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Mirrors > Home > MPE Home > Th. List > xmetge0 | Structured version Visualization version GIF version |
Description: The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetge0 | β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β π· β (βMetβπ)) | |
2 | simp2 1134 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β π΄ β π) | |
3 | simp3 1135 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β π΅ β π) | |
4 | xmettri2 24266 | . . . 4 β’ ((π· β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ π΅ β π)) β (π΅π·π΅) β€ ((π΄π·π΅) +π (π΄π·π΅))) | |
5 | 1, 2, 3, 3, 4 | syl13anc 1369 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΅π·π΅) β€ ((π΄π·π΅) +π (π΄π·π΅))) |
6 | 2re 12324 | . . . . 5 β’ 2 β β | |
7 | rexr 11298 | . . . . 5 β’ (2 β β β 2 β β*) | |
8 | xmul01 13286 | . . . . 5 β’ (2 β β* β (2 Β·e 0) = 0) | |
9 | 6, 7, 8 | mp2b 10 | . . . 4 β’ (2 Β·e 0) = 0 |
10 | xmet0 24268 | . . . . 5 β’ ((π· β (βMetβπ) β§ π΅ β π) β (π΅π·π΅) = 0) | |
11 | 10 | 3adant2 1128 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΅π·π΅) = 0) |
12 | 9, 11 | eqtr4id 2787 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) = (π΅π·π΅)) |
13 | xmetcl 24257 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
14 | x2times 13318 | . . . 4 β’ ((π΄π·π΅) β β* β (2 Β·e (π΄π·π΅)) = ((π΄π·π΅) +π (π΄π·π΅))) | |
15 | 13, 14 | syl 17 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e (π΄π·π΅)) = ((π΄π·π΅) +π (π΄π·π΅))) |
16 | 5, 12, 15 | 3brtr4d 5184 | . 2 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) β€ (2 Β·e (π΄π·π΅))) |
17 | 0xr 11299 | . . 3 β’ 0 β β* | |
18 | 2rp 13019 | . . . 4 β’ 2 β β+ | |
19 | 18 | a1i 11 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β 2 β β+) |
20 | xlemul2 13310 | . . 3 β’ ((0 β β* β§ (π΄π·π΅) β β* β§ 2 β β+) β (0 β€ (π΄π·π΅) β (2 Β·e 0) β€ (2 Β·e (π΄π·π΅)))) | |
21 | 17, 13, 19, 20 | mp3an2i 1462 | . 2 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (0 β€ (π΄π·π΅) β (2 Β·e 0) β€ (2 Β·e (π΄π·π΅)))) |
22 | 16, 21 | mpbird 256 | 1 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βcr 11145 0cc0 11146 β*cxr 11285 β€ cle 11287 2c2 12305 β+crp 13014 +π cxad 13130 Β·e cxmu 13131 βMetcxmet 21271 |
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 ax-pre-mulgt0 11223 |
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-rmo 3374 df-reu 3375 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-iun 5002 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-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 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-sub 11484 df-neg 11485 df-div 11910 df-2 12313 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-xmet 21279 |
This theorem is referenced by: metge0 24271 xmetlecl 24272 xmetrtri 24281 xmetgt0 24284 prdsxmetlem 24294 imasdsf1olem 24299 xpsdsval 24307 xblpnf 24322 blgt0 24325 xblss2 24328 xbln0 24340 xmsge0 24389 comet 24442 stdbdxmet 24444 stdbdmet 24445 xrsmopn 24748 metdsf 24784 metdstri 24787 metdscnlem 24791 iscfil2 25214 heicant 37161 |
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