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Mirrors > Home > ILE Home > Th. List > mettri2 | 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 14082 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
2 | xmettri2 14088 | . . 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 283 | . 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 14080 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) β β) | |
5 | 4 | 3adant3r3 1215 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΄) β β) |
6 | metcl 14080 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΅ β π) β (πΆπ·π΅) β β) | |
7 | 6 | 3adant3r2 1214 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΅) β β) |
8 | rexadd 9865 | . . 3 β’ (((πΆπ·π΄) β β β§ (πΆπ·π΅) β β) β ((πΆπ·π΄) +π (πΆπ·π΅)) = ((πΆπ·π΄) + (πΆπ·π΅))) | |
9 | 5, 7, 8 | syl2anc 411 | . 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΅))) |
10 | 3, 9 | breqtrd 4041 | 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 set class |
Syntax hints: β wi 4 β§ wa 104 β§ w3a 979 = wceq 1363 β wcel 2158 class class class wbr 4015 βcfv 5228 (class class class)co 5888 βcr 7823 + caddc 7827 β€ cle 8006 +π cxad 9783 βMetcxmet 13653 Metcmet 13654 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 ax-rnegex 7933 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-map 6663 df-pnf 8007 df-mnf 8008 df-xr 8009 df-xadd 9786 df-xmet 13661 df-met 13662 |
This theorem is referenced by: mettri 14100 mstri2 14198 |
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