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Mirrors > Home > MPE Home > Th. List > cfilucfil3 | 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 Cauchy filters for the metric. (Contributed by Thierry Arnoux, 15-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
cfilucfil3 | β’ ((π β β β§ π· β (βMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β πΆ β (CauFilβπ·))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetpsmet 24270 | . . 3 β’ (π· β (βMetβπ) β π· β (PsMetβπ)) | |
2 | cfilucfil2 24486 | . . . . 5 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₯)))) | |
3 | 2 | anbi2d 628 | . . . 4 β’ ((π β β β§ π· β (PsMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β (πΆ β (Filβπ) β§ (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯))))) |
4 | filfbas 23768 | . . . . . . 7 β’ (πΆ β (Filβπ) β πΆ β (fBasβπ)) | |
5 | 4 | pm4.71i 558 | . . . . . 6 β’ (πΆ β (Filβπ) β (πΆ β (Filβπ) β§ πΆ β (fBasβπ))) |
6 | 5 | anbi1i 622 | . . . . 5 β’ ((πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)) β ((πΆ β (Filβπ) β§ πΆ β (fBasβπ)) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯))) |
7 | anass 467 | . . . . 5 β’ (((πΆ β (Filβπ) β§ πΆ β (fBasβπ)) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)) β (πΆ β (Filβπ) β§ (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) | |
8 | 6, 7 | bitr2i 275 | . . . 4 β’ ((πΆ β (Filβπ) β§ (πΆ β (fBasβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯))) β (πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯))) |
9 | 3, 8 | bitrdi 286 | . . 3 β’ ((π β β β§ π· β (PsMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β (πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
10 | 1, 9 | sylan2 591 | . 2 β’ ((π β β β§ π· β (βMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β (πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
11 | iscfil 25209 | . . 3 β’ (π· β (βMetβπ) β (πΆ β (CauFilβπ·) β (πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) | |
12 | 11 | adantl 480 | . 2 β’ ((π β β β§ π· β (βMetβπ)) β (πΆ β (CauFilβπ·) β (πΆ β (Filβπ) β§ βπ₯ β β+ βπ¦ β πΆ (π· β (π¦ Γ π¦)) β (0[,)π₯)))) |
13 | 10, 12 | bitr4d 281 | 1 β’ ((π β β β§ π· β (βMetβπ)) β ((πΆ β (Filβπ) β§ πΆ β (CauFiluβ(metUnifβπ·))) β πΆ β (CauFilβπ·))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 β wne 2930 βwral 3051 βwrex 3060 β wss 3940 β c0 4318 Γ cxp 5670 β cima 5675 βcfv 6542 (class class class)co 7415 0cc0 11136 β+crp 13004 [,)cico 13356 PsMetcpsmet 21265 βMetcxmet 21266 fBascfbas 21269 metUnifcmetu 21272 Filcfil 23765 CauFiluccfilu 24207 CauFilccfil 25196 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-2 12303 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ico 13360 df-psmet 21273 df-xmet 21274 df-fbas 21278 df-fg 21279 df-metu 21280 df-fil 23766 df-ust 24121 df-cfilu 24208 df-cfil 25199 |
This theorem is referenced by: cfilucfil4 25265 |
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