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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 12884 | . . . . 5 | |
2 | relelfvdm 5512 | . . . . 5 | |
3 | 1, 2 | mpan 421 | . . . 4 |
4 | 3 | adantr 274 | . . 3 |
5 | simpr 109 | . . 3 | |
6 | 4, 5 | ssexd 4116 | . 2 |
7 | xmetf 12891 | . . . 4 | |
8 | 7 | adantr 274 | . . 3 |
9 | xpss12 4705 | . . . 4 | |
10 | 5, 9 | sylancom 417 | . . 3 |
11 | 8, 10 | fssresd 5358 | . 2 |
12 | ovres 5972 | . . . . 5 | |
13 | 12 | adantl 275 | . . . 4 |
14 | 13 | eqeq1d 2173 | . . 3 |
15 | simpll 519 | . . . 4 | |
16 | simplr 520 | . . . . 5 | |
17 | simprl 521 | . . . . 5 | |
18 | 16, 17 | sseldd 3138 | . . . 4 |
19 | simprr 522 | . . . . 5 | |
20 | 16, 19 | sseldd 3138 | . . . 4 |
21 | xmeteq0 12900 | . . . 4 | |
22 | 15, 18, 20, 21 | syl3anc 1227 | . . 3 |
23 | 14, 22 | bitrd 187 | . 2 |
24 | simpll 519 | . . . 4 | |
25 | simplr 520 | . . . . 5 | |
26 | simpr3 994 | . . . . 5 | |
27 | 25, 26 | sseldd 3138 | . . . 4 |
28 | 18 | 3adantr3 1147 | . . . 4 |
29 | 20 | 3adantr3 1147 | . . . 4 |
30 | xmettri2 12902 | . . . 4 | |
31 | 24, 27, 28, 29, 30 | syl13anc 1229 | . . 3 |
32 | 13 | 3adantr3 1147 | . . 3 |
33 | simpr1 992 | . . . . 5 | |
34 | 26, 33 | ovresd 5973 | . . . 4 |
35 | simpr2 993 | . . . . 5 | |
36 | 26, 35 | ovresd 5973 | . . . 4 |
37 | 34, 36 | oveq12d 5854 | . . 3 |
38 | 31, 32, 37 | 3brtr4d 4008 | . 2 |
39 | 6, 11, 23, 38 | isxmetd 12888 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wss 3111 class class class wbr 3976 cxp 4596 cdm 4598 cres 4600 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: metres2 12922 xmetres 12923 xmetresbl 12981 metrest 13047 divcnap 13096 cncfmet 13120 limcimolemlt 13174 cnplimcim 13177 cnplimclemr 13179 limccnpcntop 13185 limccnp2cntop 13187 |
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