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Mirrors > Home > ILE Home > Th. List > xmettri2 | Unicode version |
Description: Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmettri2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetrel 13058 | . . . . . . . 8 | |
2 | relelfvdm 5526 | . . . . . . . 8 | |
3 | 1, 2 | mpan 422 | . . . . . . 7 |
4 | isxmet 13060 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 5 | ibi 175 | . . . . 5 |
7 | simpr 109 | . . . . . 6 | |
8 | 7 | 2ralimi 2534 | . . . . 5 |
9 | 6, 8 | simpl2im 384 | . . . 4 |
10 | oveq1 5857 | . . . . . 6 | |
11 | oveq2 5858 | . . . . . . 7 | |
12 | 11 | oveq1d 5865 | . . . . . 6 |
13 | 10, 12 | breq12d 4000 | . . . . 5 |
14 | oveq2 5858 | . . . . . 6 | |
15 | oveq2 5858 | . . . . . . 7 | |
16 | 15 | oveq2d 5866 | . . . . . 6 |
17 | 14, 16 | breq12d 4000 | . . . . 5 |
18 | oveq1 5857 | . . . . . . 7 | |
19 | oveq1 5857 | . . . . . . 7 | |
20 | 18, 19 | oveq12d 5868 | . . . . . 6 |
21 | 20 | breq2d 3999 | . . . . 5 |
22 | 13, 17, 21 | rspc3v 2850 | . . . 4 |
23 | 9, 22 | syl5 32 | . . 3 |
24 | 23 | 3comr 1206 | . 2 |
25 | 24 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 class class class wbr 3987 cxp 4607 cdm 4609 wrel 4614 wf 5192 cfv 5196 (class class class)co 5850 cc0 7761 cxr 7940 cle 7942 cxad 9714 cxmet 12695 |
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 7852 ax-resscn 7853 |
This theorem depends on definitions: df-bi 116 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-ral 2453 df-rex 2454 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-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-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-map 6624 df-pnf 7943 df-mnf 7944 df-xr 7945 df-xmet 12703 |
This theorem is referenced by: mettri2 13077 xmetge0 13080 xmetsym 13083 xmetpsmet 13084 xmettri 13087 xmetres2 13094 xblss2 13120 xmstri2 13185 comet 13214 xmetxp 13222 |
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