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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 14609 |
. . . . 5
 | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1251 |
. . . 4
|
| 7 | metcl 14589 |
. . . . . . 7
 | |
| 8 | 1, 4, 2, 7 | syl3anc 1249 |
. . . . . 6
|
| 9 | 8 | recnd 8055 |
. . . . 5
 |
| 10 | metcl 14589 |
. . . . . . 7
| |
| 11 | 1, 4, 3, 10 | syl3anc 1249 |
. . . . . 6
|
| 12 | 11 | recnd 8055 |
. . . . 5
|
| 13 | metsym 14607 |
. . . . . . 7
 | |
| 14 | 1, 2, 4, 13 | syl3anc 1249 |
. . . . . 6
|
| 15 | 14 | oveq1d 5937 |
. . . . 5
|
| 16 | 9, 12, 15 | comraddd 8183 |
. . . 4
|
| 17 | 6, 16 | breqtrd 4059 |
. . 3
|
| 18 | metcl 14589 |
. . . . 5
| |
| 19 | 1, 2, 3, 18 | syl3anc 1249 |
. . . 4
|
| 20 | 19, 8, 11 | lesubaddd 8569 |
. . 3
|
| 21 | 17, 20 | mpbird 167 |
. 2
|
| 22 | mettri 14609 |
. . . 4
| |
| 23 | 1, 4, 3, 2, 22 | syl13anc 1251 |
. . 3
|
| 24 | 19 | recnd 8055 |
. . . 4
|
| 25 | 9, 24 | addcomd 8177 |
. . 3
|
| 26 | 23, 25 | breqtrd 4059 |
. 2
|
| 27 | 11, 19, 8 | absdifled 11344 |
. 2
|
| 28 | 21, 26, 27 | mpbir2and 946 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 980
wceq 1364
wcel 2167
class class class wbr 4033 cfv 5258 (class class class)co 5922
cr 7878
caddc 7882 cle 8062 cmin 8197 cabs 11162 cmet 14093 |
| 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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-map 6709 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-rp 9729 df-xadd 9848 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-xmet 14100 df-met 14101 |
| This theorem is referenced by: msrtri 14712 |
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