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| Mirrors > Home > ILE Home > Th. List > metrtri | Unicode version | ||
| Description: Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.) |
| Ref | Expression |
|---|---|
| metrtri |
        ![/\](wedge.gif "/\"){kind=small} 
  
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 108 |
. . . . 5
| |
| 2 | simpr2 988 |
. . . . 5
| |
| 3 | simpr3 989 |
. . . . 5
| |
| 4 | simpr1 987 |
. . . . 5
| |
| 5 | mettri 12542 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1218 |
. . . 4
|
| 7 | metcl 12522 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1216 |
. . . . . 6
|
| 9 | 8 | recnd 7794 |
. . . . 5
 |
| 10 | metcl 12522 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1216 |
. . . . . 6
|
| 12 | 11 | recnd 7794 |
. . . . 5
|
| 13 | metsym 12540 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1216 |
. . . . . 6
|
| 15 | 14 | oveq1d 5789 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 7919 |
. . . 4
|
| 17 | 6, 16 | breqtrd 3954 |
. . 3
|
| 18 | metcl 12522 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1216 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8304 |
. . 3
|
| 21 | 17, 20 | mpbird 166 |
. 2
|
| 22 | mettri 12542 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1218 |
. . 3
|
| 24 | 19 | recnd 7794 |
. . . 4
|
| 25 | 9, 24 | addcomd 7913 |
. . 3
|
| 26 | 23, 25 | breqtrd 3954 |
. 2
|
| 27 | 11, 19, 8 | absdifled 10951 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 928 |
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
cr 7619
caddc 7623 cle 7801 cmin 7933 cabs 10769 cmet 12150 |
| 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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
| 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-reu 2423 df-rmo 2424 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-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 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-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-xmet 12157 df-met 12158 |
| This theorem is referenced by: msrtri 12645 |
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