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Mirrors > Home > MPE Home > Th. List > metuel | Structured version Visualization version GIF version |
Description: Elementhood in the uniform structure generated by a metric π· (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
metuel | β’ ((π β β β§ π· β (PsMetβπ)) β (π β (metUnifβπ·) β (π β (π Γ π) β§ βπ€ β ran (π β β+ β¦ (β‘π· β (0[,)π)))π€ β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metuval 23928 | . . . 4 β’ (π· β (PsMetβπ) β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) | |
2 | 1 | adantl 483 | . . 3 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 | eleq2d 2820 | . 2 β’ ((π β β β§ π· β (PsMetβπ)) β (π β (metUnifβπ·) β π β ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))))) |
4 | oveq2 7369 | . . . . . . 7 β’ (π = π β (0[,)π) = (0[,)π)) | |
5 | 4 | imaeq2d 6017 | . . . . . 6 β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
6 | 5 | cbvmptv 5222 | . . . . 5 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) |
7 | 6 | rneqi 5896 | . . . 4 β’ ran (π β β+ β¦ (β‘π· β (0[,)π))) = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
8 | 7 | metustfbas 23936 | . . 3 β’ ((π β β β§ π· β (PsMetβπ)) β ran (π β β+ β¦ (β‘π· β (0[,)π))) β (fBasβ(π Γ π))) |
9 | elfg 23245 | . . 3 β’ (ran (π β β+ β¦ (β‘π· β (0[,)π))) β (fBasβ(π Γ π)) β (π β ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) β (π β (π Γ π) β§ βπ€ β ran (π β β+ β¦ (β‘π· β (0[,)π)))π€ β π))) | |
10 | 8, 9 | syl 17 | . 2 β’ ((π β β β§ π· β (PsMetβπ)) β (π β ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) β (π β (π Γ π) β§ βπ€ β ran (π β β+ β¦ (β‘π· β (0[,)π)))π€ β π))) |
11 | 3, 10 | bitrd 279 | 1 β’ ((π β β β§ π· β (PsMetβπ)) β (π β (metUnifβπ·) β (π β (π Γ π) β§ βπ€ β ran (π β β+ β¦ (β‘π· β (0[,)π)))π€ β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βwrex 3070 β wss 3914 β c0 4286 β¦ cmpt 5192 Γ cxp 5635 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 fBascfbas 20807 filGencfg 20808 metUnifcmetu 20810 |
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 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 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-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-rp 12924 df-ico 13279 df-psmet 20811 df-fbas 20816 df-fg 20817 df-metu 20818 |
This theorem is referenced by: metuel2 23944 metustbl 23945 restmetu 23949 |
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