blhalf - Intuitionistic Logic Explorer

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Theorem blhalf 15076

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 9354	. . . . . 6 $/\$ $/\$ $/\$
6	5	rexrd 8192	. . . . 5 $/\$ $/\$ $/\$
7		elbl 15059	. . . . 5 $/\$ $/\$ $/\$
8	1, 2, 6, 7	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$ $/\$
9	3, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$
10	9	simpld 112	. 2 $/\$ $/\$ $/\$
11		xmetcl 15020	. . . . 5 $/\$ $/\$
12	1, 2, 10, 11	syl3anc 1271	. . . 4 $/\$ $/\$ $/\$
13	9	simprd 114	. . . 4 $/\$ $/\$ $/\$
14	12, 6, 13	xrltled 9991	. . 3 $/\$ $/\$ $/\$
15	5	recnd 8171	. . . . 5 $/\$ $/\$ $/\$
16	15, 15	pncand 8454	. . . 4 $/\$ $/\$ $/\$
17	4	recnd 8171	. . . . . 6 $/\$ $/\$ $/\$
18	17	2halvesd 9353	. . . . 5 $/\$ $/\$ $/\$
19	18	oveq1d 6015	. . . 4 $/\$ $/\$ $/\$
20	16, 19	eqtr3d 2264	. . 3 $/\$ $/\$ $/\$
21	14, 20	breqtrd 4108	. 2 $/\$ $/\$ $/\$
22		blss2 15075	. 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 4082

cfv 5317 (class class class)co 6000

cr 7994

caddc 7998

cxr 8176

clt 8177

cle 8178

cmin 8313

cdiv 8815

c2 9157

cxmet 14494

cbl 14496

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 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113

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 3888 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-map 6795 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-2 9165 df-xneg 9964 df-xadd 9965 df-psmet 14501 df-xmet 14502 df-bl 14504

This theorem is referenced by: (None)