cdlemg12g - Mathbox for Norm Megill

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Theorem cdlemg12g 36791

Description: TODO: FIX COMMENT TODO: Combine with cdlemg12f 36790. (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 1105	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
2		hlop 35503	. . 3
3	1, 2	syl 16	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
4		hllat 35504	. . . 4
5	1, 4	syl 16	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
6		simp12l 1107	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
7		simp11 1024	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
8		simp21 1027	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
9		simp22 1028	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
10		cdlemg12.l	. . . . . 6
11		cdlemg12.a	. . . . . 6
12		cdlemg12.h	. . . . . 6
13		cdlemg12.t	. . . . . 6
14	10, 11, 12, 13	ltrncoat 36284	. . . . 5 $/\$ $/\$ $/\$ $/\$
15	7, 8, 9, 6, 14	syl121anc 1231	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
16		eqid 2454	. . . . 5
17		cdlemg12.j	. . . . 5 $.\/$
18	16, 17, 11	hlatjcl 35507	. . . 4 $/\$ $/\$ $.\/$
19	1, 6, 15, 18	syl3anc 1226	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
20		simp13l 1109	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
21	10, 11, 12, 13	ltrncoat 36284	. . . . 5 $/\$ $/\$ $/\$ $/\$
22	7, 8, 9, 20, 21	syl121anc 1231	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
23	16, 17, 11	hlatjcl 35507	. . . 4 $/\$ $/\$ $.\/$
24	1, 20, 22, 23	syl3anc 1226	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
25		cdlemg12.m	. . . 4 $./\$
26	16, 25	latmcl 15884	. . 3 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
27	5, 19, 24, 26	syl3anc 1226	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
28		simp12 1025	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
29		simp13 1026	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$
30		simp33 1032	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
31	10, 17, 25, 11, 12, 13	cdlemg11a 36779	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
32	31	necomd 2725	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
33	7, 28, 29, 8, 9, 30, 32	syl123anc 1243	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
34	10, 17, 25, 11, 12	lhpat 36183	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
35	7, 28, 15, 33, 34	syl112anc 1230	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
36	17, 11	hlatjcom 35508	. . . . 5 $/\$ $/\$ $.\/$ $.\/$
37	1, 6, 15, 36	syl3anc 1226	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
38	17, 11	hlatjcom 35508	. . . . 5 $/\$ $/\$ $.\/$ $.\/$
39	1, 20, 22, 38	syl3anc 1226	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
40	37, 39	oveq12d 6288	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
41		simp1 994	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simp2 995	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
43		simp31l 1117	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
44		simp31r 1118	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
45		simp32 1031	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
46		cdlemg12b.r	. . . . 5
47		eqid 2454	. . . . 5
48	10, 17, 25, 11, 12, 13, 46, 47	cdlemg12e 36789	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
49	41, 42, 43, 44, 45, 48	syl113anc 1238	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
50	40, 49	eqnetrd 2747	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
51	10, 17, 25, 11, 12, 13, 46	cdlemg12f 36790	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
52	16, 10, 47, 11	leat2 35435	. 2 $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
53	3, 27, 35, 50, 51, 52	syl32anc 1234	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 367 $/\$ w3a 971

wceq 1398

wcel 1823

wne 2649 class class class wbr 4439

cfv 5570 (class class class)co 6270

cbs 14719

cple 14794

cjn 15775

cmee 15776

cp0 15869

clat 15877

cops 35313

catm 35404

chlt 35491

clh 36124

cltrn 36241

ctrl 36299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1623 ax-4 1636 ax-5 1709 ax-6 1752 ax-7 1795 ax-8 1825 ax-9 1827 ax-10 1842 ax-11 1847 ax-12 1859 ax-13 2004 ax-ext 2432 ax-rep 4550 ax-sep 4560 ax-nul 4568 ax-pow 4615 ax-pr 4676 ax-un 6565 ax-riotaBAD 35100

This theorem depends on definitions: df-bi 185 df-or 368 df-an 369 df-3or 972 df-3an 973 df-tru 1401 df-ex 1618 df-nf 1622 df-sb 1745 df-eu 2288 df-mo 2289 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2651 df-nel 2652 df-ral 2809 df-rex 2810 df-reu 2811 df-rmo 2812 df-rab 2813 df-v 3108 df-sbc 3325 df-csb 3421 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-nul 3784 df-if 3930 df-pw 4001 df-sn 4017 df-pr 4019 df-op 4023 df-uni 4236 df-iun 4317 df-iin 4318 df-br 4440 df-opab 4498 df-mpt 4499 df-id 4784 df-xp 4994 df-rel 4995 df-cnv 4996 df-co 4997 df-dm 4998 df-rn 4999 df-res 5000 df-ima 5001 df-iota 5534 df-fun 5572 df-fn 5573 df-f 5574 df-f1 5575 df-fo 5576 df-f1o 5577 df-fv 5578 df-riota 6232 df-ov 6273 df-oprab 6274 df-mpt2 6275 df-1st 6773 df-2nd 6774 df-undef 6994 df-map 7414 df-preset 15759 df-poset 15777 df-plt 15790 df-lub 15806 df-glb 15807 df-join 15808 df-meet 15809 df-p0 15871 df-p1 15872 df-lat 15878 df-clat 15940 df-oposet 35317 df-ol 35319 df-oml 35320 df-covers 35407 df-ats 35408 df-atl 35439 df-cvlat 35463 df-hlat 35492 df-llines 35638 df-lplanes 35639 df-lvols 35640 df-lines 35641 df-psubsp 35643 df-pmap 35644 df-padd 35936 df-lhyp 36128 df-laut 36129 df-ldil 36244 df-ltrn 36245 df-trl 36300

This theorem is referenced by: cdlemg12 36792

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