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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 6919 | . . . 4 β’ (π· β (βMetβπ) β π β dom βMet) | |
2 | isxmet 24174 | . . . 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 1533 β wcel 2098 βwral 3053 class class class wbr 5139 Γ cxp 5665 dom cdm 5667 βΆwf 6530 βcfv 6534 (class class class)co 7402 0cc0 11107 β*cxr 11246 β€ cle 11248 +π cxad 13091 βMetcxmet 21219 |
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 ax-cnex 11163 ax-resscn 11164 |
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-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 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-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-xr 11251 df-xmet 21227 |
This theorem is referenced by: xmetcl 24181 xmetdmdm 24185 xmetpsmet 24198 xmettpos 24199 xmetres2 24211 xmetres 24214 imasdsf1olem 24223 xmeterval 24282 xmeter 24283 xmetresbl 24287 tmsval 24333 tmslem 24334 tmslemOLD 24335 tmsxms 24339 imasf1oxms 24342 comet 24366 stdbdxmet 24368 prdsxms 24383 xrsdsre 24670 xmetdcn2 24697 iscfil2 25138 caufval 25147 isbndx 37154 ssbnd 37160 ismtyval 37172 |
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