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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 109 |
. . . . 5
| |
| 2 | simpr2 1028 |
. . . . 5
| |
| 3 | simpr3 1029 |
. . . . 5
| |
| 4 | simpr1 1027 |
. . . . 5
| |
| 5 | mettri 15062 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1273 |
. . . 4
|
| 7 | metcl 15042 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1271 |
. . . . . 6
|
| 9 | 8 | recnd 8186 |
. . . . 5
 |
| 10 | metcl 15042 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1271 |
. . . . . 6
|
| 12 | 11 | recnd 8186 |
. . . . 5
|
| 13 | metsym 15060 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1271 |
. . . . . 6
|
| 15 | 14 | oveq1d 6022 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8314 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4109 |
. . 3
|
| 18 | metcl 15042 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1271 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8700 |
. . 3
|
| 21 | 17, 20 | mpbird 167 |
. 2
|
| 22 | mettri 15062 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1273 |
. . 3
|
| 24 | 19 | recnd 8186 |
. . . 4
|
| 25 | 9, 24 | addcomd 8308 |
. . 3
|
| 26 | 23, 25 | breqtrd 4109 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11705 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 950 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 1002
wceq 1395
wcel 2200
class class class wbr 4083 cfv 5318 (class class class)co 6007
cr 8009
caddc 8013 cle 8193 cmin 8328 cabs 11523 cmet 14516 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130 |
| This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-map 6805 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-rp 9862 df-xadd 9981 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-xmet 14523 df-met 14524 |
| This theorem is referenced by: msrtri 15165 |
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