢ (𝐹 ∈ ∞MetSp → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
17 | 16 | 3ad2ant2 1076 | . . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (MetOpen‘𝐷)) |
18 | 13, 14, 17 | 3eqtr4rd 2655 | . 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → (TopOpen‘𝐹) = (unifTop‘𝑈)) |
19 | xmsusp.u | . . 3 ⊢ 𝑈 = (UnifSt‘𝐹) | |
20 | 3, 19, 15 | isusp 21875 | . 2 ⊢ (𝐹 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ (TopOpen‘𝐹) = (unifTop‘𝑈))) |
21 | 11, 18, 20 | sylanbrc 695 | 1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 × cxp 5036 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 distcds 15777 TopOpenctopn 15905 PsMetcpsmet 19551 ∞Metcxmt 19552 MetOpencmopn 19557 metUnifcmetu 19558 UnifOncust 21813 unifTopcutop 21844 UnifStcuss 21867 UnifSpcusp 21868 ∞MetSpcxme 21932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-metu 19566 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-fil 21460 df-ust 21814 df-utop 21845 df-usp 21871 df-xms 21935 |
This theorem is referenced by: cmetcusp1 22957 |
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