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Mirrors > Home > ILE Home > Th. List > ismet2 | Unicode version |
Description: An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ismet2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metrel 12889 | . . 3 | |
2 | relelfvdm 5512 | . . . 4 | |
3 | 2 | elexd 2734 | . . 3 |
4 | 1, 3 | mpan 421 | . 2 |
5 | xmetrel 12890 | . . . . 5 | |
6 | relelfvdm 5512 | . . . . 5 | |
7 | 5, 6 | mpan 421 | . . . 4 |
8 | 7 | elexd 2734 | . . 3 |
9 | 8 | adantr 274 | . 2 |
10 | simpllr 524 | . . . . . . . . . . . 12 | |
11 | simpr 109 | . . . . . . . . . . . 12 | |
12 | simplrl 525 | . . . . . . . . . . . 12 | |
13 | 10, 11, 12 | fovrnd 5977 | . . . . . . . . . . 11 |
14 | simplrr 526 | . . . . . . . . . . . 12 | |
15 | 10, 11, 14 | fovrnd 5977 | . . . . . . . . . . 11 |
16 | 13, 15 | rexaddd 9781 | . . . . . . . . . 10 |
17 | 16 | breq2d 3988 | . . . . . . . . 9 |
18 | 17 | ralbidva 2460 | . . . . . . . 8 |
19 | 18 | anbi2d 460 | . . . . . . 7 |
20 | 19 | 2ralbidva 2486 | . . . . . 6 |
21 | simpr 109 | . . . . . . . 8 | |
22 | ressxr 7933 | . . . . . . . 8 | |
23 | fss 5343 | . . . . . . . 8 | |
24 | 21, 22, 23 | sylancl 410 | . . . . . . 7 |
25 | 24 | biantrurd 303 | . . . . . 6 |
26 | 20, 25 | bitr3d 189 | . . . . 5 |
27 | 26 | pm5.32da 448 | . . . 4 |
28 | ancom 264 | . . . 4 | |
29 | 27, 28 | bitrdi 195 | . . 3 |
30 | ismet 12891 | . . 3 | |
31 | isxmet 12892 | . . . 4 | |
32 | 31 | anbi1d 461 | . . 3 |
33 | 29, 30, 32 | 3bitr4d 219 | . 2 |
34 | 4, 9, 33 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 cvv 2721 wss 3111 class class class wbr 3976 cxp 4596 cdm 4598 wrel 4603 wf 5178 cfv 5182 (class class class)co 5836 cr 7743 cc0 7744 caddc 7747 cxr 7923 cle 7925 cxad 9697 cxmet 12527 cmet 12528 |
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 ax-1re 7838 ax-addrcl 7841 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 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-nel 2430 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-if 3516 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-xadd 9700 df-xmet 12535 df-met 12536 |
This theorem is referenced by: metxmet 12902 metres2 12928 xmetresbl 12987 bdmet 13049 |
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