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Mirrors > Home > MPE Home > Th. List > isxmetd | Structured version Visualization version GIF version |
Description: Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 9-Apr-2024.) |
Ref | Expression |
---|---|
isxmetd.0 | β’ (π β π β π) |
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 1351 | . . . . . 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 417 | . . . . 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 3141 | . . . 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 510 | . . 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 3196 | . 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 π) | |
10 | isxmet 24248 | . . 3 β’ (π 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¦)))))) | |
11 | 9, 10 | syl 17 | . 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 711 | 1 β’ (π β π· β (βMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3057 class class class wbr 5150 Γ cxp 5678 βΆwf 6547 βcfv 6551 (class class class)co 7424 0cc0 11144 β*cxr 11283 β€ cle 11285 +π cxad 13128 βMetcxmet 21269 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8851 df-xr 11288 df-xmet 21277 |
This theorem is referenced by: isxmet2d 24251 xmetres2 24285 comet 24440 |
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