bldisj - Intuitionistic Logic Explorer

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Theorem bldisj 15392

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 1032	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 1030	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 1031	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 10236	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 15343	. . . . . 6 $/\$ $/\$
6	5	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 8354	. . . . 5 $/\$
8	4, 6, 7	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3406	. . . 4 $/\$
11		simpl1 1027	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 1028	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 15382	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 1029	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 15382	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1274	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	14, 17	anbi12d 473	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		anandi 594	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	bitr4di 198	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	12	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simpr 110	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		xmetcl 15343	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 15343	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 10232	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1275	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 15365	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 10236	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 10158	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	34, 39	mpand 429	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	32, 40	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expimpd 363	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	20, 42	sylbid 150	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	10, 43	biimtrid 152	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
45	9, 44	mtod 669	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3557	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1005

wceq 1398

wcel 2205

cin 3213

c0 3512 class class class wbr 4114

cfv 5357 (class class class)co 6058

cxr 8323

clt 8324

cle 8325

cxad 10122

cxmet 14810

cbl 14812

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-addass 8245 ax-i2m1 8248 ax-0id 8251 ax-rnegex 8252 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-map 6897 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-xadd 10125 df-psmet 14817 df-xmet 14818 df-bl 14820

This theorem is referenced by: bl2in 15394

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