bldisj - Intuitionistic Logic Explorer

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Theorem bldisj 12942

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 994	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 992	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 993	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 9811	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 12893	. . . . . 6 $/\$ $/\$
6	5	adantr 274	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 7954	. . . . 5 $/\$
8	4, 6, 7	syl2anc 409	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 146	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3300	. . . 4 $/\$
11		simpl1 989	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 990	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 12932	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1227	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 991	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 12932	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1227	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	14, 17	anbi12d 465	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		anandi 580	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	bitr4di 197	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	11	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	12	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simpr 109	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		xmetcl 12893	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1227	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 12893	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1227	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 274	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 9807	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1228	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 12915	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1229	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 9811	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 9733	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1227	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	34, 39	mpand 426	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	32, 40	syld 45	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	41	expimpd 361	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	20, 42	sylbid 149	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	10, 43	syl5bi 151	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
45	9, 44	mtod 653	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3448	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 967

wceq 1342

wcel 2135

cin 3110

c0 3404 class class class wbr 3976

cfv 5182 (class class class)co 5836

cxr 7923

clt 7924

cle 7925

cxad 9697

cxmet 12521

cbl 12523

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-xadd 9700 df-psmet 12528 df-xmet 12529 df-bl 12531

This theorem is referenced by: bl2in 12944

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