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| Mirrors > Home > MPE Home > Th. List > xmstri3 | Structured version Visualization version 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 |
|---|---|
| xmstri3 | β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (π΅π·πΆ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | xmsxmet2 24315 | . . 3 β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
| 4 | xmettri3 24209 | . . 3 β’ (((π· βΎ (π Γ π)) β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((π΄(π· βΎ (π Γ π))πΆ) +π (π΅(π· βΎ (π Γ π))πΆ))) | |
| 5 | 3, 4 | sylan 579 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((π΄(π· βΎ (π Γ π))πΆ) +π (π΅(π· βΎ (π Γ π))πΆ))) |
| 6 | simpr1 1191 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr2 1192 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 7570 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
| 9 | simpr3 1193 | . . . 4 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 10 | 6, 9 | ovresd 7570 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))πΆ) = (π΄π·πΆ)) |
| 11 | 7, 9 | ovresd 7570 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅(π· βΎ (π Γ π))πΆ) = (π΅π·πΆ)) |
| 12 | 10, 11 | oveq12d 7422 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄(π· βΎ (π Γ π))πΆ) +π (π΅(π· βΎ (π Γ π))πΆ)) = ((π΄π·πΆ) +π (π΅π·πΆ))) |
| 13 | 5, 8, 12 | 3brtr3d 5172 | 1 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (π΅π·πΆ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 Γ cxp 5667 βΎ cres 5671 βcfv 6536 (class class class)co 7404 β€ cle 11250 +π cxad 13093 Basecbs 17150 distcds 17212 βMetcxmet 21220 βMetSpcxms 24173 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-bl 21230 df-mopn 21231 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-xms 24176 |
| This theorem is referenced by: (None) |
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