bldisj - Intuitionistic Logic Explorer

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Theorem bldisj 15425

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 1036	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr1 1034	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3		simpr2 1035	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
4	2, 3	xaddcld 10265	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		xmetcl 15376	. . . . . 6 $/\$ $/\$
6	5	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
7		xrlenlt 8380	. . . . 5 $/\$
8	4, 6, 7	syl2anc 415	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
9	1, 8	mpbid 147	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
10		elin 3412	. . . 4 $/\$
11		simpl1 1031	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl2 1032	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
13		elbl 15415	. . . . . . . 8 $/\$ $/\$ $/\$
14	11, 12, 2, 13	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl3 1033	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
16		elbl 15415	. . . . . . . 8 $/\$ $/\$ $/\$
17	11, 15, 3, 16	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	14, 17	anbi12d 477	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		anandi 598	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
20	18, 19	bitr4di 198	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	12	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simpr 110	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		xmetcl 15376	. . . . . . . . 9 $/\$ $/\$
25	21, 22, 23, 24	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	15	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		xmetcl 15376	. . . . . . . . 9 $/\$ $/\$
28	21, 26, 23, 27	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	2	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	3	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		xlt2add 10261	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
32	25, 28, 29, 30, 31	syl22anc 1279	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33		xmettri3 15398	. . . . . . . . 9 $/\$ $/\$ $/\$
34	21, 22, 26, 23, 33	syl13anc 1280	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	6	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	25, 28	xaddcld 10265	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	4	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		xrlelttr 10187	. . . . . . . . 9 $/\$ $/\$ $/\$
39	35, 36, 37, 38	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	34, 39	mpand 433	. . . . . . 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 673	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
46	45	eq0rdv 3570	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209

cin 3219

c0 3520 class class class wbr 4125

cfv 5372 (class class class)co 6075

cxr 8349

clt 8350

cle 8351

cxad 10151

cxmet 14845

cbl 14847

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-addass 8271 ax-i2m1 8274 ax-0id 8277 ax-rnegex 8278 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-map 6914 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-xadd 10154 df-psmet 14852 df-xmet 14853 df-bl 14855

This theorem is referenced by: bl2in 15427

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