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| Mirrors > Home > MPE Home > Th. List > mettri2 | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| mettri2 | β’ ((π· β (Metβπ) 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 | metxmet 23839 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 2 | xmettri2 23845 | . . 3 β’ ((π· β (βMetβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) | |
| 3 | 1, 2 | sylan 580 | . 2 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) |
| 4 | metcl 23837 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) β β) | |
| 5 | 4 | 3adant3r3 1184 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΄) β β) |
| 6 | metcl 23837 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΅ β π) β (πΆπ·π΅) β β) | |
| 7 | 6 | 3adant3r2 1183 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΅) β β) |
| 8 | 5, 7 | rexaddd 13212 | . 2 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β ((πΆπ·π΄) +π (πΆπ·π΅)) = ((πΆπ·π΄) + (πΆπ·π΅))) |
| 9 | 3, 8 | breqtrd 5174 | 1 β’ ((π· β (Metβπ) 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 setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 βcr 11108 + caddc 11112 β€ cle 11248 +π cxad 13089 βMetcxmet 20928 Metcmet 20929 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-mulcl 11171 ax-i2m1 11177 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-xadd 13092 df-xmet 20936 df-met 20937 |
| This theorem is referenced by: mettri 23857 mstri2 23972 metf1o 36618 isbnd3 36647 heibor1lem 36672 bfplem2 36686 |
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