![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mettri3 | Structured version Visualization version GIF version |
Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 13-Mar-2007.) |
Ref | Expression |
---|---|
mettri3 | β’ ((π· β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) + (π΅π·πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mettri 24278 | . 2 β’ ((π· β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) + (πΆπ·π΅))) | |
2 | metsym 24276 | . . . 4 β’ ((π· β (Metβπ) β§ π΅ β π β§ πΆ β π) β (π΅π·πΆ) = (πΆπ·π΅)) | |
3 | 2 | 3adant3r1 1179 | . . 3 β’ ((π· β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΅π·πΆ) = (πΆπ·π΅)) |
4 | 3 | oveq2d 7442 | . 2 β’ ((π· β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄π·πΆ) + (π΅π·πΆ)) = ((π΄π·πΆ) + (πΆπ·π΅))) |
5 | 1, 4 | breqtrrd 5180 | 1 β’ ((π· β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) + (π΅π·πΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 + caddc 11149 β€ cle 11287 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-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-po 5594 df-so 5595 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-ltxr 11291 df-le 11292 df-xadd 13133 df-xmet 21279 df-met 21280 |
This theorem is referenced by: mstri3 24397 |
Copyright terms: Public domain | W3C validator |