bldisj - Intuitionistic Logic Explorer

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Theorem bldisj 14948

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 1008	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 1006	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 1007	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 10026	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 14899	. . . . . 6 $/\$ $/\$
6	5	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 8157	. . . . 5 $/\$
8	4, 6, 7	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3360	. . . 4 $/\$
11		simpl1 1003	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 1004	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 14938	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 1005	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 14938	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	14, 17	anbi12d 473	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		anandi 590	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	bitr4di 198	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	12	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simpr 110	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		xmetcl 14899	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1250	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 14899	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1250	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 10022	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1251	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 14921	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1252	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 10026	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 9948	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1250	. . . . . . . 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 665	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3509	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 981

wceq 1373

wcel 2177

cin 3169

c0 3464 class class class wbr 4051

cfv 5280 (class class class)co 5957

cxr 8126

clt 8127

cle 8128

cxad 9912

cxmet 14373

cbl 14375

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-addcom 8045 ax-addass 8047 ax-i2m1 8050 ax-0id 8053 ax-rnegex 8054 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-fv 5288 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-map 6750 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-xadd 9915 df-psmet 14380 df-xmet 14381 df-bl 14383

This theorem is referenced by: bl2in 14950

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