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Mirrors > Home > ILE Home > Th. List > mettri | Unicode version |
Description: Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
mettri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mettri2 12534 | . . . . 5 | |
2 | 1 | expcom 115 | . . . 4 |
3 | 2 | 3coml 1188 | . . 3 |
4 | 3 | impcom 124 | . 2 |
5 | metsym 12543 | . . . 4 | |
6 | 5 | 3adant3r2 1191 | . . 3 |
7 | 6 | oveq1d 5789 | . 2 |
8 | 4, 7 | breqtrrd 3956 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 caddc 7626 cle 7804 cmet 12153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1re 7717 ax-addrcl 7720 ax-0id 7731 ax-rnegex 7732 ax-pre-ltirr 7735 ax-pre-apti 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-xadd 9563 df-xmet 12160 df-met 12161 |
This theorem is referenced by: mettri3 12547 metrtri 12549 mstri 12645 |
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