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Mirrors > Home > ILE Home > Th. List > xmetge0 | Unicode version |
Description: The distance function of a metric space is nonnegative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmet0 13122 | . . . . 5 | |
2 | 1 | 3adant2 1011 | . . . 4 |
3 | 0xr 7959 | . . . . 5 | |
4 | xaddid1 9812 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | 2, 5 | eqtr4di 2221 | . . 3 |
7 | simp1 992 | . . . 4 | |
8 | simp2 993 | . . . 4 | |
9 | simp3 994 | . . . 4 | |
10 | xmettri2 13120 | . . . 4 | |
11 | 7, 8, 9, 9, 10 | syl13anc 1235 | . . 3 |
12 | 6, 11 | eqbrtrrd 4011 | . 2 |
13 | xmetcl 13111 | . . 3 | |
14 | xleaddadd 9837 | . . 3 | |
15 | 3, 13, 14 | sylancr 412 | . 2 |
16 | 12, 15 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cfv 5196 (class class class)co 5851 cc0 7767 cxr 7946 cle 7948 cxad 9720 cxmet 12739 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-lttrn 7881 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-map 6626 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-2 8930 df-xadd 9723 df-xmet 12747 |
This theorem is referenced by: metge0 13125 xmetlecl 13126 xmetrtri 13135 xblpnf 13158 blgt0 13161 xblss2 13164 xblm 13176 xmsge0 13226 comet 13258 bdxmet 13260 bdmet 13261 xmetxp 13266 |
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