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Mirrors > Home > ILE Home > Th. List > xmsxmet2 | GIF version |
Description: The distance function, suitably truncated, is an extended metric on π. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
mscl.x | β’ π = (Baseβπ) |
mscl.d | β’ π· = (distβπ) |
Ref | Expression |
---|---|
xmsxmet2 | β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mscl.x | . 2 β’ π = (Baseβπ) | |
2 | mscl.d | . . 3 β’ π· = (distβπ) | |
3 | 2 | reseq1i 4903 | . 2 β’ (π· βΎ (π Γ π)) = ((distβπ) βΎ (π Γ π)) |
4 | 1, 3 | xmsxmet 13853 | 1 β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Γ cxp 4624 βΎ cres 4628 βcfv 5216 Basecbs 12456 distcds 12539 βMetcxmet 13331 βMetSpcxms 13729 |
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-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-stab 831 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-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-map 6649 df-sup 6982 df-inf 6983 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-xneg 9770 df-xadd 9771 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-ndx 12459 df-slot 12460 df-base 12462 df-tset 12549 df-rest 12680 df-topn 12681 df-topgen 12699 df-psmet 13338 df-xmet 13339 df-bl 13341 df-mopn 13342 df-top 13389 df-topon 13402 df-topsp 13422 df-bases 13434 df-xms 13732 |
This theorem is referenced by: xmscl 13859 xmsge0 13860 xmseq0 13861 xmssym 13862 xmstri2 13863 xmstri 13865 xmstri3 13867 |
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