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| Mirrors > Home > MPE Home > Th. List > xmettri | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| xmettri | β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π· β (βMetβπ)) | |
| 2 | simpr3 1195 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 3 | simpr1 1193 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 4 | simpr2 1194 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 5 | xmettri2 24167 | . . 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΅))) | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1371 | . 2 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) |
| 7 | xmetsym 24174 | . . . 4 β’ ((π· β (βMetβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) = (π΄π·πΆ)) | |
| 8 | 1, 2, 3, 7 | syl3anc 1370 | . . 3 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πΆπ·π΄) = (π΄π·πΆ)) |
| 9 | 8 | oveq1d 7427 | . 2 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((πΆπ·π΄) +π (πΆπ·π΅)) = ((π΄π·πΆ) +π (πΆπ·π΅))) |
| 10 | 6, 9 | breqtrd 5174 | 1 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (πΆπ·π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 β€ cle 11256 +π cxad 13097 βMetcxmet 21219 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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-po 5588 df-so 5589 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 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-xadd 13100 df-xmet 21227 |
| This theorem is referenced by: xmettri3 24180 xmetrtri 24182 imasdsf1olem 24200 xmeter 24260 xmstri 24295 metdcnlem 24673 iscau3 25127 heicant 36990 |
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