Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > blhalf | Unicode version |
Description: A ball of radius is contained in a ball of radius centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.) |
Ref | Expression |
---|---|
blhalf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . 2 | |
2 | simplr 520 | . 2 | |
3 | simprr 522 | . . . 4 | |
4 | simprl 521 | . . . . . . 7 | |
5 | 4 | rehalfcld 9084 | . . . . . 6 |
6 | 5 | rexrd 7929 | . . . . 5 |
7 | elbl 12861 | . . . . 5 | |
8 | 1, 2, 6, 7 | syl3anc 1220 | . . . 4 |
9 | 3, 8 | mpbid 146 | . . 3 |
10 | 9 | simpld 111 | . 2 |
11 | xmetcl 12822 | . . . . 5 | |
12 | 1, 2, 10, 11 | syl3anc 1220 | . . . 4 |
13 | 9 | simprd 113 | . . . 4 |
14 | 12, 6, 13 | xrltled 9712 | . . 3 |
15 | 5 | recnd 7908 | . . . . 5 |
16 | 15, 15 | pncand 8191 | . . . 4 |
17 | 4 | recnd 7908 | . . . . . 6 |
18 | 17 | 2halvesd 9083 | . . . . 5 |
19 | 18 | oveq1d 5841 | . . . 4 |
20 | 16, 19 | eqtr3d 2192 | . . 3 |
21 | 14, 20 | breqtrd 3992 | . 2 |
22 | blss2 12877 | . 2 | |
23 | 1, 2, 10, 5, 4, 21, 22 | syl33anc 1235 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 wss 3102 class class class wbr 3967 cfv 5172 (class class class)co 5826 cr 7733 caddc 7737 cxr 7913 clt 7914 cle 7915 cmin 8050 cdiv 8549 c2 8889 cxmet 12450 cbl 12452 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-po 4258 df-iso 4259 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-map 6597 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-2 8897 df-xneg 9685 df-xadd 9686 df-psmet 12457 df-xmet 12458 df-bl 12460 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |