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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 24196 | . . . 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 12287 | . . . . 5 β’ 2 β β | |
7 | rexr 11261 | . . . . 5 β’ (2 β β β 2 β β*) | |
8 | xmul01 13249 | . . . . 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 24198 | . . . . 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 2785 | . . 3 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) = (π΅π·π΅)) |
13 | xmetcl 24187 | . . . 4 β’ ((π· β (βMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
14 | x2times 13281 | . . . 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 5173 | . 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 11262 | . . 3 β’ 0 β β* | |
18 | 2rp 12982 | . . . 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 13273 | . . 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 257 | 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 5141 βcfv 6536 (class class class)co 7404 βcr 11108 0cc0 11109 β*cxr 11248 β€ cle 11250 2c2 12268 β+crp 12977 +π cxad 13093 Β·e cxmu 13094 βMetcxmet 21220 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-2 12276 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-xmet 21228 |
This theorem is referenced by: metge0 24201 xmetlecl 24202 xmetrtri 24211 xmetgt0 24214 prdsxmetlem 24224 imasdsf1olem 24229 xpsdsval 24237 xblpnf 24252 blgt0 24255 xblss2 24258 xbln0 24270 xmsge0 24319 comet 24372 stdbdxmet 24374 stdbdmet 24375 xrsmopn 24678 metdsf 24714 metdstri 24717 metdscnlem 24721 iscfil2 25144 heicant 37035 |
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