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Mirrors > Home > ILE Home > Th. List > isxmetd | Unicode version |
Description: Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
isxmetd.0 | |
isxmetd.1 | |
isxmetd.2 | |
isxmetd.3 |
Ref | Expression |
---|---|
isxmetd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isxmetd.1 | . 2 | |
2 | isxmetd.2 | . . . 4 | |
3 | isxmetd.3 | . . . . . . 7 | |
4 | 3 | 3exp2 1207 | . . . . . 6 |
5 | 4 | imp32 255 | . . . . 5 |
6 | 5 | ralrimiv 2529 | . . . 4 |
7 | 2, 6 | jca 304 | . . 3 |
8 | 7 | ralrimivva 2539 | . 2 |
9 | isxmetd.0 | . . 3 | |
10 | isxmet 12841 | . . 3 | |
11 | 9, 10 | syl 14 | . 2 |
12 | 1, 8, 11 | mpbir2and 929 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wral 2435 cvv 2712 class class class wbr 3967 cxp 4587 wf 5169 cfv 5173 (class class class)co 5827 cc0 7735 cxr 7914 cle 7916 cxad 9684 cxmet 12476 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-map 6598 df-pnf 7917 df-mnf 7918 df-xr 7919 df-xmet 12484 |
This theorem is referenced by: isxmet2d 12844 xmetres2 12875 comet 12995 xmetxp 13003 |
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