Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > xmstri2 | GIF version | ||
| Description: Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| xmstri2 | β’ ((π β βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | xmsxmet2 14048 | . . 3 β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
| 4 | xmettri2 13946 | . . 3 β’ (((π· βΎ (π Γ π)) β (βMetβπ) 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΄) +π (πΆ(π· βΎ (π Γ π))π΅))) | |
| 5 | 3, 4 | sylan 283 | . 2 β’ ((π β βMetSp 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΄) +π (πΆ(π· βΎ (π Γ π))π΅))) |
| 6 | simpr2 1004 | . . 3 β’ ((π β βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΄ β π) | |
| 7 | simpr3 1005 | . . 3 β’ ((π β βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 6017 | . 2 β’ ((π β βMetSp 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 βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β πΆ β π) | |
| 10 | 9, 6 | ovresd 6017 | . . 3 β’ ((π β βMetSp 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 6017 | . . 3 β’ ((π β βMetSp 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 5895 | . 2 β’ ((π β βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β ((πΆ(π· βΎ (π Γ π))π΄) +π (πΆ(π· βΎ (π Γ π))π΅)) = ((πΆπ·π΄) +π (πΆπ·π΅))) |
| 13 | 5, 8, 12 | 3brtr3d 4036 | 1 β’ ((π β βMetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +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 4005 Γ cxp 4626 βΎ cres 4630 βcfv 5218 (class class class)co 5877 β€ cle 7995 +π cxad 9772 Basecbs 12464 distcds 12547 βMetcxmet 13525 βMetSpcxms 13921 |
| 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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 |
| 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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-map 6652 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-ndx 12467 df-slot 12468 df-base 12470 df-tset 12557 df-rest 12695 df-topn 12696 df-topgen 12714 df-psmet 13532 df-xmet 13533 df-bl 13535 df-mopn 13536 df-top 13583 df-topon 13596 df-topsp 13616 df-bases 13628 df-xms 13924 |
| This theorem is referenced by: (None) |
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