blhalf - Intuitionistic Logic Explorer

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Theorem blhalf 13993

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 9167	. . . . . 6 $/\$ $/\$ $/\$
6	5	rexrd 8009	. . . . 5 $/\$ $/\$ $/\$
7		elbl 13976	. . . . 5 $/\$ $/\$ $/\$
8	1, 2, 6, 7	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$
9	3, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$
10	9	simpld 112	. 2 $/\$ $/\$ $/\$
11		xmetcl 13937	. . . . 5 $/\$ $/\$
12	1, 2, 10, 11	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$
13	9	simprd 114	. . . 4 $/\$ $/\$ $/\$
14	12, 6, 13	xrltled 9801	. . 3 $/\$ $/\$ $/\$
15	5	recnd 7988	. . . . 5 $/\$ $/\$ $/\$
16	15, 15	pncand 8271	. . . 4 $/\$ $/\$ $/\$
17	4	recnd 7988	. . . . . 6 $/\$ $/\$ $/\$
18	17	2halvesd 9166	. . . . 5 $/\$ $/\$ $/\$
19	18	oveq1d 5892	. . . 4 $/\$ $/\$ $/\$
20	16, 19	eqtr3d 2212	. . 3 $/\$ $/\$ $/\$
21	14, 20	breqtrd 4031	. 2 $/\$ $/\$ $/\$
22		blss2 13992	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
23	1, 2, 10, 5, 4, 21, 22	syl33anc 1253	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wcel 2148

wss 3131 class class class wbr 4005

cfv 5218 (class class class)co 5877

cr 7812

caddc 7816

cxr 7993

clt 7994

cle 7995

cmin 8130

cdiv 8631

c2 8972

cxmet 13525

cbl 13527

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 df-xneg 9774 df-xadd 9775 df-psmet 13532 df-xmet 13533 df-bl 13535

This theorem is referenced by: (None)