cmtbr3N - Mathbox for Norm Megill

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Theorem cmtbr3N 34541

Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 28467 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cmtbr2.b
cmtbr2.j	$.\/$
cmtbr2.m	$./\$
cmtbr2.o
cmtbr2.c

**Assertion**
Ref	Expression
cmtbr3N	$/\$ $/\$ $./\$ $.\/$ $./\$

**Proof of Theorem cmtbr3N**
Step	Hyp	Ref	Expression
1		cmtbr2.b	. . . . 5
2		cmtbr2.c	. . . . 5
3	1, 2	cmtcomN 34536	. . . 4 $/\$ $/\$
4		cmtbr2.j	. . . . . 6 $.\/$
5		cmtbr2.m	. . . . . 6 $./\$
6		cmtbr2.o	. . . . . 6
7	1, 4, 5, 6, 2	cmtbr2N 34540	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$
8	7	3com23 1271	. . . 4 $/\$ $/\$ $.\/$ $./\$ $.\/$
9	3, 8	bitrd 268	. . 3 $/\$ $/\$ $.\/$ $./\$ $.\/$
10		oveq2 6658	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
11	10	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$ $.\/$
12		omlol 34527	. . . . . . . . 9
13	12	3ad2ant1 1082	. . . . . . . 8 $/\$ $/\$
14		simp2 1062	. . . . . . . 8 $/\$ $/\$
15		omllat 34529	. . . . . . . . . 10
16	15	3ad2ant1 1082	. . . . . . . . 9 $/\$ $/\$
17		simp3 1063	. . . . . . . . 9 $/\$ $/\$
18	1, 4	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$
19	16, 17, 14, 18	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $.\/$
20		omlop 34528	. . . . . . . . . . 11
21	20	3ad2ant1 1082	. . . . . . . . . 10 $/\$ $/\$
22	1, 6	opoccl 34481	. . . . . . . . . 10 $/\$
23	21, 14, 22	syl2anc 693	. . . . . . . . 9 $/\$ $/\$
24	1, 4	latjcl 17051	. . . . . . . . 9 $/\$ $/\$ $.\/$
25	16, 17, 23, 24	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $.\/$
26	1, 5	latmassOLD 34516	. . . . . . . 8 $/\$ $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
27	13, 14, 19, 25, 26	syl13anc 1328	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
28	1, 4	latjcom 17059	. . . . . . . . . . 11 $/\$ $/\$ $.\/$ $.\/$
29	16, 17, 14, 28	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $.\/$ $.\/$
30	29	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$
31	1, 4, 5	latabs2 17088	. . . . . . . . . 10 $/\$ $/\$ $./\$ $.\/$
32	15, 31	syl3an1 1359	. . . . . . . . 9 $/\$ $/\$ $./\$ $.\/$
33	30, 32	eqtrd 2656	. . . . . . . 8 $/\$ $/\$ $./\$ $.\/$
34	1, 4	latjcom 17059	. . . . . . . . 9 $/\$ $/\$ $.\/$ $.\/$
35	16, 17, 23, 34	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $.\/$ $.\/$
36	33, 35	oveq12d 6668	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
37	27, 36	eqtr3d 2658	. . . . . 6 $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
38	37	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$
39	11, 38	eqtr2d 2657	. . . 4 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
40	39	ex 450	. . 3 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
41	9, 40	sylbid 230	. 2 $/\$ $/\$ $./\$ $.\/$ $./\$
42		simp1 1061	. . . . . . . . 9 $/\$ $/\$
43	1, 6	opoccl 34481	. . . . . . . . . . 11 $/\$
44	21, 17, 43	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$
45	1, 5	latmcl 17052	. . . . . . . . . 10 $/\$ $/\$ $./\$
46	16, 14, 44, 45	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $./\$
47	42, 46, 14	3jca 1242	. . . . . . . 8 $/\$ $/\$ $/\$ $./\$ $/\$
48		eqid 2622	. . . . . . . . . 10
49	1, 48, 5	latmle1 17076	. . . . . . . . 9 $/\$ $/\$ $./\$
50	16, 14, 44, 49	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $./\$
51	1, 48, 4, 5, 6	omllaw2N 34531	. . . . . . . 8 $/\$ $./\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$
52	47, 50, 51	sylc 65	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$
53	1, 6	opoccl 34481	. . . . . . . . . 10 $/\$ $./\$ $./\$
54	21, 46, 53	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $./\$
55	1, 5	latmcl 17052	. . . . . . . . 9 $/\$ $./\$ $/\$ $./\$ $./\$
56	16, 54, 14, 55	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $./\$ $./\$
57	1, 4	latjcom 17059	. . . . . . . 8 $/\$ $./\$ $/\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$
58	16, 46, 56, 57	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$
59	52, 58	eqtr3d 2658	. . . . . 6 $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$
60	59	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $.\/$ $./\$
61	1, 4, 5, 6	oldmm3N 34506	. . . . . . . . . . 11 $/\$ $/\$ $./\$ $.\/$
62	12, 61	syl3an1 1359	. . . . . . . . . 10 $/\$ $/\$ $./\$ $.\/$
63	62	oveq2d 6666	. . . . . . . . 9 $/\$ $/\$ $./\$ $./\$ $./\$ $.\/$
64	1, 5	latmcom 17075	. . . . . . . . . 10 $/\$ $/\$ $./\$ $./\$ $./\$ $./\$ $./\$
65	16, 14, 54, 64	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$ $./\$ $./\$ $./\$ $./\$
66	63, 65	eqtr3d 2658	. . . . . . . 8 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$
67	66	eqeq1d 2624	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $./\$
68		oveq1 6657	. . . . . . 7 $./\$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
69	67, 68	syl6bi 243	. . . . . 6 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
70	69	imp 445	. . . . 5 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
71	60, 70	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
72	71	ex 450	. . 3 $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
73	1, 4, 5, 6, 2	cmtvalN 34498	. . 3 $/\$ $/\$ $./\$ $.\/$ $./\$
74	72, 73	sylibrd 249	. 2 $/\$ $/\$ $./\$ $.\/$ $./\$
75	41, 74	impbid 202	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

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-lat 17046 df-oposet 34463 df-cmtN 34464 df-ol 34465 df-oml 34466

This theorem is referenced by: cmtbr4N 34542 omlfh1N 34545

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