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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 6879 | . . . 4 β’ (π· β (βMetβπ) β π β dom βMet) | |
2 | isxmet 23677 | . . . 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 266 | . 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 495 | 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 396 = wceq 1541 β wcel 2106 βwral 3064 class class class wbr 5105 Γ cxp 5631 dom cdm 5633 βΆwf 6492 βcfv 6496 (class class class)co 7357 0cc0 11051 β*cxr 11188 β€ cle 11190 +π cxad 13031 βMetcxmet 20781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-xr 11193 df-xmet 20789 |
This theorem is referenced by: xmetcl 23684 xmetdmdm 23688 xmetpsmet 23701 xmettpos 23702 xmetres2 23714 xmetres 23717 imasdsf1olem 23726 xmeterval 23785 xmeter 23786 xmetresbl 23790 tmsval 23836 tmslem 23837 tmslemOLD 23838 tmsxms 23842 imasf1oxms 23845 comet 23869 stdbdxmet 23871 prdsxms 23886 xrsdsre 24173 xmetdcn2 24200 iscfil2 24630 caufval 24639 isbndx 36241 ssbnd 36247 ismtyval 36259 |
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