dia2dimlem3 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem3		Structured version Visualization version Unicode version

Theorem dia2dimlem3 36355

Description: Lemma for dia2dim 36366. Define a translation

whose trace is atom

. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem3.l
dia2dimlem3.j	$.\/$
dia2dimlem3.m	$./\$
dia2dimlem3.a
dia2dimlem3.h
dia2dimlem3.t
dia2dimlem3.r
dia2dimlem3.q	$.\/$ $./\$ $.\/$
dia2dimlem3.k	$/\$
dia2dimlem3.u	$/\$
dia2dimlem3.v	$/\$
dia2dimlem3.p	$/\$
dia2dimlem3.f	$/\$
dia2dimlem3.rf	$.\/$
dia2dimlem3.uv
dia2dimlem3.ru
dia2dimlem3.rv
dia2dimlem3.d
dia2dimlem3.dv

**Assertion**
Ref	Expression
dia2dimlem3

**Proof of Theorem dia2dimlem3**
Step	Hyp	Ref	Expression
1		dia2dimlem3.k	. . . . . . 7 $/\$
2	1	simpld 475	. . . . . 6
3		dia2dimlem3.f	. . . . . . . . 9 $/\$
4	3	simpld 475	. . . . . . . 8
5		dia2dimlem3.p	. . . . . . . 8 $/\$
6		dia2dimlem3.l	. . . . . . . . 9
7		dia2dimlem3.a	. . . . . . . . 9
8		dia2dimlem3.h	. . . . . . . . 9
9		dia2dimlem3.t	. . . . . . . . 9
10	6, 7, 8, 9	ltrnel 35425	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
11	1, 4, 5, 10	syl3anc 1326	. . . . . . 7 $/\$
12	11	simpld 475	. . . . . 6
13		dia2dimlem3.v	. . . . . . 7 $/\$
14	13	simpld 475	. . . . . 6
15		dia2dimlem3.j	. . . . . . 7 $.\/$
16	6, 15, 7	hlatlej2 34662	. . . . . 6 $/\$ $/\$ $.\/$
17	2, 12, 14, 16	syl3anc 1326	. . . . 5 $.\/$
18		hllat 34650	. . . . . . 7
19	2, 18	syl 17	. . . . . 6
20		eqid 2622	. . . . . . . 8
21	20, 7	atbase 34576	. . . . . . 7
22	14, 21	syl 17	. . . . . 6
23	20, 15, 7	hlatjcl 34653	. . . . . . 7 $/\$ $/\$ $.\/$
24	2, 12, 14, 23	syl3anc 1326	. . . . . 6 $.\/$
25		dia2dimlem3.r	. . . . . . . . 9
26	6, 7, 8, 9, 25	trlat 35456	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
27	1, 5, 3, 26	syl3anc 1326	. . . . . . 7
28		dia2dimlem3.u	. . . . . . . 8 $/\$
29	28	simpld 475	. . . . . . 7
30	20, 15, 7	hlatjcl 34653	. . . . . . 7 $/\$ $/\$ $.\/$
31	2, 27, 29, 30	syl3anc 1326	. . . . . 6 $.\/$
32		dia2dimlem3.m	. . . . . . 7 $./\$
33	20, 6, 32	latmlem2 17082	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
34	19, 22, 24, 31, 33	syl13anc 1328	. . . . 5 $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
35	17, 34	mpd 15	. . . 4 $.\/$ $./\$ $.\/$ $./\$ $.\/$
36		dia2dimlem3.rf	. . . . . . 7 $.\/$
37	15, 7	hlatjcom 34654	. . . . . . . 8 $/\$ $/\$ $.\/$ $.\/$
38	2, 29, 14, 37	syl3anc 1326	. . . . . . 7 $.\/$ $.\/$
39	36, 38	breqtrd 4679	. . . . . 6 $.\/$
40		dia2dimlem3.ru	. . . . . . 7
41	6, 15, 7	hlatexch2 34682	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
42	2, 27, 14, 29, 40, 41	syl131anc 1339	. . . . . 6 $.\/$ $.\/$
43	39, 42	mpd 15	. . . . 5 $.\/$
44	20, 6, 32	latleeqm2 17080	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$
45	19, 22, 31, 44	syl3anc 1326	. . . . 5 $.\/$ $.\/$ $./\$
46	43, 45	mpbid 222	. . . 4 $.\/$ $./\$
47		dia2dimlem3.d	. . . . . 6
48		dia2dimlem3.q	. . . . . . 7 $.\/$ $./\$ $.\/$
49		dia2dimlem3.uv	. . . . . . 7
50	6, 15, 32, 7, 8, 9, 25, 48, 1, 28, 13, 5, 3, 36, 49, 40	dia2dimlem1 36353	. . . . . 6 $/\$
51	6, 15, 32, 7, 8, 9, 25	trlval2 35450	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
52	1, 47, 50, 51	syl3anc 1326	. . . . 5 $.\/$ $./\$
53	48	a1i 11	. . . . . . . . 9 $.\/$ $./\$ $.\/$
54		dia2dimlem3.dv	. . . . . . . . 9
55	53, 54	oveq12d 6668	. . . . . . . 8 $.\/$ $.\/$ $./\$ $.\/$ $.\/$
56	5	simpld 475	. . . . . . . . . 10
57	20, 15, 7	hlatjcl 34653	. . . . . . . . . 10 $/\$ $/\$ $.\/$
58	2, 56, 29, 57	syl3anc 1326	. . . . . . . . 9 $.\/$
59	6, 15, 7	hlatlej1 34661	. . . . . . . . . 10 $/\$ $/\$ $.\/$
60	2, 12, 14, 59	syl3anc 1326	. . . . . . . . 9 $.\/$
61	20, 6, 15, 32, 7	atmod4i1 35152	. . . . . . . . 9 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
62	2, 12, 58, 24, 60, 61	syl131anc 1339	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
63	15, 7	hlatj32 34658	. . . . . . . . . 10 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
64	2, 56, 29, 12, 63	syl13anc 1328	. . . . . . . . 9 $.\/$ $.\/$ $.\/$ $.\/$
65	64	oveq1d 6665	. . . . . . . 8 $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$
66	55, 62, 65	3eqtrd 2660	. . . . . . 7 $.\/$ $.\/$ $.\/$ $./\$ $.\/$
67	66	oveq1d 6665	. . . . . 6 $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
68		hlol 34648	. . . . . . . 8
69	2, 68	syl 17	. . . . . . 7
70	20, 15, 7	hlatjcl 34653	. . . . . . . . 9 $/\$ $/\$ $.\/$
71	2, 56, 12, 70	syl3anc 1326	. . . . . . . 8 $.\/$
72	20, 7	atbase 34576	. . . . . . . . 9
73	29, 72	syl 17	. . . . . . . 8
74	20, 15	latjcl 17051	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $.\/$
75	19, 71, 73, 74	syl3anc 1326	. . . . . . 7 $.\/$ $.\/$
76	1	simprd 479	. . . . . . . 8
77	20, 8	lhpbase 35284	. . . . . . . 8
78	76, 77	syl 17	. . . . . . 7
79	20, 32	latm32 34518	. . . . . . 7 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$
80	69, 75, 24, 78, 79	syl13anc 1328	. . . . . 6 $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$
81	6, 15, 32, 7, 8, 9, 25	trlval2 35450	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
82	1, 4, 5, 81	syl3anc 1326	. . . . . . . . 9 $.\/$ $./\$
83	82	oveq1d 6665	. . . . . . . 8 $.\/$ $.\/$ $./\$ $.\/$
84	28	simprd 479	. . . . . . . . 9
85	20, 6, 15, 32, 7	atmod4i1 35152	. . . . . . . . 9 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
86	2, 29, 71, 78, 84, 85	syl131anc 1339	. . . . . . . 8 $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
87	83, 86	eqtr2d 2657	. . . . . . 7 $.\/$ $.\/$ $./\$ $.\/$
88	87	oveq1d 6665	. . . . . 6 $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
89	67, 80, 88	3eqtrd 2660	. . . . 5 $.\/$ $./\$ $.\/$ $./\$ $.\/$
90	52, 89	eqtr2d 2657	. . . 4 $.\/$ $./\$ $.\/$
91	35, 46, 90	3brtr3d 4684	. . 3
92		hlatl 34647	. . . . 5
93	2, 92	syl 17	. . . 4
94		hlop 34649	. . . . . . . . . 10
95	2, 94	syl 17	. . . . . . . . 9
96		eqid 2622	. . . . . . . . . 10
97		eqid 2622	. . . . . . . . . 10
98	96, 97, 7	0ltat 34578	. . . . . . . . 9 $/\$
99	95, 14, 98	syl2anc 693	. . . . . . . 8
100		hlpos 34652	. . . . . . . . . 10
101	2, 100	syl 17	. . . . . . . . 9
102	20, 96	op0cl 34471	. . . . . . . . . 10
103	95, 102	syl 17	. . . . . . . . 9
104	20, 8, 9, 25	trlcl 35451	. . . . . . . . . 10 $/\$ $/\$
105	1, 47, 104	syl2anc 693	. . . . . . . . 9
106	20, 6, 97	pltletr 16971	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
107	101, 103, 22, 105, 106	syl13anc 1328	. . . . . . . 8 $/\$
108	99, 91, 107	mp2and 715	. . . . . . 7
109	20, 97, 96	opltn0 34477	. . . . . . . 8 $/\$
110	95, 105, 109	syl2anc 693	. . . . . . 7
111	108, 110	mpbid 222	. . . . . 6
112	111	neneqd 2799	. . . . 5
113	96, 7, 8, 9, 25	trlator0 35458	. . . . . . . 8 $/\$ $/\$ $\/$
114	1, 47, 113	syl2anc 693	. . . . . . 7 $\/$
115	114	orcomd 403	. . . . . 6 $\/$
116	115	ord 392	. . . . 5
117	112, 116	mpd 15	. . . 4
118	6, 7	atcmp 34598	. . . 4 $/\$ $/\$
119	93, 14, 117, 118	syl3anc 1326	. . 3
120	91, 119	mpbid 222	. 2
121	120	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-llines 34784 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

W3C validator