dalem2 - Mathbox for Norm Megill

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Theorem dalem2 34858

Description: Lemma for dath 34933. Show the lines

and

form a plane. (Contributed by NM, 11-Aug-2012.)

**Hypotheses**
Ref	Expression
dalema.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
dalemc.l
dalemc.j	$.\/$
dalemc.a
dalem1.o
dalem1.y	$.\/$ $.\/$

**Assertion**
Ref	Expression
dalem2	$.\/$ $.\/$ $.\/$

**Proof of Theorem dalem2**
Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
2	1	dalemkehl 34820	. . 3
3	1	dalempea 34823	. . 3
4	1	dalemqea 34824	. . 3
5	1	dalemsea 34826	. . 3
6	1	dalemtea 34827	. . 3
7		dalemc.j	. . . 4 $.\/$
8		dalemc.a	. . . 4
9	7, 8	hlatj4 34571	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
10	2, 3, 4, 5, 6, 9	syl122anc 1237	. 2 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
11		dalemc.l	. . . . 5
12		dalem1.o	. . . . 5
13		dalem1.y	. . . . 5 $.\/$ $.\/$
14	1, 11, 7, 8, 12, 13	dalempjsen 34850	. . . 4 $.\/$
15	1, 11, 7, 8, 12, 13	dalemqnet 34849	. . . . 5
16		eqid 2467	. . . . . 6
17	7, 8, 16	llni2 34709	. . . . 5 $/\$ $/\$ $/\$ $.\/$
18	2, 4, 6, 15, 17	syl31anc 1231	. . . 4 $.\/$
19	1, 11, 7, 8, 12, 13	dalem1 34856	. . . 4 $.\/$ $.\/$
20	1, 11, 7, 8, 12, 13	dalemcea 34857	. . . . 5
21	1	dalemclpjs 34831	. . . . 5 $.\/$
22	1	dalemclqjt 34832	. . . . 5 $.\/$
23		eqid 2467	. . . . . 6
24		eqid 2467	. . . . . 6
25	11, 23, 24, 8, 16	2llnm4 34767	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
26	2, 20, 14, 18, 21, 22, 25	syl132anc 1246	. . . 4 $.\/$ $.\/$
27	23, 24, 8, 16	2llnmat 34721	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
28	2, 14, 18, 19, 26, 27	syl32anc 1236	. . 3 $.\/$ $.\/$
29	7, 23, 8, 16, 12	2llnmj 34757	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
30	2, 14, 18, 29	syl3anc 1228	. . 3 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
31	28, 30	mpbid 210	. 2 $.\/$ $.\/$ $.\/$
32	10, 31	eqeltrd 2555	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

cp0 15541

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-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

This theorem is referenced by: dalemdea 34859

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