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Mirrors > Home > ILE Home > Th. List > metres | GIF version |
Description: A restriction of a metric is a metric. (Contributed by NM, 26-Aug-2007.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
metres | β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) β (Metβ(π β© π ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metf 13936 | . . 3 β’ (π· β (Metβπ) β π·:(π Γ π)βΆβ) | |
2 | fdm 5373 | . . 3 β’ (π·:(π Γ π)βΆβ β dom π· = (π Γ π)) | |
3 | metreslem 13965 | . . 3 β’ (dom π· = (π Γ π) β (π· βΎ (π Γ π )) = (π· βΎ ((π β© π ) Γ (π β© π )))) | |
4 | 1, 2, 3 | 3syl 17 | . 2 β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) = (π· βΎ ((π β© π ) Γ (π β© π )))) |
5 | inss1 3357 | . . 3 β’ (π β© π ) β π | |
6 | metres2 13966 | . . 3 β’ ((π· β (Metβπ) β§ (π β© π ) β π) β (π· βΎ ((π β© π ) Γ (π β© π ))) β (Metβ(π β© π ))) | |
7 | 5, 6 | mpan2 425 | . 2 β’ (π· β (Metβπ) β (π· βΎ ((π β© π ) Γ (π β© π ))) β (Metβ(π β© π ))) |
8 | 4, 7 | eqeltrd 2254 | 1 β’ (π· β (Metβπ) β (π· βΎ (π Γ π )) β (Metβ(π β© π ))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 β© cin 3130 β wss 3131 Γ cxp 4626 dom cdm 4628 βΎ cres 4630 βΆwf 5214 βcfv 5218 βcr 7812 Metcmet 13526 |
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 ax-1re 7907 ax-addrcl 7910 ax-rnegex 7922 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-nel 2443 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-if 3537 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-xadd 9775 df-xmet 13533 df-met 13534 |
This theorem is referenced by: (None) |
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