cdlemg12g - Mathbox for Norm Megill

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Theorem cdlemg12g 35445

Description: TODO: FIX COMMENT TODO: Combine with cdlemg12f 35444. (Contributed by NM, 6-May-2013.)

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

**Assertion**
Ref	Expression
cdlemg12g	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

**Proof of Theorem cdlemg12g**
Step	Hyp	Ref	Expression
1		simp11l 1107	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
2		hlop 34159	. . 3
3	1, 2	syl 16	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
4		hllat 34160	. . . 4
5	1, 4	syl 16	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
6		simp12l 1109	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
7		simp11 1026	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
8		simp21 1029	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
9		simp22 1030	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
10		cdlemg12.l	. . . . . 6
11		cdlemg12.a	. . . . . 6
12		cdlemg12.h	. . . . . 6
13		cdlemg12.t	. . . . . 6
14	10, 11, 12, 13	ltrncoat 34940	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	7, 8, 9, 6, 14	syl121anc 1233	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
16		eqid 2467	. . . . 5
17		cdlemg12.j	. . . . 5 $.\/$
18	16, 17, 11	hlatjcl 34163	. . . 4 $/\$ $/\$ $.\/$
19	1, 6, 15, 18	syl3anc 1228	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
20		simp13l 1111	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
21	10, 11, 12, 13	ltrncoat 34940	. . . . 5 $/\$ $/\$ $/\$ $/\$
22	7, 8, 9, 20, 21	syl121anc 1233	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
23	16, 17, 11	hlatjcl 34163	. . . 4 $/\$ $/\$ $.\/$
24	1, 20, 22, 23	syl3anc 1228	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
25		cdlemg12.m	. . . 4 $./\$
26	16, 25	latmcl 15535	. . 3 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
27	5, 19, 24, 26	syl3anc 1228	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
28		simp12 1027	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
29		simp13 1028	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
30		simp33 1034	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
31	10, 17, 25, 11, 12, 13	cdlemg11a 35433	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
32	31	necomd 2738	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
33	7, 28, 29, 8, 9, 30, 32	syl123anc 1245	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
34	10, 17, 25, 11, 12	lhpat 34839	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
35	7, 28, 15, 33, 34	syl112anc 1232	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
36	17, 11	hlatjcom 34164	. . . . 5 $/\$ $/\$ $.\/$ $.\/$
37	1, 6, 15, 36	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
38	17, 11	hlatjcom 34164	. . . . 5 $/\$ $/\$ $.\/$ $.\/$
39	1, 20, 22, 38	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
40	37, 39	oveq12d 6300	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
41		simp1 996	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simp2 997	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
43		simp31l 1119	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
44		simp31r 1120	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
45		simp32 1033	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
46		cdlemg12b.r	. . . . 5
47		eqid 2467	. . . . 5
48	10, 17, 25, 11, 12, 13, 46, 47	cdlemg12e 35443	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
49	41, 42, 43, 44, 45, 48	syl113anc 1240	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
50	40, 49	eqnetrd 2760	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
51	10, 17, 25, 11, 12, 13, 46	cdlemg12f 35444	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
52	16, 10, 47, 11	leat2 34091	. 2 $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
53	3, 27, 35, 50, 51, 52	syl32anc 1236	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767

wne 2662 class class class wbr 4447

cfv 5586 (class class class)co 6282

cbs 14486

cple 14558

cjn 15427

cmee 15428

cp0 15520

clat 15528

cops 33969

catm 34060

chlt 34147

clh 34780

cltrn 34897

ctrl 34954

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 4558 ax-sep 4568 ax-nul 4576 ax-pow 4625 ax-pr 4686 ax-un 6574 ax-riotaBAD 33756

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 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-nel 2665 df-ral 2819 df-rex 2820 df-reu 2821 df-rmo 2822 df-rab 2823 df-v 3115 df-sbc 3332 df-csb 3436 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-nul 3786 df-if 3940 df-pw 4012 df-sn 4028 df-pr 4030 df-op 4034 df-uni 4246 df-iun 4327 df-iin 4328 df-br 4448 df-opab 4506 df-mpt 4507 df-id 4795 df-xp 5005 df-rel 5006 df-cnv 5007 df-co 5008 df-dm 5009 df-rn 5010 df-res 5011 df-ima 5012 df-iota 5549 df-fun 5588 df-fn 5589 df-f 5590 df-f1 5591 df-fo 5592 df-f1o 5593 df-fv 5594 df-riota 6243 df-ov 6285 df-oprab 6286 df-mpt2 6287 df-1st 6781 df-2nd 6782 df-undef 6999 df-map 7419 df-poset 15429 df-plt 15441 df-lub 15457 df-glb 15458 df-join 15459 df-meet 15460 df-p0 15522 df-p1 15523 df-lat 15529 df-clat 15591 df-oposet 33973 df-ol 33975 df-oml 33976 df-covers 34063 df-ats 34064 df-atl 34095 df-cvlat 34119 df-hlat 34148 df-llines 34294 df-lplanes 34295 df-lvols 34296 df-lines 34297 df-psubsp 34299 df-pmap 34300 df-padd 34592 df-lhyp 34784 df-laut 34785 df-ldil 34900 df-ltrn 34901 df-trl 34955

This theorem is referenced by: cdlemg12 35446

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