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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 24260 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
| 2 | xmettri2 24266 | . . 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 578 | . 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 24258 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) β β) | |
| 5 | 4 | 3adant3r3 1181 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΄) β β) |
| 6 | metcl 24258 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΅ β π) β (πΆπ·π΅) β β) | |
| 7 | 6 | 3adant3r2 1180 | . . 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 13253 | . 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 5178 | 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 394 β§ w3a 1084 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βcr 11145 + caddc 11149 β€ cle 11287 +π cxad 13130 βMetcxmet 21271 Metcmet 21272 |
| 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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-mulcl 11208 ax-i2m1 11214 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-xadd 13133 df-xmet 21279 df-met 21280 |
| This theorem is referenced by: mettri 24278 mstri2 24393 metf1o 37261 isbnd3 37290 heibor1lem 37315 bfplem2 37329 |
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