![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 14252 | . . 3 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
2 | xmettri2 14258 | . . 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 14250 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) β β) | |
5 | 4 | 3adant3r3 1216 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΄) β β) |
6 | metcl 14250 | . . . 4 β’ ((π· β (Metβπ) β§ πΆ β π β§ π΅ β π) β (πΆπ·π΅) β β) | |
7 | 6 | 3adant3r2 1215 | . . 3 β’ ((π· β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆπ·π΅) β β) |
8 | rexadd 9870 | . . 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 4044 | 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 980 = wceq 1364 β wcel 2160 class class class wbr 4018 βcfv 5231 (class class class)co 5891 βcr 7828 + caddc 7832 β€ cle 8011 +π cxad 9788 βMetcxmet 13810 Metcmet 13811 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 ax-rnegex 7938 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8012 df-mnf 8013 df-xr 8014 df-xadd 9791 df-xmet 13818 df-met 13819 |
This theorem is referenced by: mettri 14270 mstri2 14368 |
Copyright terms: Public domain | W3C validator |