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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 994 |
. . . . 5
| |
| 3 | simpr3 995 |
. . . . 5
| |
| 4 | simpr1 993 |
. . . . 5
| |
| 5 | mettri 13013 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1230 |
. . . 4
|
| 7 | metcl 12993 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1228 |
. . . . . 6
|
| 9 | 8 | recnd 7927 |
. . . . 5
 |
| 10 | metcl 12993 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1228 |
. . . . . 6
|
| 12 | 11 | recnd 7927 |
. . . . 5
|
| 13 | metsym 13011 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1228 |
. . . . . 6
|
| 15 | 14 | oveq1d 5857 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8055 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4008 |
. . 3
|
| 18 | metcl 12993 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1228 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8440 |
. . 3
|
| 21 | 17, 20 | mpbird 166 |
. 2
|
| 22 | mettri 13013 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1230 |
. . 3
|
| 24 | 19 | recnd 7927 |
. . . 4
|
| 25 | 9, 24 | addcomd 8049 |
. . 3
|
| 26 | 23, 25 | breqtrd 4008 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11121 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 934 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
w3a 968
wceq 1343
wcel 2136
class class class wbr 3982 cfv 5188 (class class class)co 5842
cr 7752
caddc 7756 cle 7934 cmin 8069 cabs 10939 cmet 12621 |
| 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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 |
| This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-map 6616 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-rp 9590 df-xadd 9709 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-xmet 12628 df-met 12629 |
| This theorem is referenced by: msrtri 13116 |
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