![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24170 | . . . 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 12290 | . . . . 5 β’ 2 β β | |
7 | rexr 11264 | . . . . 5 β’ (2 β β β 2 β β*) | |
8 | xmul01 13252 | . . . . 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 24169 | . . . . 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 2785 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (2 Β·e 0) = (π΅π·π΅)) |
13 | psmetcl 24168 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) β β*) | |
14 | x2times 13284 | . . . 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 11265 | . . 3 β’ 0 β β* | |
18 | 2rp 12985 | . . . 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 13276 | . . 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 257 | 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 6537 (class class class)co 7405 βcr 11111 0cc0 11112 β*cxr 11251 β€ cle 11253 2c2 12271 β+crp 12980 +π cxad 13096 Β·e cxmu 13097 PsMetcpsmet 21224 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-psmet 21232 |
This theorem is referenced by: psmetxrge0 24174 psmetlecl 24176 distspace 24177 xblpnfps 24256 xblss2ps 24262 metustexhalf 24420 blval2 24426 metuel2 24429 metider 33404 |
Copyright terms: Public domain | W3C validator |