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Mirrors > Home > ILE Home > Th. List > isxmetd | GIF version |
Description: Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
isxmetd.0 | β’ (π β π β V) |
isxmetd.1 | β’ (π β π·:(π Γ π)βΆβ*) |
isxmetd.2 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β ((π₯π·π¦) = 0 β π₯ = π¦)) |
isxmetd.3 | β’ ((π β§ (π₯ β π β§ π¦ β π β§ π§ β π)) β (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))) |
Ref | Expression |
---|---|
isxmetd | β’ (π β π· β (βMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isxmetd.1 | . 2 β’ (π β π·:(π Γ π)βΆβ*) | |
2 | isxmetd.2 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β ((π₯π·π¦) = 0 β π₯ = π¦)) | |
3 | isxmetd.3 | . . . . . . 7 β’ ((π β§ (π₯ β π β§ π¦ β π β§ π§ β π)) β (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))) | |
4 | 3 | 3exp2 1225 | . . . . . 6 β’ (π β (π₯ β π β (π¦ β π β (π§ β π β (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))))) |
5 | 4 | imp32 257 | . . . . 5 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π§ β π β (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))) |
6 | 5 | ralrimiv 2549 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦))) |
7 | 2, 6 | jca 306 | . . 3 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (((π₯π·π¦) = 0 β π₯ = π¦) β§ βπ§ β π (π₯π·π¦) β€ ((π§π·π₯) +π (π§π·π¦)))) |
8 | 7 | ralrimivva 2559 | . 2 β’ (π 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¦)))) |
9 | isxmetd.0 | . . 3 β’ (π β π β V) | |
10 | isxmet 13930 | . . 3 β’ (π β V 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¦)))))) | |
11 | 9, 10 | syl 14 | . 2 β’ (π 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¦)))))) |
12 | 1, 8, 11 | mpbir2and 944 | 1 β’ (π β π· β (βMetβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 978 = wceq 1353 β wcel 2148 βwral 2455 Vcvv 2739 class class class wbr 4005 Γ cxp 4626 βΆwf 5214 βcfv 5218 (class class class)co 5877 0cc0 7813 β*cxr 7993 β€ cle 7995 +π cxad 9772 βMetcxmet 13525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-pnf 7996 df-mnf 7997 df-xr 7998 df-xmet 13533 |
This theorem is referenced by: isxmet2d 13933 xmetres2 13964 comet 14084 xmetxp 14092 |
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