Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > mstri2 | GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| mstri2 | β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | msmet2 13967 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | mettri2 13865 | . . 3 β’ (((π· βΎ (π Γ π)) β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅))) | |
| 5 | 3, 4 | sylan 283 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅))) |
| 6 | simpr2 1004 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΄ β π) | |
| 7 | simpr3 1005 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 6015 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
| 9 | simpr1 1003 | . . . 4 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β πΆ β π) | |
| 10 | 9, 6 | ovresd 6015 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆ(π· βΎ (π Γ π))π΄) = (πΆπ·π΄)) |
| 11 | 9, 7 | ovresd 6015 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆ(π· βΎ (π Γ π))π΅) = (πΆπ·π΅)) |
| 12 | 10, 11 | oveq12d 5893 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅)) = ((πΆπ·π΄) + (πΆπ·π΅))) |
| 13 | 5, 8, 12 | 3brtr3d 4035 | 1 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 = wceq 1353 β wcel 2148 class class class wbr 4004 Γ cxp 4625 βΎ cres 4629 βcfv 5217 (class class class)co 5875 + caddc 7814 β€ cle 7993 Basecbs 12462 distcds 12545 Metcmet 13444 MetSpcms 13840 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931 |
| This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-map 6650 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-5 8981 df-6 8982 df-7 8983 df-8 8984 df-9 8985 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-xneg 9772 df-xadd 9773 df-seqfrec 10446 df-exp 10520 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-ndx 12465 df-slot 12466 df-base 12468 df-tset 12555 df-rest 12690 df-topn 12691 df-topgen 12709 df-psmet 13450 df-xmet 13451 df-met 13452 df-bl 13453 df-mopn 13454 df-top 13501 df-topon 13514 df-topsp 13534 df-bases 13546 df-xms 13842 df-ms 13843 |
| This theorem is referenced by: (None) |
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