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Mirrors > Home > MPE Home > Th. List > cfilucfil2 | Structured version Visualization version GIF version |
Description: Given a metric π· and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 24535. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
cfilucfil2 | β’ ((π β β β§ π· β (PsMetβπ)) β (πΆ β (CauFiluβ(metUnifβπ·)) β (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metuval 23811 | . . . . 5 β’ (π· β (PsMetβπ) β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) | |
2 | 1 | adantl 482 | . . . 4 β’ ((π β β β§ π· β (PsMetβπ)) β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) |
3 | 2 | fveq2d 6829 | . . 3 β’ ((π β β β§ π· β (PsMetβπ)) β (CauFiluβ(metUnifβπ·)) = (CauFiluβ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))))) |
4 | 3 | eleq2d 2822 | . 2 β’ ((π β β β§ π· β (PsMetβπ)) β (πΆ β (CauFiluβ(metUnifβπ·)) β πΆ β (CauFiluβ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))))) |
5 | oveq2 7345 | . . . . . 6 β’ (π = π β (0[,)π) = (0[,)π)) | |
6 | 5 | imaeq2d 5999 | . . . . 5 β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
7 | 6 | cbvmptv 5205 | . . . 4 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) |
8 | 7 | rneqi 5878 | . . 3 β’ ran (π β β+ β¦ (β‘π· β (0[,)π))) = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
9 | 8 | cfilucfil 23821 | . 2 β’ ((π β β β§ π· β (PsMetβπ)) β (πΆ β (CauFiluβ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) β (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
10 | 4, 9 | bitrd 278 | 1 β’ ((π β β β§ π· β (PsMetβπ)) β (πΆ β (CauFiluβ(metUnifβπ·)) β (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 β wne 2940 βwral 3061 βwrex 3070 β wss 3898 β c0 4269 β¦ cmpt 5175 Γ cxp 5618 β‘ccnv 5619 ran crn 5621 β cima 5623 βcfv 6479 (class class class)co 7337 0cc0 10972 β+crp 12831 [,)cico 13182 PsMetcpsmet 20687 fBascfbas 20691 filGencfg 20692 metUnifcmetu 20694 CauFiluccfilu 23544 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-2 12137 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-ico 13186 df-psmet 20695 df-fbas 20700 df-fg 20701 df-metu 20702 df-fil 23103 df-ust 23458 df-cfilu 23545 |
This theorem is referenced by: cfilucfil3 24590 cmetcusp 24624 |
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