Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 12884 | . . . . . . . 8 | |
2 | relelfvdm 5512 | . . . . . . . 8 | |
3 | 1, 2 | mpan 421 | . . . . . . 7 |
4 | isxmet 12886 | . . . . . . 7 | |
5 | 3, 4 | syl 14 | . . . . . 6 |
6 | 5 | ibi 175 | . . . . 5 |
7 | simpr 109 | . . . . . 6 | |
8 | 7 | 2ralimi 2528 | . . . . 5 |
9 | 6, 8 | simpl2im 384 | . . . 4 |
10 | oveq1 5843 | . . . . . 6 | |
11 | oveq2 5844 | . . . . . . 7 | |
12 | 11 | oveq1d 5851 | . . . . . 6 |
13 | 10, 12 | breq12d 3989 | . . . . 5 |
14 | oveq2 5844 | . . . . . 6 | |
15 | oveq2 5844 | . . . . . . 7 | |
16 | 15 | oveq2d 5852 | . . . . . 6 |
17 | 14, 16 | breq12d 3989 | . . . . 5 |
18 | oveq1 5843 | . . . . . . 7 | |
19 | oveq1 5843 | . . . . . . 7 | |
20 | 18, 19 | oveq12d 5854 | . . . . . 6 |
21 | 20 | breq2d 3988 | . . . . 5 |
22 | 13, 17, 21 | rspc3v 2841 | . . . 4 |
23 | 9, 22 | syl5 32 | . . 3 |
24 | 23 | 3comr 1200 | . 2 |
25 | 24 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 cxp 4596 cdm 4598 wrel 4603 wf 5178 cfv 5182 (class class class)co 5836 cc0 7744 cxr 7923 cle 7925 cxad 9697 cxmet 12521 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-xmet 12529 |
This theorem is referenced by: mettri2 12903 xmetge0 12906 xmetsym 12909 xmetpsmet 12910 xmettri 12913 xmetres2 12920 xblss2 12946 xmstri2 13011 comet 13040 xmetxp 13048 |
Copyright terms: Public domain | W3C validator |