dalem16 - Mathbox for Norm Megill

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Theorem dalem16 34876

Description: Lemma for dath 34933. The atoms

, and

form a line of perspectivity. This is Desargue's Theorem for the special case where planes

and

are different. (Contributed by NM, 7-Aug-2012.)

**Hypotheses**
Ref	Expression
dalema.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
dalemc.l
dalemc.j	$.\/$
dalemc.a
dalem16.m	$./\$
dalem16.o
dalem16.y	$.\/$ $.\/$
dalem16.z	$.\/$ $.\/$
dalem16.d	$.\/$ $./\$ $.\/$
dalem16.e	$.\/$ $./\$ $.\/$
dalem16.f	$.\/$ $./\$ $.\/$

**Assertion**
Ref	Expression
dalem16	$/\$ $.\/$

**Proof of Theorem dalem16**
Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
2		dalemc.l	. . . 4
3		dalemc.j	. . . 4 $.\/$
4		dalemc.a	. . . 4
5		dalem16.m	. . . 4 $./\$
6		dalem16.o	. . . 4
7		dalem16.y	. . . 4 $.\/$ $.\/$
8		dalem16.z	. . . 4 $.\/$ $.\/$
9		eqid 2467	. . . 4 $./\$ $./\$
10		dalem16.f	. . . 4 $.\/$ $./\$ $.\/$
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem12 34872	. . 3 $./\$
12	11	adantr 465	. 2 $/\$ $./\$
13		dalem16.d	. . . . . 6 $.\/$ $./\$ $.\/$
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 13	dalem10 34870	. . . . 5 $./\$
15		dalem16.e	. . . . . 6 $.\/$ $./\$ $.\/$
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15	dalem11 34871	. . . . 5 $./\$
17	1	dalemkelat 34821	. . . . . 6
18	1, 2, 3, 4, 5, 6, 7, 8, 13	dalemdea 34859	. . . . . . 7
19		eqid 2467	. . . . . . . 8
20	19, 4	atbase 34487	. . . . . . 7
21	18, 20	syl 16	. . . . . 6
22	1, 2, 3, 4, 5, 6, 7, 8, 15	dalemeea 34860	. . . . . . 7
23	19, 4	atbase 34487	. . . . . . 7
24	22, 23	syl 16	. . . . . 6
25	1, 6	dalemyeb 34846	. . . . . . 7
26	1	dalemzeo 34830	. . . . . . . 8
27	19, 6	lplnbase 34731	. . . . . . . 8
28	26, 27	syl 16	. . . . . . 7
29	19, 5	latmcl 15556	. . . . . . 7 $/\$ $/\$ $./\$
30	17, 25, 28, 29	syl3anc 1228	. . . . . 6 $./\$
31	19, 2, 3	latjle12 15566	. . . . . 6 $/\$ $/\$ $/\$ $./\$ $./\$ $/\$ $./\$ $.\/$ $./\$
32	17, 21, 24, 30, 31	syl13anc 1230	. . . . 5 $./\$ $/\$ $./\$ $.\/$ $./\$
33	14, 16, 32	mpbi2and 919	. . . 4 $.\/$ $./\$
34	33	adantr 465	. . 3 $/\$ $.\/$ $./\$
35	1	dalemkehl 34820	. . . . 5
36	35	adantr 465	. . . 4 $/\$
37	1, 2, 3, 4, 5, 6, 7, 8, 13, 15	dalemdnee 34863	. . . . . 6
38		eqid 2467	. . . . . . 7
39	3, 4, 38	llni2 34709	. . . . . 6 $/\$ $/\$ $/\$ $.\/$
40	35, 18, 22, 37, 39	syl31anc 1231	. . . . 5 $.\/$
41	40	adantr 465	. . . 4 $/\$ $.\/$
42	1, 2, 3, 4, 5, 38, 6, 7, 8, 9	dalem15 34875	. . . 4 $/\$ $./\$
43	2, 38	llncmp 34719	. . . 4 $/\$ $.\/$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
44	36, 41, 42, 43	syl3anc 1228	. . 3 $/\$ $.\/$ $./\$ $.\/$ $./\$
45	34, 44	mpbid 210	. 2 $/\$ $.\/$ $./\$
46	12, 45	breqtrrd 4479	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 4453

cfv 5594 (class class class)co 6295

cbs 14507

cple 14579

cjn 15448

cmee 15449

clat 15549

catm 34461

chlt 34548

clln 34688

clpl 34689

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 4564 ax-sep 4574 ax-nul 4582 ax-pow 4631 ax-pr 4692 ax-un 6587

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 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 2822 df-rex 2823 df-reu 2824 df-rab 2826 df-v 3120 df-sbc 3337 df-csb 3441 df-dif 3484 df-un 3486 df-in 3488 df-ss 3495 df-nul 3791 df-if 3946 df-pw 4018 df-sn 4034 df-pr 4036 df-op 4040 df-uni 4252 df-iun 4333 df-br 4454 df-opab 4512 df-mpt 4513 df-id 4801 df-xp 5011 df-rel 5012 df-cnv 5013 df-co 5014 df-dm 5015 df-rn 5016 df-res 5017 df-ima 5018 df-iota 5557 df-fun 5596 df-fn 5597 df-f 5598 df-f1 5599 df-fo 5600 df-f1o 5601 df-fv 5602 df-riota 6256 df-ov 6298 df-oprab 6299 df-poset 15450 df-plt 15462 df-lub 15478 df-glb 15479 df-join 15480 df-meet 15481 df-p0 15543 df-lat 15550 df-clat 15612 df-oposet 34374 df-ol 34376 df-oml 34377 df-covers 34464 df-ats 34465 df-atl 34496 df-cvlat 34520 df-hlat 34549 df-llines 34695 df-lplanes 34696 df-lvols 34697

This theorem is referenced by: dalem63 34932

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