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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 2817 | . 2 β’ (π΅ β πΉ β π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π)))) |
3 | elex 3485 | . . . 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 7909 | . . . . 5 β’ (π· β (PsMetβπ) β β‘π· β V) | |
6 | imaexg 7900 | . . . . 5 β’ (β‘π· β V β (β‘π· β (0[,)π)) β V) | |
7 | eleq1a 2820 | . . . . 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 3152 | . . 3 β’ (π· β (PsMetβπ) β (βπ β β+ π΅ = (β‘π· β (0[,)π)) β π΅ β V)) |
10 | eqid 2724 | . . . . 5 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) | |
11 | 10 | elrnmpt 5946 | . . . 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 379 | . 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 1533 β wcel 2098 βwrex 3062 Vcvv 3466 β¦ cmpt 5222 β‘ccnv 5666 ran crn 5668 β cima 5670 βcfv 6534 (class class class)co 7402 0cc0 11107 β+crp 12975 [,)cico 13327 PsMetcpsmet 21218 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-xp 5673 df-rel 5674 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 |
This theorem is referenced by: metustto 24406 metustid 24407 metustexhalf 24409 metustfbas 24410 cfilucfil 24412 metucn 24424 |
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