dalem17 - Mathbox for Norm Megill

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Theorem dalem17 33627

Description: Lemma for dath 33683. When planes

and

are equal, the center of perspectivity

is in

. (Contributed by NM, 1-Aug-2012.)

**Hypotheses**
Ref	Expression
dalema.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
dalemc.l
dalemc.j	$.\/$
dalemc.a
dalem17.o
dalem17.y	$.\/$ $.\/$
dalem17.z	$.\/$ $.\/$

**Assertion**
Ref	Expression
dalem17	$/\$

**Proof of Theorem dalem17**
Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
2	1	dalemclrju 33583	. . 3 $.\/$
3	2	adantr 465	. 2 $/\$ $.\/$
4	1	dalemkelat 33571	. . . . . 6
5		dalemc.j	. . . . . . 7 $.\/$
6		dalemc.a	. . . . . . 7
7	1, 5, 6	dalempjqeb 33592	. . . . . 6 $.\/$
8	1, 6	dalemreb 33588	. . . . . 6
9		eqid 2451	. . . . . . 7
10		dalemc.l	. . . . . . 7
11	9, 10, 5	latlej2 15330	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $.\/$
12	4, 7, 8, 11	syl3anc 1219	. . . . 5 $.\/$ $.\/$
13		dalem17.y	. . . . 5 $.\/$ $.\/$
14	12, 13	syl6breqr 4427	. . . 4
15	14	adantr 465	. . 3 $/\$
16	1, 5, 6	dalemsjteb 33593	. . . . . . 7 $.\/$
17	1, 6	dalemueb 33591	. . . . . . 7
18	9, 10, 5	latlej2 15330	. . . . . . 7 $/\$ $.\/$ $/\$ $.\/$ $.\/$
19	4, 16, 17, 18	syl3anc 1219	. . . . . 6 $.\/$ $.\/$
20		dalem17.z	. . . . . 6 $.\/$ $.\/$
21	19, 20	syl6breqr 4427	. . . . 5
22	21	adantr 465	. . . 4 $/\$
23		simpr 461	. . . 4 $/\$
24	22, 23	breqtrrd 4413	. . 3 $/\$
25		dalem17.o	. . . . . 6
26	1, 25	dalemyeb 33596	. . . . 5
27	9, 10, 5	latjle12 15331	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
28	4, 8, 17, 26, 27	syl13anc 1221	. . . 4 $/\$ $.\/$
29	28	adantr 465	. . 3 $/\$ $/\$ $.\/$
30	15, 24, 29	mpbi2and 912	. 2 $/\$ $.\/$
31	1, 6	dalemceb 33585	. . . 4
32	1	dalemkehl 33570	. . . . 5
33	1	dalemrea 33575	. . . . 5
34	1	dalemuea 33578	. . . . 5
35	9, 5, 6	hlatjcl 33314	. . . . 5 $/\$ $/\$ $.\/$
36	32, 33, 34, 35	syl3anc 1219	. . . 4 $.\/$
37	9, 10	lattr 15325	. . . 4 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
38	4, 31, 36, 26, 37	syl13anc 1221	. . 3 $.\/$ $/\$ $.\/$
39	38	adantr 465	. 2 $/\$ $.\/$ $/\$ $.\/$
40	3, 30, 39	mp2and 679	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1370

wcel 1758 class class class wbr 4387

cfv 5513 (class class class)co 6187

cbs 14273

cple 14344

cjn 15213

clat 15314

catm 33211

chlt 33298

clpl 33439

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 1952 ax-ext 2430 ax-rep 4498 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 ax-un 6469

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-reu 2800 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-pw 3957 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fn 5516 df-f 5517 df-f1 5518 df-fo 5519 df-f1o 5520 df-fv 5521 df-riota 6148 df-ov 6190 df-oprab 6191 df-poset 15215 df-lub 15243 df-glb 15244 df-join 15245 df-meet 15246 df-lat 15315 df-ats 33215 df-atl 33246 df-cvlat 33270 df-hlat 33299 df-lplanes 33446

This theorem is referenced by: dalem19 33629 dalem25 33645

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