![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24451 | . . . 4 β’ (π· β (PsMetβπ) β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) | |
2 | 1 | adantl 481 | . . 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 2815 | . 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 7422 | . . . . . . 7 β’ (π = π β (0[,)π) = (0[,)π)) | |
5 | 4 | imaeq2d 6057 | . . . . . 6 β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
6 | 5 | cbvmptv 5255 | . . . . 5 β’ (π β β+ β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) |
7 | 6 | rneqi 5933 | . . . 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 24459 | . . 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 23768 | . . 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 395 = wceq 1534 β wcel 2099 β wne 2936 βwrex 3066 β wss 3945 β c0 4318 β¦ cmpt 5225 Γ cxp 5670 β‘ccnv 5671 ran crn 5673 β cima 5675 βcfv 6542 (class class class)co 7414 0cc0 11132 β+crp 13000 [,)cico 13352 PsMetcpsmet 21256 fBascfbas 21260 filGencfg 21261 metUnifcmetu 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 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 5143 df-opab 5205 df-mpt 5226 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-rp 13001 df-ico 13356 df-psmet 21264 df-fbas 21269 df-fg 21270 df-metu 21271 |
This theorem is referenced by: metuel2 24467 metustbl 24468 restmetu 24472 |
Copyright terms: Public domain | W3C validator |