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Mirrors > Home > ILE Home > Th. List > xmetresbl | Unicode version |
Description: An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 12984, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance from each other. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xmetresbl.1 |
Ref | Expression |
---|---|
xmetresbl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 986 | . . 3 | |
2 | xmetresbl.1 | . . . 4 | |
3 | blssm 12968 | . . . 4 | |
4 | 2, 3 | eqsstrid 3183 | . . 3 |
5 | xmetres2 12926 | . . 3 | |
6 | 1, 4, 5 | syl2anc 409 | . 2 |
7 | xmetf 12897 | . . . . . 6 | |
8 | 1, 7 | syl 14 | . . . . 5 |
9 | xpss12 4705 | . . . . . 6 | |
10 | 4, 4, 9 | syl2anc 409 | . . . . 5 |
11 | 8, 10 | fssresd 5358 | . . . 4 |
12 | 11 | ffnd 5332 | . . 3 |
13 | ovres 5972 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | simpl1 989 | . . . . . . . . 9 | |
16 | eqid 2164 | . . . . . . . . . 10 | |
17 | 16 | xmeter 12983 | . . . . . . . . 9 |
18 | 15, 17 | syl 14 | . . . . . . . 8 |
19 | 16 | blssec 12985 | . . . . . . . . . . . 12 |
20 | 2, 19 | eqsstrid 3183 | . . . . . . . . . . 11 |
21 | 20 | sselda 3137 | . . . . . . . . . 10 |
22 | 21 | adantrr 471 | . . . . . . . . 9 |
23 | simpl2 990 | . . . . . . . . . 10 | |
24 | elecg 6530 | . . . . . . . . . 10 | |
25 | 22, 23, 24 | syl2anc 409 | . . . . . . . . 9 |
26 | 22, 25 | mpbid 146 | . . . . . . . 8 |
27 | 20 | sselda 3137 | . . . . . . . . . 10 |
28 | 27 | adantrl 470 | . . . . . . . . 9 |
29 | elecg 6530 | . . . . . . . . . 10 | |
30 | 28, 23, 29 | syl2anc 409 | . . . . . . . . 9 |
31 | 28, 30 | mpbid 146 | . . . . . . . 8 |
32 | 18, 26, 31 | ertr3d 6510 | . . . . . . 7 |
33 | 16 | xmeterval 12982 | . . . . . . . 8 |
34 | 15, 33 | syl 14 | . . . . . . 7 |
35 | 32, 34 | mpbid 146 | . . . . . 6 |
36 | 35 | simp3d 1000 | . . . . 5 |
37 | 14, 36 | eqeltrd 2241 | . . . 4 |
38 | 37 | ralrimivva 2546 | . . 3 |
39 | ffnov 5937 | . . 3 | |
40 | 12, 38, 39 | sylanbrc 414 | . 2 |
41 | ismet2 12901 | . 2 | |
42 | 6, 40, 41 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 wss 3111 class class class wbr 3976 cxp 4596 ccnv 4597 cres 4600 cima 4601 wfn 5177 wf 5178 cfv 5182 (class class class)co 5836 wer 6489 cec 6490 cr 7743 cxr 7923 cxmet 12527 cmet 12528 cbl 12529 |
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-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-stab 821 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-reu 2449 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-po 4268 df-iso 4269 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-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-er 6492 df-ec 6494 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-2 8907 df-xneg 9699 df-xadd 9700 df-psmet 12534 df-xmet 12535 df-met 12536 df-bl 12537 |
This theorem is referenced by: (None) |
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