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Mirrors > Home > ILE Home > Th. List > blval | Unicode version |
Description: The ball around a point is the set of all points whose distance from is less than the ball's radius . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
blval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blfval 12544 | . . 3 | |
2 | 1 | 3ad2ant1 1002 | . 2 |
3 | simprl 520 | . . . . 5 | |
4 | 3 | oveq1d 5782 | . . . 4 |
5 | simprr 521 | . . . 4 | |
6 | 4, 5 | breq12d 3937 | . . 3 |
7 | 6 | rabbidv 2670 | . 2 |
8 | simp2 982 | . 2 | |
9 | simp3 983 | . 2 | |
10 | xmetrel 12501 | . . . . 5 | |
11 | relelfvdm 5446 | . . . . 5 | |
12 | 10, 11 | mpan 420 | . . . 4 |
13 | 12 | 3ad2ant1 1002 | . . 3 |
14 | rabexg 4066 | . . 3 | |
15 | 13, 14 | syl 14 | . 2 |
16 | 2, 7, 8, 9, 15 | ovmpod 5891 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 crab 2418 cvv 2681 class class class wbr 3924 cdm 4534 wrel 4539 cfv 5118 (class class class)co 5767 cmpo 5769 cxr 7792 clt 7793 cxmet 12138 cbl 12140 |
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-psmet 12145 df-xmet 12146 df-bl 12148 |
This theorem is referenced by: elbl 12549 metss2lem 12655 bdbl 12661 xmetxpbl 12666 |
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