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Mirrors > Home > MPE Home > Th. List > psmetge0 | Structured version Visualization version GIF version |
Description: The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
psmetge0 | β’ ((π· β (PsMetβπ) 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 (PsMetβπ) β§ π΄ β π β§ π΅ β π) β π· β (PsMetβπ)) | |
2 | simp2 1134 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β π΄ β π) | |
3 | simp3 1135 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β π΅ β π) | |
4 | psmettri2 24231 | . . . 4 β’ ((π· β (PsMetβπ) 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 (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΅π·π΅) β€ ((π΄π·π΅) +π (π΄π·π΅))) |
6 | 2re 12314 | . . . . 5 β’ 2 β β | |
7 | rexr 11288 | . . . . 5 β’ (2 β β β 2 β β*) | |
8 | xmul01 13276 | . . . . 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 | psmet0 24230 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π΅ β π) β (π΅π·π΅) = 0) | |
11 | 10 | 3adant2 1128 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΅π·π΅) = 0) |
12 | 9, 11 | eqtr4id 2784 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) = (π΅π·π΅)) |
13 | psmetcl 24229 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
14 | x2times 13308 | . . . 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 (PsMetβπ) 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 (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) β€ (2 Β·e (π΄π·π΅))) |
17 | 0xr 11289 | . . 3 β’ 0 β β* | |
18 | 2rp 13009 | . . . 4 β’ 2 β β+ | |
19 | 18 | a1i 11 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 2 β β+) |
20 | xlemul2 13300 | . . 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 (PsMetβπ) 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 (PsMetβπ) 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 6541 (class class class)co 7414 βcr 11135 0cc0 11136 β*cxr 11275 β€ cle 11277 2c2 12295 β+crp 13004 +π cxad 13120 Β·e cxmu 13121 PsMetcpsmet 21265 |
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 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-2 12303 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-psmet 21273 |
This theorem is referenced by: psmetxrge0 24235 psmetlecl 24237 distspace 24238 xblpnfps 24317 xblss2ps 24323 metustexhalf 24481 blval2 24487 metuel2 24490 metider 33524 |
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