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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 1030 |
. . . . 5
| |
| 3 | simpr3 1031 |
. . . . 5
| |
| 4 | simpr1 1029 |
. . . . 5
| |
| 5 | mettri 15096 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1275 |
. . . 4
|
| 7 | metcl 15076 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1273 |
. . . . . 6
|
| 9 | 8 | recnd 8207 |
. . . . 5
 |
| 10 | metcl 15076 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1273 |
. . . . . 6
|
| 12 | 11 | recnd 8207 |
. . . . 5
|
| 13 | metsym 15094 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1273 |
. . . . . 6
|
| 15 | 14 | oveq1d 6032 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8335 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4114 |
. . 3
|
| 18 | metcl 15076 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1273 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8721 |
. . 3
|
| 21 | 17, 20 | mpbird 167 |
. 2
|
| 22 | mettri 15096 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1275 |
. . 3
|
| 24 | 19 | recnd 8207 |
. . . 4
|
| 25 | 9, 24 | addcomd 8329 |
. . 3
|
| 26 | 23, 25 | breqtrd 4114 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11739 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 952 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 1004
wceq 1397
wcel 2202
class class class wbr 4088 cfv 5326 (class class class)co 6017
cr 8030
caddc 8034 cle 8214 cmin 8349 cabs 11557 cmet 14550 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151 |
| This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-map 6818 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-rp 9888 df-xadd 10007 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-xmet 14557 df-met 14558 |
| This theorem is referenced by: msrtri 15199 |
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