omlfh1N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > omlfh1N		Structured version Visualization version Unicode version

Theorem omlfh1N 34545

Description: Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 28477 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
omlfh1.b
omlfh1.j	$.\/$
omlfh1.m	$./\$
omlfh1.c

**Assertion**
Ref	Expression
omlfh1N	$/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

**Proof of Theorem omlfh1N**
Step	Hyp	Ref	Expression
1		omllat 34529	. . . . 5
2		omlfh1.b	. . . . . 6
3		eqid 2622	. . . . . 6
4		omlfh1.j	. . . . . 6 $.\/$
5		omlfh1.m	. . . . . 6 $./\$
6	2, 3, 4, 5	latledi 17089	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
7	1, 6	sylan 488	. . . 4 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
8	7	3adant3 1081	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
9	1	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$
10		simpr1 1067	. . . . . . 7 $/\$ $/\$ $/\$
11		simpr2 1068	. . . . . . . 8 $/\$ $/\$ $/\$
12		simpr3 1069	. . . . . . . 8 $/\$ $/\$ $/\$
13	2, 4	latjcl 17051	. . . . . . . 8 $/\$ $/\$ $.\/$
14	9, 11, 12, 13	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$
15	2, 5	latmcom 17075	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
16	9, 10, 14, 15	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $./\$ $.\/$ $.\/$ $./\$
17		omlol 34527	. . . . . . . . 9
18	17	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$
19	2, 5	latmcl 17052	. . . . . . . . 9 $/\$ $/\$ $./\$
20	9, 10, 11, 19	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$
21	2, 5	latmcl 17052	. . . . . . . . 9 $/\$ $/\$ $./\$
22	9, 10, 12, 21	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$
23		eqid 2622	. . . . . . . . 9
24	2, 4, 5, 23	oldmj1 34508	. . . . . . . 8 $/\$ $./\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$
25	18, 20, 22, 24	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$
26	2, 4, 5, 23	oldmm1 34504	. . . . . . . . 9 $/\$ $/\$ $./\$ $.\/$
27	18, 10, 11, 26	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $.\/$
28	2, 4, 5, 23	oldmm1 34504	. . . . . . . . 9 $/\$ $/\$ $./\$ $.\/$
29	18, 10, 12, 28	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $.\/$
30	27, 29	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $.\/$
31	25, 30	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
32	16, 31	oveq12d 6668	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
33	32	3adant3 1081	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
34		omlop 34528	. . . . . . . . . . 11
35	34	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$
36	2, 23	opoccl 34481	. . . . . . . . . 10 $/\$
37	35, 10, 36	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
38	2, 23	opoccl 34481	. . . . . . . . . 10 $/\$
39	35, 11, 38	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
40	2, 4	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$
41	9, 37, 39, 40	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $.\/$
42	2, 23	opoccl 34481	. . . . . . . . . 10 $/\$
43	35, 12, 42	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
44	2, 4	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$
45	9, 37, 43, 44	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $.\/$
46	2, 5	latmcl 17052	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
47	9, 41, 45, 46	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
48	2, 5	latmassOLD 34516	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
49	18, 14, 10, 47, 48	syl13anc 1328	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
50	49	3adant3 1081	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
51		omlfh1.c	. . . . . . . . . . . . . 14
52	2, 23, 51	cmt2N 34537	. . . . . . . . . . . . 13 $/\$ $/\$
53	52	3adant3r3 1276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
54		simpl 473	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
55	2, 4, 5, 23, 51	cmtbr3N 34541	. . . . . . . . . . . . 13 $/\$ $/\$ $./\$ $.\/$ $./\$
56	54, 10, 39, 55	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
57	53, 56	bitrd 268	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
58	57	biimpa 501	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
59	58	adantrr 753	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
60	59	3impa 1259	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
61	2, 23, 51	cmt2N 34537	. . . . . . . . . . . . 13 $/\$ $/\$
62	61	3adant3r2 1275	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
63	2, 4, 5, 23, 51	cmtbr3N 34541	. . . . . . . . . . . . 13 $/\$ $/\$ $./\$ $.\/$ $./\$
64	54, 10, 43, 63	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
65	62, 64	bitrd 268	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
66	65	biimpa 501	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
67	66	adantrl 752	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
68	67	3impa 1259	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
69	60, 68	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $./\$
70	2, 5	latmmdiN 34521	. . . . . . . . 9 $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
71	18, 10, 41, 45, 70	syl13anc 1328	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
72	71	3adant3 1081	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
73	2, 5	latmmdiN 34521	. . . . . . . . 9 $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$
74	18, 10, 39, 43, 73	syl13anc 1328	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$
75	74	3adant3 1081	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$
76	69, 72, 75	3eqtr4d 2666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$
77	76	oveq2d 6666	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $./\$
78	2, 5	latmcl 17052	. . . . . . . 8 $/\$ $/\$ $./\$
79	9, 39, 43, 78	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $./\$
80	2, 5	latm12 34517	. . . . . . 7 $/\$ $.\/$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$
81	18, 14, 10, 79, 80	syl13anc 1328	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$
82	81	3adant3 1081	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$
83	50, 77, 82	3eqtrd 2660	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$
84	2, 4, 5, 23	oldmj1 34508	. . . . . . . . . 10 $/\$ $/\$ $.\/$ $./\$
85	18, 11, 12, 84	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $/\$ $.\/$ $./\$
86	85	oveq2d 6666	. . . . . . . 8 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$
87		eqid 2622	. . . . . . . . . 10
88	2, 23, 5, 87	opnoncon 34495	. . . . . . . . 9 $/\$ $.\/$ $.\/$ $./\$ $.\/$
89	35, 14, 88	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
90	86, 89	eqtr3d 2658	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
91	90	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$
92	2, 5, 87	olm01 34523	. . . . . . 7 $/\$ $./\$
93	18, 10, 92	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$ $./\$
94	91, 93	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$
95	94	3adant3 1081	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$
96	33, 83, 95	3eqtrd 2660	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
97	2, 4	latjcl 17051	. . . . . 6 $/\$ $./\$ $/\$ $./\$ $./\$ $.\/$ $./\$
98	9, 20, 22, 97	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
99	2, 5	latmcl 17052	. . . . . 6 $/\$ $/\$ $.\/$ $./\$ $.\/$
100	9, 10, 14, 99	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$
101	2, 3, 5, 23, 87	omllaw3 34532	. . . . 5 $/\$ $./\$ $.\/$ $./\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
102	54, 98, 100, 101	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
103	102	3adant3 1081	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
104	8, 96, 103	mp2and 715	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
105	104	eqcomd 2628	1 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

coc 15949

cjn 16944

cmee 16945

cp0 17037

clat 17045

cops 34459

ccmtN 34460

col 34461

coml 34462

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-preset 16928 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-oposet 34463 df-cmtN 34464 df-ol 34465 df-oml 34466

This theorem is referenced by: omlfh3N 34546 omlmod1i2N 34547

W3C validator