blhalf - Intuitionistic Logic Explorer

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Theorem blhalf 15097

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.)

**Assertion**
Ref	Expression
blhalf	$/\$ $/\$ $/\$

**Proof of Theorem blhalf**
Step	Hyp	Ref	Expression
1		simpll 527	. 2 $/\$ $/\$ $/\$
2		simplr 528	. 2 $/\$ $/\$ $/\$
3		simprr 531	. . . 4 $/\$ $/\$ $/\$
4		simprl 529	. . . . . . 7 $/\$ $/\$ $/\$
5	4	rehalfcld 9369	. . . . . 6 $/\$ $/\$ $/\$
6	5	rexrd 8207	. . . . 5 $/\$ $/\$ $/\$
7		elbl 15080	. . . . 5 $/\$ $/\$ $/\$
8	1, 2, 6, 7	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
9	3, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$
10	9	simpld 112	. 2 $/\$ $/\$ $/\$
11		xmetcl 15041	. . . . 5 $/\$ $/\$
12	1, 2, 10, 11	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$
13	9	simprd 114	. . . 4 $/\$ $/\$ $/\$
14	12, 6, 13	xrltled 10007	. . 3 $/\$ $/\$ $/\$
15	5	recnd 8186	. . . . 5 $/\$ $/\$ $/\$
16	15, 15	pncand 8469	. . . 4 $/\$ $/\$ $/\$
17	4	recnd 8186	. . . . . 6 $/\$ $/\$ $/\$
18	17	2halvesd 9368	. . . . 5 $/\$ $/\$ $/\$
19	18	oveq1d 6022	. . . 4 $/\$ $/\$ $/\$
20	16, 19	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$
21	14, 20	breqtrd 4109	. 2 $/\$ $/\$ $/\$
22		blss2 15096	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
23	1, 2, 10, 5, 4, 21, 22	syl33anc 1286	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wcel 2200

wss 3197 class class class wbr 4083

cfv 5318 (class class class)co 6007

cr 8009

caddc 8013

cxr 8191

clt 8192

cle 8193

cmin 8328

cdiv 8830

c2 9172

cxmet 14515

cbl 14517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-map 6805 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-2 9180 df-xneg 9980 df-xadd 9981 df-psmet 14522 df-xmet 14523 df-bl 14525

This theorem is referenced by: (None)