blhalf - Intuitionistic Logic Explorer

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Theorem blhalf 14305

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 9183	. . . . . 6 $/\$ $/\$ $/\$
6	5	rexrd 8025	. . . . 5 $/\$ $/\$ $/\$
7		elbl 14288	. . . . 5 $/\$ $/\$ $/\$
8	1, 2, 6, 7	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$
9	3, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$
10	9	simpld 112	. 2 $/\$ $/\$ $/\$
11		xmetcl 14249	. . . . 5 $/\$ $/\$
12	1, 2, 10, 11	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$
13	9	simprd 114	. . . 4 $/\$ $/\$ $/\$
14	12, 6, 13	xrltled 9817	. . 3 $/\$ $/\$ $/\$
15	5	recnd 8004	. . . . 5 $/\$ $/\$ $/\$
16	15, 15	pncand 8287	. . . 4 $/\$ $/\$ $/\$
17	4	recnd 8004	. . . . . 6 $/\$ $/\$ $/\$
18	17	2halvesd 9182	. . . . 5 $/\$ $/\$ $/\$
19	18	oveq1d 5906	. . . 4 $/\$ $/\$ $/\$
20	16, 19	eqtr3d 2224	. . 3 $/\$ $/\$ $/\$
21	14, 20	breqtrd 4044	. 2 $/\$ $/\$ $/\$
22		blss2 14304	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
23	1, 2, 10, 5, 4, 21, 22	syl33anc 1264	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wcel 2160

wss 3144 class class class wbr 4018

cfv 5231 (class class class)co 5891

cr 7828

caddc 7832

cxr 8009

clt 8010

cle 8011

cmin 8146

cdiv 8647

c2 8988

cxmet 13810

cbl 13812

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-map 6668 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-2 8996 df-xneg 9790 df-xadd 9791 df-psmet 13817 df-xmet 13818 df-bl 13820

This theorem is referenced by: (None)