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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 418 | . . . . 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 3139 | . . . 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 511 | . . 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 3194 | . 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 24181 | . . 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 710 | 1 β’ (π β π· β (βMetβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 class class class wbr 5141 Γ cxp 5667 βΆwf 6532 βcfv 6536 (class class class)co 7404 0cc0 11109 β*cxr 11248 β€ cle 11250 +π cxad 13093 βMetcxmet 21221 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-xr 11253 df-xmet 21229 |
This theorem is referenced by: isxmet2d 24184 xmetres2 24218 comet 24373 |
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