bldisj - Intuitionistic Logic Explorer

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Theorem bldisj 14815

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 1007	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 1005	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 1006	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 10005	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 14766	. . . . . 6 $/\$ $/\$
6	5	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 8136	. . . . 5 $/\$
8	4, 6, 7	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3355	. . . 4 $/\$
11		simpl1 1002	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 1003	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 14805	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 1004	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 14805	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1249	. . . . . . 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 14766	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 14766	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1249	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 10001	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1250	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 14788	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1251	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 10005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 9927	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1249	. . . . . . . 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 664	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3504	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1372

wcel 2175

cin 3164

c0 3459 class class class wbr 4043

cfv 5270 (class class class)co 5943

cxr 8105

clt 8106

cle 8107

cxad 9891

cxmet 14240

cbl 14242

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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-i2m1 8029 ax-0id 8032 ax-rnegex 8033 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-po 4342 df-iso 4343 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-fv 5278 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-map 6736 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-xadd 9894 df-psmet 14247 df-xmet 14248 df-bl 14250

This theorem is referenced by: bl2in 14817

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