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Mirrors > Home > MPE Home > Th. List > xmetf | Structured version Visualization version GIF version |
Description: Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetf | β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6934 | . . . 4 β’ (π· β (βMetβπ) β π β dom βMet) | |
2 | isxmet 24243 | . . . 4 β’ (π β dom βMet β (π· β (βMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π βπ¦ β π (((π₯π·π¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π· β (βMetβπ) β (π· β (βMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π βπ¦ β π (((π₯π·π¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))))) |
4 | 3 | ibi 267 | . 2 β’ (π· β (βMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ₯ β π βπ¦ β π (((π₯π·π¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))))) |
5 | 4 | simpld 494 | 1 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 Γ cxp 5676 dom cdm 5678 βΆwf 6544 βcfv 6548 (class class class)co 7420 0cc0 11139 β*cxr 11278 β€ cle 11280 +π cxad 13123 βMetcxmet 21264 |
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 ax-cnex 11195 ax-resscn 11196 |
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-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 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-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-xr 11283 df-xmet 21272 |
This theorem is referenced by: xmetcl 24250 xmetdmdm 24254 xmetpsmet 24267 xmettpos 24268 xmetres2 24280 xmetres 24283 imasdsf1olem 24292 xmeterval 24351 xmeter 24352 xmetresbl 24356 tmsval 24402 tmslem 24403 tmslemOLD 24404 tmsxms 24408 imasf1oxms 24411 comet 24435 stdbdxmet 24437 prdsxms 24452 xrsdsre 24739 xmetdcn2 24766 iscfil2 25207 caufval 25216 isbndx 37255 ssbnd 37261 ismtyval 37273 |
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