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Mirrors > Home > MPE Home > Th. List > metustel | Structured version Visualization version GIF version |
Description: Define a filter base πΉ generated by a metric π·. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
metust.1 | β’ πΉ = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
Ref | Expression |
---|---|
metustel | β’ (π· β (PsMetβπ) β (π΅ β πΉ β βπ β β+ π΅ = (β‘π· β (0[,)π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metust.1 | . . 3 β’ πΉ = ran (π β β+ β¦ (β‘π· β (0[,)π))) | |
2 | 1 | eleq2i 2826 | . 2 β’ (π΅ β πΉ β π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π)))) |
3 | elex 3465 | . . . 4 β’ (π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β π΅ β V) | |
4 | 3 | a1i 11 | . . 3 β’ (π· β (PsMetβπ) β (π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β π΅ β V)) |
5 | cnvexg 7865 | . . . . 5 β’ (π· β (PsMetβπ) β β‘π· β V) | |
6 | imaexg 7856 | . . . . 5 β’ (β‘π· β V β (β‘π· β (0[,)π)) β V) | |
7 | eleq1a 2829 | . . . . 5 β’ ((β‘π· β (0[,)π)) β V β (π΅ = (β‘π· β (0[,)π)) β π΅ β V)) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 β’ (π· β (PsMetβπ) β (π΅ = (β‘π· β (0[,)π)) β π΅ β V)) |
9 | 8 | rexlimdvw 3154 | . . 3 β’ (π· β (PsMetβπ) β (βπ β β+ π΅ = (β‘π· β (0[,)π)) β π΅ β V)) |
10 | eqid 2733 | . . . . 5 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) | |
11 | 10 | elrnmpt 5915 | . . . 4 β’ (π΅ β V β (π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ π΅ = (β‘π· β (0[,)π)))) |
12 | 11 | a1i 11 | . . 3 β’ (π· β (PsMetβπ) β (π΅ β V β (π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ π΅ = (β‘π· β (0[,)π))))) |
13 | 4, 9, 12 | pm5.21ndd 381 | . 2 β’ (π· β (PsMetβπ) β (π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ π΅ = (β‘π· β (0[,)π)))) |
14 | 2, 13 | bitrid 283 | 1 β’ (π· β (PsMetβπ) β (π΅ β πΉ β βπ β β+ π΅ = (β‘π· β (0[,)π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwrex 3070 Vcvv 3447 β¦ cmpt 5192 β‘ccnv 5636 ran crn 5638 β cima 5640 βcfv 6500 (class class class)co 7361 0cc0 11059 β+crp 12923 [,)cico 13275 PsMetcpsmet 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: metustto 23932 metustid 23933 metustexhalf 23935 metustfbas 23936 cfilucfil 23938 metucn 23950 |
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