dalem16 - Mathbox for Norm Megill

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Theorem dalem16 33686

Description: Lemma for dath 33743. 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 2454	. . . 4 $./\$ $./\$
10		dalem16.f	. . . 4 $.\/$ $./\$ $.\/$
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem12 33682	. . 3 $./\$
12	11	adantr 465	. 2 $/\$ $./\$
13		dalem16.d	. . . . . 6 $.\/$ $./\$ $.\/$
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 13	dalem10 33680	. . . . 5 $./\$
15		dalem16.e	. . . . . 6 $.\/$ $./\$ $.\/$
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15	dalem11 33681	. . . . 5 $./\$
17	1	dalemkelat 33631	. . . . . 6
18	1, 2, 3, 4, 5, 6, 7, 8, 13	dalemdea 33669	. . . . . . 7
19		eqid 2454	. . . . . . . 8
20	19, 4	atbase 33297	. . . . . . 7
21	18, 20	syl 16	. . . . . 6
22	1, 2, 3, 4, 5, 6, 7, 8, 15	dalemeea 33670	. . . . . . 7
23	19, 4	atbase 33297	. . . . . . 7
24	22, 23	syl 16	. . . . . 6
25	1, 6	dalemyeb 33656	. . . . . . 7
26	1	dalemzeo 33640	. . . . . . . 8
27	19, 6	lplnbase 33541	. . . . . . . 8
28	26, 27	syl 16	. . . . . . 7
29	19, 5	latmcl 15345	. . . . . . 7 $/\$ $/\$ $./\$
30	17, 25, 28, 29	syl3anc 1219	. . . . . 6 $./\$
31	19, 2, 3	latjle12 15355	. . . . . 6 $/\$ $/\$ $/\$ $./\$ $./\$ $/\$ $./\$ $.\/$ $./\$
32	17, 21, 24, 30, 31	syl13anc 1221	. . . . 5 $./\$ $/\$ $./\$ $.\/$ $./\$
33	14, 16, 32	mpbi2and 912	. . . 4 $.\/$ $./\$
34	33	adantr 465	. . 3 $/\$ $.\/$ $./\$
35	1	dalemkehl 33630	. . . . 5
36	35	adantr 465	. . . 4 $/\$
37	1, 2, 3, 4, 5, 6, 7, 8, 13, 15	dalemdnee 33673	. . . . . 6
38		eqid 2454	. . . . . . 7
39	3, 4, 38	llni2 33519	. . . . . 6 $/\$ $/\$ $/\$ $.\/$
40	35, 18, 22, 37, 39	syl31anc 1222	. . . . 5 $.\/$
41	40	adantr 465	. . . 4 $/\$ $.\/$
42	1, 2, 3, 4, 5, 38, 6, 7, 8, 9	dalem15 33685	. . . 4 $/\$ $./\$
43	2, 38	llncmp 33529	. . . 4 $/\$ $.\/$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$
44	36, 41, 42, 43	syl3anc 1219	. . 3 $/\$ $.\/$ $./\$ $.\/$ $./\$
45	34, 44	mpbid 210	. 2 $/\$ $.\/$ $./\$
46	12, 45	breqtrrd 4429	1 $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wcel 1758

wne 2648 class class class wbr 4403

cfv 5529 (class class class)co 6203

cbs 14296

cple 14368

cjn 15237

cmee 15238

clat 15338

catm 33271

chlt 33358

clln 33498

clpl 33499

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-poset 15239 df-plt 15251 df-lub 15267 df-glb 15268 df-join 15269 df-meet 15270 df-p0 15332 df-lat 15339 df-clat 15401 df-oposet 33184 df-ol 33186 df-oml 33187 df-covers 33274 df-ats 33275 df-atl 33306 df-cvlat 33330 df-hlat 33359 df-llines 33505 df-lplanes 33506 df-lvols 33507

This theorem is referenced by: dalem63 33742

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