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Mirrors > Home > MPE Home > Th. List > psmetge0 | Structured version Visualization version Unicode version |
Description: The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
psmetge0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 |
. . . 4
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2 | simp2 1062 |
. . . 4
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3 | simp3 1063 |
. . . 4
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4 | psmettri2 22114 |
. . . 4
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5 | 1, 2, 3, 3, 4 | syl13anc 1328 |
. . 3
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6 | psmet0 22113 |
. . . . 5
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7 | 6 | 3adant2 1080 |
. . . 4
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8 | 2re 11090 |
. . . . 5
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9 | rexr 10085 |
. . . . 5
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10 | xmul01 12097 |
. . . . 5
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11 | 8, 9, 10 | mp2b 10 |
. . . 4
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12 | 7, 11 | syl6reqr 2675 |
. . 3
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13 | psmetcl 22112 |
. . . 4
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14 | x2times 12129 |
. . . 4
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15 | 13, 14 | syl 17 |
. . 3
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16 | 5, 12, 15 | 3brtr4d 4685 |
. 2
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17 | 0xr 10086 |
. . . 4
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18 | 17 | a1i 11 |
. . 3
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19 | 2rp 11837 |
. . . 4
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20 | 19 | a1i 11 |
. . 3
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21 | xlemul2 12121 |
. . 3
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22 | 18, 13, 20, 21 | syl3anc 1326 |
. 2
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23 | 16, 22 | mpbird 247 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-psmet 19738 |
This theorem is referenced by: psmetxrge0 22118 psmetlecl 22120 distspace 22121 xblpnfps 22200 xblss2ps 22206 metustexhalf 22361 blval2 22367 metuel2 22370 metider 29937 |
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