trlval3 - Mathbox for Norm Megill

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Theorem trlval3 35474

Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)

**Hypotheses**
Ref	Expression
trlval3.l
trlval3.j	$.\/$
trlval3.m	$./\$
trlval3.a
trlval3.h
trlval3.t
trlval3.r

**Assertion**
Ref	Expression
trlval3	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

**Proof of Theorem trlval3**
Step	Hyp	Ref	Expression
1		simpl1 1064	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
2		simpl31 1142	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
3		simpl2 1065	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
4		simpr 477	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
5		trlval3.l	. . . . 5
6		eqid 2622	. . . . 5
7		trlval3.a	. . . . 5
8		trlval3.h	. . . . 5
9		trlval3.t	. . . . 5
10		trlval3.r	. . . . 5
11	5, 6, 7, 8, 9, 10	trl0 35457	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
12	1, 2, 3, 4, 11	syl112anc 1330	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
13		simpl33 1144	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
14		simpl1l 1112	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
15		hlatl 34647	. . . . . 6
16	14, 15	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
17	4	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
18		simp31l 1184	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
19	18	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
20		trlval3.j	. . . . . . . . 9 $.\/$
21	20, 7	hlatjidm 34655	. . . . . . . 8 $/\$ $.\/$
22	14, 19, 21	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
23	17, 22	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
24	23, 19	eqeltrd 2701	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
25		simp1 1061	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
26		simp2 1062	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
27		simp31 1097	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
28		simp32 1098	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
29	5, 7, 8, 9	ltrn2ateq 35467	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	25, 26, 27, 28, 29	syl13anc 1328	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31	30	biimpa 501	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
32	31	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
33		simp32l 1186	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
34	33	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
35	20, 7	hlatjidm 34655	. . . . . . . 8 $/\$ $.\/$
36	14, 34, 35	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
37	32, 36	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
38	37, 34	eqeltrd 2701	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
39		trlval3.m	. . . . . 6 $./\$
40	39, 6, 7	atnem0 34605	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
41	16, 24, 38, 40	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
42	13, 41	mpbid 222	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
43	12, 42	eqtr4d 2659	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
44		simpl1 1064	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
45		simpl2 1065	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
46		simpl31 1142	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
47	5, 20, 39, 7, 8, 9, 10	trlval2 35450	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
48	44, 45, 46, 47	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$
49		simpl1l 1112	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
50		hllat 34650	. . . . . . 7
51	49, 50	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
52	18	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
53	5, 7, 8, 9	ltrnat 35426	. . . . . . . 8 $/\$ $/\$ $/\$
54	44, 45, 52, 53	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
55		eqid 2622	. . . . . . . 8
56	55, 20, 7	hlatjcl 34653	. . . . . . 7 $/\$ $/\$ $.\/$
57	49, 52, 54, 56	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
58		simpl1r 1113	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
59	55, 8	lhpbase 35284	. . . . . . 7
60	58, 59	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
61	55, 5, 39	latmle1 17076	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
62	51, 57, 60, 61	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
63	48, 62	eqbrtrd 4675	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
64		simpl32 1143	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
65	5, 20, 39, 7, 8, 9, 10	trlval2 35450	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
66	44, 45, 64, 65	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$
67	33	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
68	5, 7, 8, 9	ltrnat 35426	. . . . . . . 8 $/\$ $/\$ $/\$
69	44, 45, 67, 68	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
70	55, 20, 7	hlatjcl 34653	. . . . . . 7 $/\$ $/\$ $.\/$
71	49, 67, 69, 70	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
72	55, 5, 39	latmle1 17076	. . . . . 6 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
73	51, 71, 60, 72	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
74	66, 73	eqbrtrd 4675	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
75	55, 8, 9, 10	trlcl 35451	. . . . . 6 $/\$ $/\$
76	44, 45, 75	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
77	55, 5, 39	latlem12 17078	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
78	51, 76, 57, 71, 77	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
79	63, 74, 78	mpbi2and 956	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
80	49, 15	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
81		simpr 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
82	5, 7, 8, 9, 10	trlat 35456	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
83	44, 46, 45, 81, 82	syl112anc 1330	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
84	55, 39	latmcl 17052	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
85	51, 57, 71, 84	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
86	55, 5, 6, 7	atlen0 34597	. . . . . . 7 $/\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
87	80, 85, 83, 79, 86	syl31anc 1329	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
88	87	neneqd 2799	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
89		simpl33 1144	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
90	20, 39, 6, 7	2atmat0 34812	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$
91	49, 52, 54, 67, 69, 89, 90	syl33anc 1341	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $\/$ $.\/$ $./\$ $.\/$
92	91	ord 392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
93	88, 92	mt3d 140	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
94	5, 7	atcmp 34598	. . . 4 $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
95	80, 83, 93, 94	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
96	79, 95	mpbid 222	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
97	43, 96	pm2.61dane 2881	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ 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

cjn 16944

cmee 16945

cp0 17037

clat 17045

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-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-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-llines 34784 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446

This theorem is referenced by: trlval4 35475

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