cdlemg17a - Mathbox for Norm Megill

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Theorem cdlemg17a 34299

Description: TODO: FIX COMMENT. (Contributed by NM, 8-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg12.l
cdlemg12.j	$.\/$
cdlemg12.m	$./\$
cdlemg12.a
cdlemg12.h
cdlemg12.t
cdlemg12b.r

**Assertion**
Ref	Expression
cdlemg17a	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

**Proof of Theorem cdlemg17a**
Step	Hyp	Ref	Expression
1		eqid 2471	. 2
2		cdlemg12.l	. 2
3		simp1l 1054	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
4		hllat 33000	. . 3
5	3, 4	syl 17	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
6		simp1 1030	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
7		simp3l 1058	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
8		simp2ll 1097	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
9		cdlemg12.a	. . . . 5
10		cdlemg12.h	. . . . 5
11		cdlemg12.t	. . . . 5
12	2, 9, 10, 11	ltrnat 33776	. . . 4 $/\$ $/\$ $/\$
13	6, 7, 8, 12	syl3anc 1292	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
14	1, 9	atbase 32926	. . 3
15	13, 14	syl 17	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
16		cdlemg12.j	. . . 4 $.\/$
17	1, 16, 9	hlatjcl 33003	. . 3 $/\$ $/\$ $.\/$
18	3, 8, 13, 17	syl3anc 1292	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
19		simp2rl 1099	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20	1, 16, 9	hlatjcl 33003	. . 3 $/\$ $/\$ $.\/$
21	3, 8, 19, 20	syl3anc 1292	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
22	2, 16, 9	hlatlej2 33012	. . 3 $/\$ $/\$ $.\/$
23	3, 8, 13, 22	syl3anc 1292	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
24		simp2l 1056	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
25		cdlemg12.m	. . . . 5 $./\$
26		eqid 2471	. . . . 5 $.\/$ $./\$ $.\/$ $./\$
27	2, 16, 25, 9, 10, 26	cdleme0cp 33851	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
28	6, 24, 13, 27	syl12anc 1290	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
29	2, 16, 9	hlatlej1 33011	. . . . 5 $/\$ $/\$ $.\/$
30	3, 8, 19, 29	syl3anc 1292	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31		cdlemg12b.r	. . . . . . 7
32	2, 16, 25, 9, 10, 11, 31	trlval2 33800	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
33	6, 7, 24, 32	syl3anc 1292	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
34		simp3r 1059	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
35	33, 34	eqbrtrrd 4418	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
36	1, 9	atbase 32926	. . . . . 6
37	8, 36	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
38		simp1r 1055	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
39	1, 10	lhpbase 33634	. . . . . . 7
40	38, 39	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
41	1, 25	latmcl 16376	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$
42	5, 18, 40, 41	syl3anc 1292	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
43	1, 2, 16	latjle12 16386	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
44	5, 37, 42, 21, 43	syl13anc 1294	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
45	30, 35, 44	mpbi2and 935	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
46	28, 45	eqbrtrrd 4418	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
47	1, 2, 5, 15, 18, 21, 23, 46	lattrd 16382	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 376 $/\$ w3a 1007

wceq 1452

wcel 1904 class class class wbr 4395

cfv 5589 (class class class)co 6308

cbs 15199

cple 15275

cjn 16267

cmee 16268

clat 16369

catm 32900

chlt 32987

clh 33620

cltrn 33737

ctrl 33795

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602

This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-iun 4271 df-iin 4272 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-1st 6812 df-2nd 6813 df-map 7492 df-preset 16251 df-poset 16269 df-plt 16282 df-lub 16298 df-glb 16299 df-join 16300 df-meet 16301 df-p0 16363 df-p1 16364 df-lat 16370 df-clat 16432 df-oposet 32813 df-ol 32815 df-oml 32816 df-covers 32903 df-ats 32904 df-atl 32935 df-cvlat 32959 df-hlat 32988 df-psubsp 33139 df-pmap 33140 df-padd 33432 df-lhyp 33624 df-laut 33625 df-ldil 33740 df-ltrn 33741 df-trl 33796

This theorem is referenced by: cdlemg17b 34300

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