dia2dimlem2 - Mathbox for Norm Megill

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Theorem dia2dimlem2 36354

Description: Lemma for dia2dim 36366. Define a translation

whose trace is atom

. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem2.l
dia2dimlem2.j	$.\/$
dia2dimlem2.m	$./\$
dia2dimlem2.a
dia2dimlem2.h
dia2dimlem2.t
dia2dimlem2.r
dia2dimlem2.q	$.\/$ $./\$ $.\/$
dia2dimlem2.k	$/\$
dia2dimlem2.u	$/\$
dia2dimlem2.v	$/\$
dia2dimlem2.p	$/\$
dia2dimlem2.f	$/\$
dia2dimlem2.rf	$.\/$
dia2dimlem2.rv
dia2dimlem2.g
dia2dimlem2.gv

**Assertion**
Ref	Expression
dia2dimlem2

**Proof of Theorem dia2dimlem2**
Step	Hyp	Ref	Expression
1		dia2dimlem2.k	. . . . . . . . 9 $/\$
2	1	simpld 475	. . . . . . . 8
3		hllat 34650	. . . . . . . 8
4	2, 3	syl 17	. . . . . . 7
5		dia2dimlem2.p	. . . . . . . . 9 $/\$
6	5	simpld 475	. . . . . . . 8
7		eqid 2622	. . . . . . . . 9
8		dia2dimlem2.a	. . . . . . . . 9
9	7, 8	atbase 34576	. . . . . . . 8
10	6, 9	syl 17	. . . . . . 7
11		dia2dimlem2.u	. . . . . . . . 9 $/\$
12	11	simpld 475	. . . . . . . 8
13	7, 8	atbase 34576	. . . . . . . 8
14	12, 13	syl 17	. . . . . . 7
15		dia2dimlem2.l	. . . . . . . 8
16		dia2dimlem2.j	. . . . . . . 8 $.\/$
17	7, 15, 16	latlej2 17061	. . . . . . 7 $/\$ $/\$ $.\/$
18	4, 10, 14, 17	syl3anc 1326	. . . . . 6 $.\/$
19	7, 16, 8	hlatjcl 34653	. . . . . . . 8 $/\$ $/\$ $.\/$
20	2, 6, 12, 19	syl3anc 1326	. . . . . . 7 $.\/$
21		dia2dimlem2.m	. . . . . . . 8 $./\$
22	7, 15, 21	latleeqm2 17080	. . . . . . 7 $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
23	4, 14, 20, 22	syl3anc 1326	. . . . . 6 $.\/$ $.\/$ $./\$
24	18, 23	mpbid 222	. . . . 5 $.\/$ $./\$
25		dia2dimlem2.rf	. . . . . . . 8 $.\/$
26		dia2dimlem2.f	. . . . . . . . . 10 $/\$
27		dia2dimlem2.h	. . . . . . . . . . 11
28		dia2dimlem2.t	. . . . . . . . . . 11
29		dia2dimlem2.r	. . . . . . . . . . 11
30	15, 8, 27, 28, 29	trlat 35456	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
31	1, 5, 26, 30	syl3anc 1326	. . . . . . . . 9
32		dia2dimlem2.v	. . . . . . . . . 10 $/\$
33	32	simpld 475	. . . . . . . . 9
34		dia2dimlem2.rv	. . . . . . . . 9
35	15, 16, 8	hlatexch2 34682	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
36	2, 31, 12, 33, 34, 35	syl131anc 1339	. . . . . . . 8 $.\/$ $.\/$
37	25, 36	mpd 15	. . . . . . 7 $.\/$
38	26	simpld 475	. . . . . . . . . 10
39	15, 16, 21, 8, 27, 28, 29	trlval2 35450	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
40	1, 38, 5, 39	syl3anc 1326	. . . . . . . . 9 $.\/$ $./\$
41	40	oveq1d 6665	. . . . . . . 8 $.\/$ $.\/$ $./\$ $.\/$
42	15, 8, 27, 28	ltrnel 35425	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
43	1, 38, 5, 42	syl3anc 1326	. . . . . . . . . . . 12 $/\$
44	43	simpld 475	. . . . . . . . . . 11
45	7, 16, 8	hlatjcl 34653	. . . . . . . . . . 11 $/\$ $/\$ $.\/$
46	2, 6, 44, 45	syl3anc 1326	. . . . . . . . . 10 $.\/$
47	1	simprd 479	. . . . . . . . . . 11
48	7, 27	lhpbase 35284	. . . . . . . . . . 11
49	47, 48	syl 17	. . . . . . . . . 10
50	32	simprd 479	. . . . . . . . . 10
51	7, 15, 16, 21, 8	atmod4i1 35152	. . . . . . . . . 10 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
52	2, 33, 46, 49, 50, 51	syl131anc 1339	. . . . . . . . 9 $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
53	16, 8	hlatjass 34656	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
54	2, 6, 44, 33, 53	syl13anc 1328	. . . . . . . . . 10 $.\/$ $.\/$ $.\/$ $.\/$
55	54	oveq1d 6665	. . . . . . . . 9 $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
56	52, 55	eqtrd 2656	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
57	41, 56	eqtrd 2656	. . . . . . 7 $.\/$ $.\/$ $.\/$ $./\$
58	37, 57	breqtrd 4679	. . . . . 6 $.\/$ $.\/$ $./\$
59	7, 16, 8	hlatjcl 34653	. . . . . . . . . 10 $/\$ $/\$ $.\/$
60	2, 44, 33, 59	syl3anc 1326	. . . . . . . . 9 $.\/$
61	7, 16	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$ $.\/$ $.\/$
62	4, 10, 60, 61	syl3anc 1326	. . . . . . . 8 $.\/$ $.\/$
63	7, 21	latmcl 17052	. . . . . . . 8 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$
64	4, 62, 49, 63	syl3anc 1326	. . . . . . 7 $.\/$ $.\/$ $./\$
65	7, 15, 21	latmlem2 17082	. . . . . . 7 $/\$ $/\$ $.\/$ $.\/$ $./\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
66	4, 14, 64, 20, 65	syl13anc 1328	. . . . . 6 $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
67	58, 66	mpd 15	. . . . 5 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
68	24, 67	eqbrtrrd 4677	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
69		dia2dimlem2.g	. . . . . . 7
70	15, 16, 21, 8, 27, 28, 29	trlval2 35450	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
71	1, 69, 5, 70	syl3anc 1326	. . . . . 6 $.\/$ $./\$
72		dia2dimlem2.gv	. . . . . . . . . 10
73		dia2dimlem2.q	. . . . . . . . . 10 $.\/$ $./\$ $.\/$
74	72, 73	syl6eq 2672	. . . . . . . . 9 $.\/$ $./\$ $.\/$
75	74	oveq2d 6666	. . . . . . . 8 $.\/$ $.\/$ $.\/$ $./\$ $.\/$
76	75	oveq1d 6665	. . . . . . 7 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
77	15, 16, 8	hlatlej1 34661	. . . . . . . . . . 11 $/\$ $/\$ $.\/$
78	2, 6, 12, 77	syl3anc 1326	. . . . . . . . . 10 $.\/$
79	7, 15, 16, 21, 8	atmod3i1 35150	. . . . . . . . . 10 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$
80	2, 6, 20, 60, 78, 79	syl131anc 1339	. . . . . . . . 9 $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$
81	80	oveq1d 6665	. . . . . . . 8 $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
82		hlol 34648	. . . . . . . . . 10
83	2, 82	syl 17	. . . . . . . . 9
84	7, 21	latmassOLD 34516	. . . . . . . . 9 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
85	83, 20, 62, 49, 84	syl13anc 1328	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
86	81, 85	eqtrd 2656	. . . . . . 7 $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
87	76, 86	eqtrd 2656	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
88	71, 87	eqtrd 2656	. . . . 5 $.\/$ $./\$ $.\/$ $.\/$ $./\$
89	88	eqcomd 2628	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
90	68, 89	breqtrd 4679	. . 3
91		hlatl 34647	. . . . 5
92	2, 91	syl 17	. . . 4
93		hlop 34649	. . . . . . . . . 10
94	2, 93	syl 17	. . . . . . . . 9
95		eqid 2622	. . . . . . . . . 10
96		eqid 2622	. . . . . . . . . 10
97	95, 96, 8	0ltat 34578	. . . . . . . . 9 $/\$
98	94, 12, 97	syl2anc 693	. . . . . . . 8
99		hlpos 34652	. . . . . . . . . 10
100	2, 99	syl 17	. . . . . . . . 9
101	7, 95	op0cl 34471	. . . . . . . . . 10
102	94, 101	syl 17	. . . . . . . . 9
103	7, 27, 28, 29	trlcl 35451	. . . . . . . . . 10 $/\$ $/\$
104	1, 69, 103	syl2anc 693	. . . . . . . . 9
105	7, 15, 96	pltletr 16971	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
106	100, 102, 14, 104, 105	syl13anc 1328	. . . . . . . 8 $/\$
107	98, 90, 106	mp2and 715	. . . . . . 7
108	7, 96, 95	opltn0 34477	. . . . . . . 8 $/\$
109	94, 104, 108	syl2anc 693	. . . . . . 7
110	107, 109	mpbid 222	. . . . . 6
111	110	neneqd 2799	. . . . 5
112	95, 8, 27, 28, 29	trlator0 35458	. . . . . . . 8 $/\$ $/\$ $\/$
113	1, 69, 112	syl2anc 693	. . . . . . 7 $\/$
114	113	orcomd 403	. . . . . 6 $\/$
115	114	ord 392	. . . . 5
116	111, 115	mpd 15	. . . 4
117	15, 8	atcmp 34598	. . . 4 $/\$ $/\$
118	92, 12, 116, 117	syl3anc 1326	. . 3
119	90, 118	mpbid 222	. 2
120	119	eqcomd 2628	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

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

col 34461

catm 34550

cal 34551

chlt 34637

clh 35270

cltrn 35387

ctrl 35445

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-iin 4523 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-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 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-p1 17040 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-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446

This theorem is referenced by: dia2dimlem5 36357

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