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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 12609, 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 981 | . . 3 | |
2 | xmetresbl.1 | . . . 4 | |
3 | blssm 12593 | . . . 4 | |
4 | 2, 3 | eqsstrid 3143 | . . 3 |
5 | xmetres2 12551 | . . 3 | |
6 | 1, 4, 5 | syl2anc 408 | . 2 |
7 | xmetf 12522 | . . . . . 6 | |
8 | 1, 7 | syl 14 | . . . . 5 |
9 | xpss12 4646 | . . . . . 6 | |
10 | 4, 4, 9 | syl2anc 408 | . . . . 5 |
11 | 8, 10 | fssresd 5299 | . . . 4 |
12 | 11 | ffnd 5273 | . . 3 |
13 | ovres 5910 | . . . . . 6 | |
14 | 13 | adantl 275 | . . . . 5 |
15 | simpl1 984 | . . . . . . . . 9 | |
16 | eqid 2139 | . . . . . . . . . 10 | |
17 | 16 | xmeter 12608 | . . . . . . . . 9 |
18 | 15, 17 | syl 14 | . . . . . . . 8 |
19 | 16 | blssec 12610 | . . . . . . . . . . . 12 |
20 | 2, 19 | eqsstrid 3143 | . . . . . . . . . . 11 |
21 | 20 | sselda 3097 | . . . . . . . . . 10 |
22 | 21 | adantrr 470 | . . . . . . . . 9 |
23 | simpl2 985 | . . . . . . . . . 10 | |
24 | elecg 6467 | . . . . . . . . . 10 | |
25 | 22, 23, 24 | syl2anc 408 | . . . . . . . . 9 |
26 | 22, 25 | mpbid 146 | . . . . . . . 8 |
27 | 20 | sselda 3097 | . . . . . . . . . 10 |
28 | 27 | adantrl 469 | . . . . . . . . 9 |
29 | elecg 6467 | . . . . . . . . . 10 | |
30 | 28, 23, 29 | syl2anc 408 | . . . . . . . . 9 |
31 | 28, 30 | mpbid 146 | . . . . . . . 8 |
32 | 18, 26, 31 | ertr3d 6447 | . . . . . . 7 |
33 | 16 | xmeterval 12607 | . . . . . . . 8 |
34 | 15, 33 | syl 14 | . . . . . . 7 |
35 | 32, 34 | mpbid 146 | . . . . . 6 |
36 | 35 | simp3d 995 | . . . . 5 |
37 | 14, 36 | eqeltrd 2216 | . . . 4 |
38 | 37 | ralrimivva 2514 | . . 3 |
39 | ffnov 5875 | . . 3 | |
40 | 12, 38, 39 | sylanbrc 413 | . 2 |
41 | ismet2 12526 | . 2 | |
42 | 6, 40, 41 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 wss 3071 class class class wbr 3929 cxp 4537 ccnv 4538 cres 4541 cima 4542 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 wer 6426 cec 6427 cr 7622 cxr 7802 cxmet 12152 cmet 12153 cbl 12154 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-er 6429 df-ec 6431 df-map 6544 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-2 8782 df-xneg 9562 df-xadd 9563 df-psmet 12159 df-xmet 12160 df-met 12161 df-bl 12162 |
This theorem is referenced by: (None) |
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