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Mirrors > Home > ILE Home > Th. List > psmetge0 | Unicode version |
Description: The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.) |
Ref | Expression |
---|---|
psmetge0 | PsMet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7805 | . . . 4 | |
2 | xaddid1 9638 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | psmet0 12485 | . . . . 5 PsMet | |
5 | 4 | 3adant2 1000 | . . . 4 PsMet |
6 | simp1 981 | . . . . 5 PsMet PsMet | |
7 | simp2 982 | . . . . 5 PsMet | |
8 | simp3 983 | . . . . 5 PsMet | |
9 | psmettri2 12486 | . . . . 5 PsMet | |
10 | 6, 7, 8, 8, 9 | syl13anc 1218 | . . . 4 PsMet |
11 | 5, 10 | eqbrtrrd 3947 | . . 3 PsMet |
12 | 3, 11 | eqbrtrid 3958 | . 2 PsMet |
13 | psmetcl 12484 | . . 3 PsMet | |
14 | xleaddadd 9663 | . . 3 | |
15 | 1, 13, 14 | sylancr 410 | . 2 PsMet |
16 | 12, 15 | mpbird 166 | 1 PsMet |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 962 wceq 1331 wcel 1480 class class class wbr 3924 cfv 5118 (class class class)co 5767 cc0 7613 cxr 7792 cle 7794 cxad 9550 PsMetcpsmet 12137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-2 8772 df-xadd 9553 df-psmet 12145 |
This theorem is referenced by: psmetxrge0 12490 psmetlecl 12492 distspace 12493 xblpnfps 12556 xblss2ps 12562 |
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