dalem2 - Mathbox for Norm Megill

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Theorem dalem2 33271

Description: Lemma for dath 33346. 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 33233	. . 3
3	1	dalempea 33236	. . 3
4	1	dalemqea 33237	. . 3
5	1	dalemsea 33239	. . 3
6	1	dalemtea 33240	. . 3
7		dalemc.j	. . . 4 $.\/$
8		dalemc.a	. . . 4
9	7, 8	hlatj4 32984	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
10	2, 3, 4, 5, 6, 9	syl122anc 1285	. 2 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
11		dalemc.l	. . . . 5
12		dalem1.o	. . . . 5
13		dalem1.y	. . . . 5 $.\/$ $.\/$
14	1, 11, 7, 8, 12, 13	dalempjsen 33263	. . . 4 $.\/$
15	1, 11, 7, 8, 12, 13	dalemqnet 33262	. . . . 5
16		eqid 2462	. . . . . 6
17	7, 8, 16	llni2 33122	. . . . 5 $/\$ $/\$ $/\$ $.\/$
18	2, 4, 6, 15, 17	syl31anc 1279	. . . 4 $.\/$
19	1, 11, 7, 8, 12, 13	dalem1 33269	. . . 4 $.\/$ $.\/$
20	1, 11, 7, 8, 12, 13	dalemcea 33270	. . . . 5
21	1	dalemclpjs 33244	. . . . 5 $.\/$
22	1	dalemclqjt 33245	. . . . 5 $.\/$
23		eqid 2462	. . . . . 6
24		eqid 2462	. . . . . 6
25	11, 23, 24, 8, 16	2llnm4 33180	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
26	2, 20, 14, 18, 21, 22, 25	syl132anc 1294	. . . 4 $.\/$ $.\/$
27	23, 24, 8, 16	2llnmat 33134	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
28	2, 14, 18, 19, 26, 27	syl32anc 1284	. . 3 $.\/$ $.\/$
29	7, 23, 8, 16, 12	2llnmj 33170	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
30	2, 14, 18, 29	syl3anc 1276	. . 3 $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
31	28, 30	mpbid 215	. 2 $.\/$ $.\/$ $.\/$
32	10, 31	eqeltrd 2540	1 $.\/$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wcel 1898

wne 2633 class class class wbr 4416

cfv 5601 (class class class)co 6315

cbs 15170

cple 15246

cjn 16238

cmee 16239

cp0 16332

catm 32874

chlt 32961

clln 33101

clpl 33102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-preset 16222 df-poset 16240 df-plt 16253 df-lub 16269 df-glb 16270 df-join 16271 df-meet 16272 df-p0 16334 df-lat 16341 df-clat 16403 df-oposet 32787 df-ol 32789 df-oml 32790 df-covers 32877 df-ats 32878 df-atl 32909 df-cvlat 32933 df-hlat 32962 df-llines 33108 df-lplanes 33109

This theorem is referenced by: dalemdea 33272

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