dalemcea - Mathbox for Norm Megill

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Theorem dalemcea 34946

Description: Lemma for dath 35022. Frequently-used utility lemma. Here we show that

must be an atom. This is an assumption in most presentations of Desargue's theorem; instead, we assume only the

is a lattice element, in order to make later substitutions for

easier. (Contributed by NM, 23-Sep-2012.)

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

**Assertion**
Ref	Expression
dalemcea

**Proof of Theorem dalemcea**
Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
2	1	dalemkeop 34911	. . 3
3		dalemc.a	. . . 4
4	1, 3	dalemceb 34924	. . 3
5	1	dalemkehl 34909	. . . 4
6		dalemc.l	. . . . 5
7		dalemc.j	. . . . 5 $.\/$
8		dalem1.o	. . . . 5
9		dalem1.y	. . . . 5 $.\/$ $.\/$
10	1, 6, 7, 3, 8, 9	dalempjsen 34939	. . . 4 $.\/$
11	1	dalemqea 34913	. . . . 5
12	1	dalemtea 34916	. . . . 5
13	1, 6, 7, 3, 8, 9	dalemqnet 34938	. . . . 5
14		eqid 2622	. . . . . 6
15	7, 3, 14	llni2 34798	. . . . 5 $/\$ $/\$ $/\$ $.\/$
16	5, 11, 12, 13, 15	syl31anc 1329	. . . 4 $.\/$
17	1, 6, 7, 3, 8, 9	dalem1 34945	. . . 4 $.\/$ $.\/$
18	1	dalem-clpjq 34923	. . . . . . . 8 $.\/$
19	1, 7, 3	dalempjqeb 34931	. . . . . . . . . . 11 $.\/$
20		eqid 2622	. . . . . . . . . . . 12
21		eqid 2622	. . . . . . . . . . . 12
22	20, 6, 21	op0le 34473	. . . . . . . . . . 11 $/\$ $.\/$ $.\/$
23	2, 19, 22	syl2anc 693	. . . . . . . . . 10 $.\/$
24		breq1 4656	. . . . . . . . . 10 $.\/$ $.\/$
25	23, 24	syl5ibrcom 237	. . . . . . . . 9 $.\/$
26	25	necon3bd 2808	. . . . . . . 8 $.\/$
27	18, 26	mpd 15	. . . . . . 7
28		eqid 2622	. . . . . . . . 9
29	20, 28, 21	opltn0 34477	. . . . . . . 8 $/\$
30	2, 4, 29	syl2anc 693	. . . . . . 7
31	27, 30	mpbird 247	. . . . . 6
32	1	dalemclpjs 34920	. . . . . . 7 $.\/$
33	1	dalemclqjt 34921	. . . . . . 7 $.\/$
34	1	dalemkelat 34910	. . . . . . . 8
35	1	dalempea 34912	. . . . . . . . 9
36	1	dalemsea 34915	. . . . . . . . 9
37	20, 7, 3	hlatjcl 34653	. . . . . . . . 9 $/\$ $/\$ $.\/$
38	5, 35, 36, 37	syl3anc 1326	. . . . . . . 8 $.\/$
39	20, 7, 3	hlatjcl 34653	. . . . . . . . 9 $/\$ $/\$ $.\/$
40	5, 11, 12, 39	syl3anc 1326	. . . . . . . 8 $.\/$
41		eqid 2622	. . . . . . . . 9
42	20, 6, 41	latlem12 17078	. . . . . . . 8 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
43	34, 4, 38, 40, 42	syl13anc 1328	. . . . . . 7 $.\/$ $/\$ $.\/$ $.\/$ $.\/$
44	32, 33, 43	mpbi2and 956	. . . . . 6 $.\/$ $.\/$
45		opposet 34468	. . . . . . . 8
46	2, 45	syl 17	. . . . . . 7
47	20, 21	op0cl 34471	. . . . . . . 8
48	2, 47	syl 17	. . . . . . 7
49	20, 41	latmcl 17052	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
50	34, 38, 40, 49	syl3anc 1326	. . . . . . 7 $.\/$ $.\/$
51	20, 6, 28	pltletr 16971	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
52	46, 48, 4, 50, 51	syl13anc 1328	. . . . . 6 $/\$ $.\/$ $.\/$ $.\/$ $.\/$
53	31, 44, 52	mp2and 715	. . . . 5 $.\/$ $.\/$
54	20, 28, 21	opltn0 34477	. . . . . 6 $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
55	2, 50, 54	syl2anc 693	. . . . 5 $.\/$ $.\/$ $.\/$ $.\/$
56	53, 55	mpbid 222	. . . 4 $.\/$ $.\/$
57	41, 21, 3, 14	2llnmat 34810	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
58	5, 10, 16, 17, 56, 57	syl32anc 1334	. . 3 $.\/$ $.\/$
59	20, 6, 21, 3	leat2 34581	. . 3 $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
60	2, 4, 58, 27, 44, 59	syl32anc 1334	. 2 $.\/$ $.\/$
61	60, 58	eqeltrd 2701	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cpo 16940

cplt 16941

cjn 16944

cmee 16945

cp0 17037

clat 17045

cops 34459

catm 34550

chlt 34637

clln 34777

clpl 34778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785

This theorem is referenced by: dalem2 34947 dalem5 34953 dalem-cly 34957 dalem9 34958 dalem19 34968 dalem21 34980 dalem25 34984

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