dalem5 - Mathbox for Norm Megill

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Theorem dalem5 34463

Description: Lemma for dath 34532. Atom

(in plane

) belongs to the 3-dimensional volume formed by

and

. (Contributed by NM, 21-Jul-2012.)

**Hypotheses**
Ref	Expression
dalema.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
dalemc.l
dalemc.j	$.\/$
dalemc.a
dalem5.o
dalem5.y	$.\/$ $.\/$
dalem5.w	$.\/$

**Assertion**
Ref	Expression
dalem5

**Proof of Theorem dalem5**
Step	Hyp	Ref	Expression
1		eqid 2467	. 2
2		dalemc.l	. 2
3		dalema.ph	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
4	3	dalemkelat 34420	. 2
5		dalemc.a	. . 3
6	3, 5	dalemueb 34440	. 2
7	3	dalemkehl 34419	. . 3
8	3	dalemrea 34424	. . 3
9		dalemc.j	. . . 4 $.\/$
10		dalem5.o	. . . 4
11		dalem5.y	. . . 4 $.\/$ $.\/$
12	3, 2, 9, 5, 10, 11	dalemcea 34456	. . 3
13	1, 9, 5	hlatjcl 34163	. . 3 $/\$ $/\$ $.\/$
14	7, 8, 12, 13	syl3anc 1228	. 2 $.\/$
15		dalem5.w	. . 3 $.\/$
16	3, 10	dalemyeb 34445	. . . 4
17	3, 5	dalemceb 34434	. . . 4
18	1, 9	latjcl 15534	. . . 4 $/\$ $/\$ $.\/$
19	4, 16, 17, 18	syl3anc 1228	. . 3 $.\/$
20	15, 19	syl5eqel 2559	. 2
21	3	dalemclrju 34432	. . 3 $.\/$
22	3	dalemuea 34427	. . . 4
23	3	dalempea 34422	. . . . 5
24		simp313 1145	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
25	3, 24	sylbi 195	. . . . 5 $.\/$
26	2, 9, 5	atnlej1 34175	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
27	7, 12, 8, 23, 25, 26	syl131anc 1241	. . . 4
28	2, 9, 5	hlatexch1 34191	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
29	7, 12, 22, 8, 27, 28	syl131anc 1241	. . 3 $.\/$ $.\/$
30	21, 29	mpd 15	. 2 $.\/$
31	3, 9, 5	dalempjqeb 34441	. . . . . 6 $.\/$
32	3, 5	dalemreb 34437	. . . . . 6
33	1, 2, 9	latlej2 15544	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $.\/$
34	4, 31, 32, 33	syl3anc 1228	. . . . 5 $.\/$ $.\/$
35	34, 11	syl6breqr 4487	. . . 4
36	1, 2, 9	latjlej1 15548	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$
37	4, 32, 16, 17, 36	syl13anc 1230	. . . 4 $.\/$ $.\/$
38	35, 37	mpd 15	. . 3 $.\/$ $.\/$
39	38, 15	syl6breqr 4487	. 2 $.\/$
40	1, 2, 4, 6, 14, 20, 30, 39	lattrd 15541	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767

wne 2662 class class class wbr 4447

cfv 5586 (class class class)co 6282

cbs 14486

cple 14558

cjn 15427

clat 15528

catm 34060

chlt 34147

clpl 34288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-8 1769 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-rep 4558 ax-sep 4568 ax-nul 4576 ax-pow 4625 ax-pr 4686 ax-un 6574

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-ral 2819 df-rex 2820 df-reu 2821 df-rab 2823 df-v 3115 df-sbc 3332 df-csb 3436 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-nul 3786 df-if 3940 df-pw 4012 df-sn 4028 df-pr 4030 df-op 4034 df-uni 4246 df-iun 4327 df-br 4448 df-opab 4506 df-mpt 4507 df-id 4795 df-xp 5005 df-rel 5006 df-cnv 5007 df-co 5008 df-dm 5009 df-rn 5010 df-res 5011 df-ima 5012 df-iota 5549 df-fun 5588 df-fn 5589 df-f 5590 df-f1 5591 df-fo 5592 df-f1o 5593 df-fv 5594 df-riota 6243 df-ov 6285 df-oprab 6286 df-poset 15429 df-plt 15441 df-lub 15457 df-glb 15458 df-join 15459 df-meet 15460 df-p0 15522 df-lat 15529 df-clat 15591 df-oposet 33973 df-ol 33975 df-oml 33976 df-covers 34063 df-ats 34064 df-atl 34095 df-cvlat 34119 df-hlat 34148 df-llines 34294 df-lplanes 34295

This theorem is referenced by: dalem6 34464 dalem8 34466

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