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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 993 |
. . . . 5
| |
| 3 | simpr3 994 |
. . . . 5
| |
| 4 | simpr1 992 |
. . . . 5
| |
| 5 | mettri 12914 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1229 |
. . . 4
|
| 7 | metcl 12894 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1227 |
. . . . . 6
|
| 9 | 8 | recnd 7918 |
. . . . 5
 |
| 10 | metcl 12894 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1227 |
. . . . . 6
|
| 12 | 11 | recnd 7918 |
. . . . 5
|
| 13 | metsym 12912 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1227 |
. . . . . 6
|
| 15 | 14 | oveq1d 5851 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8046 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4002 |
. . 3
|
| 18 | metcl 12894 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1227 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8431 |
. . 3
|
| 21 | 17, 20 | mpbird 166 |
. 2
|
| 22 | mettri 12914 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1229 |
. . 3
|
| 24 | 19 | recnd 7918 |
. . . 4
|
| 25 | 9, 24 | addcomd 8040 |
. . 3
|
| 26 | 23, 25 | breqtrd 4002 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11107 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 933 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
w3a 967
wceq 1342
wcel 2135
class class class wbr 3976 cfv 5182 (class class class)co 5836
cr 7743
caddc 7747 cle 7925 cmin 8060 cabs 10925 cmet 12522 |
| 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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
| This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-xadd 9700 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-xmet 12529 df-met 12530 |
| This theorem is referenced by: msrtri 13017 |
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