![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2821 | . 2 β’ (π΅ β πΉ β π΅ β ran (π β β+ β¦ (β‘π· β (0[,)π)))) |
3 | elex 3490 | . . . 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 7932 | . . . . 5 β’ (π· β (PsMetβπ) β β‘π· β V) | |
6 | imaexg 7921 | . . . . 5 β’ (β‘π· β V β (β‘π· β (0[,)π)) β V) | |
7 | eleq1a 2824 | . . . . 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 3157 | . . 3 β’ (π· β (PsMetβπ) β (βπ β β+ π΅ = (β‘π· β (0[,)π)) β π΅ β V)) |
10 | eqid 2728 | . . . . 5 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) | |
11 | 10 | elrnmpt 5958 | . . . 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 1534 β wcel 2099 βwrex 3067 Vcvv 3471 β¦ cmpt 5231 β‘ccnv 5677 ran crn 5679 β cima 5681 βcfv 6548 (class class class)co 7420 0cc0 11139 β+crp 13007 [,)cico 13359 PsMetcpsmet 21263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 |
This theorem is referenced by: metustto 24475 metustid 24476 metustexhalf 24478 metustfbas 24479 cfilucfil 24481 metucn 24493 |
Copyright terms: Public domain | W3C validator |