bldisj - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > bldisj		Unicode version

Theorem bldisj 12570

Description: Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)

**Assertion**
Ref	Expression
bldisj	$/\$ $/\$ $/\$ $/\$ $/\$

Proof of Theorem bldisj Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 989	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 987	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 988	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 9667	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 12521	. . . . . 6 $/\$ $/\$
6	5	adantr 274	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 7829	. . . . 5 $/\$
8	4, 6, 7	syl2anc 408	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 146	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3259	. . . 4 $/\$
11		simpl1 984	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 985	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 12560	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1216	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 986	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 12560	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1216	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	14, 17	anbi12d 464	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		anandi 579	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	syl6bbr 197	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	11	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	12	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simpr 109	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		xmetcl 12521	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1216	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 12521	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1216	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 9663	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1217	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 12543	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1218	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 9667	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 9589	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1216	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	34, 39	mpand 425	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	32, 40	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expimpd 360	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	20, 42	sylbid 149	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	10, 43	syl5bi 151	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
45	9, 44	mtod 652	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3407	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 962

wceq 1331

wcel 1480

cin 3070

c0 3363 class class class wbr 3929

cfv 5123 (class class class)co 5774

cxr 7799

clt 7800

cle 7801

cxad 9557

cxmet 12149

cbl 12151

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-xadd 9560 df-psmet 12156 df-xmet 12157 df-bl 12159

This theorem is referenced by: bl2in 12572

W3C validator