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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 1006 |
. . . . 5
| |
| 3 | simpr3 1007 |
. . . . 5
| |
| 4 | simpr1 1005 |
. . . . 5
| |
| 5 | mettri 14787 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1251 |
. . . 4
|
| 7 | metcl 14767 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1249 |
. . . . . 6
|
| 9 | 8 | recnd 8100 |
. . . . 5
 |
| 10 | metcl 14767 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1249 |
. . . . . 6
|
| 12 | 11 | recnd 8100 |
. . . . 5
|
| 13 | metsym 14785 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1249 |
. . . . . 6
|
| 15 | 14 | oveq1d 5958 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8228 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4069 |
. . 3
|
| 18 | metcl 14767 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1249 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8614 |
. . 3
|
| 21 | 17, 20 | mpbird 167 |
. 2
|
| 22 | mettri 14787 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1251 |
. . 3
|
| 24 | 19 | recnd 8100 |
. . . 4
|
| 25 | 9, 24 | addcomd 8222 |
. . 3
|
| 26 | 23, 25 | breqtrd 4069 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11432 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 946 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 980
wceq 1372
wcel 2175
class class class wbr 4043 cfv 5270 (class class class)co 5943
cr 7923
caddc 7927 cle 8107 cmin 8242 cabs 11250 cmet 14241 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 ax-arch 8043 ax-caucvg 8044 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-po 4342 df-iso 4343 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-map 6736 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 df-div 8745 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-n0 9295 df-z 9372 df-uz 9648 df-rp 9775 df-xadd 9894 df-seqfrec 10591 df-exp 10682 df-cj 11095 df-re 11096 df-im 11097 df-rsqrt 11251 df-abs 11252 df-xmet 14248 df-met 14249 |
| This theorem is referenced by: msrtri 14890 |
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