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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 13411 | . . 3 | |
2 | relelfvdm 5539 | . . . 4 | |
3 | 2 | elexd 2748 | . . 3 |
4 | 1, 3 | mpan 424 | . 2 |
5 | xmetrel 13412 | . . . . 5 | |
6 | relelfvdm 5539 | . . . . 5 | |
7 | 5, 6 | mpan 424 | . . . 4 |
8 | 7 | elexd 2748 | . . 3 |
9 | 8 | adantr 276 | . 2 |
10 | simpllr 534 | . . . . . . . . . . . 12 | |
11 | simpr 110 | . . . . . . . . . . . 12 | |
12 | simplrl 535 | . . . . . . . . . . . 12 | |
13 | 10, 11, 12 | fovcdmd 6009 | . . . . . . . . . . 11 |
14 | simplrr 536 | . . . . . . . . . . . 12 | |
15 | 10, 11, 14 | fovcdmd 6009 | . . . . . . . . . . 11 |
16 | 13, 15 | rexaddd 9823 | . . . . . . . . . 10 |
17 | 16 | breq2d 4010 | . . . . . . . . 9 |
18 | 17 | ralbidva 2471 | . . . . . . . 8 |
19 | 18 | anbi2d 464 | . . . . . . 7 |
20 | 19 | 2ralbidva 2497 | . . . . . 6 |
21 | simpr 110 | . . . . . . . 8 | |
22 | ressxr 7975 | . . . . . . . 8 | |
23 | fss 5369 | . . . . . . . 8 | |
24 | 21, 22, 23 | sylancl 413 | . . . . . . 7 |
25 | 24 | biantrurd 305 | . . . . . 6 |
26 | 20, 25 | bitr3d 190 | . . . . 5 |
27 | 26 | pm5.32da 452 | . . . 4 |
28 | ancom 266 | . . . 4 | |
29 | 27, 28 | bitrdi 196 | . . 3 |
30 | ismet 13413 | . . 3 | |
31 | isxmet 13414 | . . . 4 | |
32 | 31 | anbi1d 465 | . . 3 |
33 | 29, 30, 32 | 3bitr4d 220 | . 2 |
34 | 4, 9, 33 | pm5.21nii 704 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 cvv 2735 wss 3127 class class class wbr 3998 cxp 4618 cdm 4620 wrel 4625 wf 5204 cfv 5208 (class class class)co 5865 cr 7785 cc0 7786 caddc 7789 cxr 7965 cle 7967 cxad 9739 cxmet 13049 cmet 13050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 ax-rnegex 7895 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-map 6640 df-pnf 7968 df-mnf 7969 df-xr 7970 df-xadd 9742 df-xmet 13057 df-met 13058 |
This theorem is referenced by: metxmet 13424 metres2 13450 xmetresbl 13509 bdmet 13571 |
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