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Mirrors > Home > ILE Home > Th. List > xmetres2 | Unicode version |
Description: Restriction of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetres2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetrel 12501 | . . . . 5 | |
2 | relelfvdm 5446 | . . . . 5 | |
3 | 1, 2 | mpan 420 | . . . 4 |
4 | 3 | adantr 274 | . . 3 |
5 | simpr 109 | . . 3 | |
6 | 4, 5 | ssexd 4063 | . 2 |
7 | xmetf 12508 | . . . 4 | |
8 | 7 | adantr 274 | . . 3 |
9 | xpss12 4641 | . . . 4 | |
10 | 5, 9 | sylancom 416 | . . 3 |
11 | 8, 10 | fssresd 5294 | . 2 |
12 | ovres 5903 | . . . . 5 | |
13 | 12 | adantl 275 | . . . 4 |
14 | 13 | eqeq1d 2146 | . . 3 |
15 | simpll 518 | . . . 4 | |
16 | simplr 519 | . . . . 5 | |
17 | simprl 520 | . . . . 5 | |
18 | 16, 17 | sseldd 3093 | . . . 4 |
19 | simprr 521 | . . . . 5 | |
20 | 16, 19 | sseldd 3093 | . . . 4 |
21 | xmeteq0 12517 | . . . 4 | |
22 | 15, 18, 20, 21 | syl3anc 1216 | . . 3 |
23 | 14, 22 | bitrd 187 | . 2 |
24 | simpll 518 | . . . 4 | |
25 | simplr 519 | . . . . 5 | |
26 | simpr3 989 | . . . . 5 | |
27 | 25, 26 | sseldd 3093 | . . . 4 |
28 | 18 | 3adantr3 1142 | . . . 4 |
29 | 20 | 3adantr3 1142 | . . . 4 |
30 | xmettri2 12519 | . . . 4 | |
31 | 24, 27, 28, 29, 30 | syl13anc 1218 | . . 3 |
32 | 13 | 3adantr3 1142 | . . 3 |
33 | simpr1 987 | . . . . 5 | |
34 | 26, 33 | ovresd 5904 | . . . 4 |
35 | simpr2 988 | . . . . 5 | |
36 | 26, 35 | ovresd 5904 | . . . 4 |
37 | 34, 36 | oveq12d 5785 | . . 3 |
38 | 31, 32, 37 | 3brtr4d 3955 | . 2 |
39 | 6, 11, 23, 38 | isxmetd 12505 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wss 3066 class class class wbr 3924 cxp 4532 cdm 4534 cres 4536 wrel 4539 wf 5114 cfv 5118 (class class class)co 5767 cc0 7613 cxr 7792 cle 7794 cxad 9550 cxmet 12138 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-map 6537 df-pnf 7795 df-mnf 7796 df-xr 7797 df-xmet 12146 |
This theorem is referenced by: metres2 12539 xmetres 12540 xmetresbl 12598 metrest 12664 divcnap 12713 cncfmet 12737 limcimolemlt 12791 cnplimcim 12794 cnplimclemr 12796 limccnpcntop 12802 limccnp2cntop 12804 |
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