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Theorem latjass 17095

Description: Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 28392 analog.) (Contributed by NM, 17-Sep-2011.)

**Hypotheses**
Ref	Expression
latjass.b
latjass.j	$.\/$

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

**Proof of Theorem latjass**
Step	Hyp	Ref	Expression
1		latjass.b	. 2
2		eqid 2622	. 2
3		simpl 473	. 2 $/\$ $/\$ $/\$
4		latjass.j	. . . . 5 $.\/$
5	1, 4	latjcl 17051	. . . 4 $/\$ $/\$ $.\/$
6	5	3adant3r3 1276	. . 3 $/\$ $/\$ $/\$ $.\/$
7		simpr3 1069	. . 3 $/\$ $/\$ $/\$
8	1, 4	latjcl 17051	. . 3 $/\$ $.\/$ $/\$ $.\/$ $.\/$
9	3, 6, 7, 8	syl3anc 1326	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$
10		simpr1 1067	. . 3 $/\$ $/\$ $/\$
11	1, 4	latjcl 17051	. . . 4 $/\$ $/\$ $.\/$
12	11	3adant3r1 1274	. . 3 $/\$ $/\$ $/\$ $.\/$
13	1, 4	latjcl 17051	. . 3 $/\$ $/\$ $.\/$ $.\/$ $.\/$
14	3, 10, 12, 13	syl3anc 1326	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$
15	1, 2, 4	latlej1 17060	. . . . 5 $/\$ $/\$ $.\/$ $.\/$ $.\/$
16	3, 10, 12, 15	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simpr2 1068	. . . . 5 $/\$ $/\$ $/\$
18	1, 2, 4	latlej1 17060	. . . . . 6 $/\$ $/\$ $.\/$
19	18	3adant3r1 1274	. . . . 5 $/\$ $/\$ $/\$ $.\/$
20	1, 2, 4	latlej2 17061	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
21	3, 10, 12, 20	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
22	1, 2, 3, 17, 12, 14, 19, 21	lattrd 17058	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$
23	1, 2, 4	latjle12 17062	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
24	3, 10, 17, 14, 23	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
25	16, 22, 24	mpbi2and 956	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
26	1, 2, 4	latlej2 17061	. . . . 5 $/\$ $/\$ $.\/$
27	26	3adant3r1 1274	. . . 4 $/\$ $/\$ $/\$ $.\/$
28	1, 2, 3, 7, 12, 14, 27, 21	lattrd 17058	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$
29	1, 2, 4	latjle12 17062	. . . 4 $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
30	3, 6, 7, 14, 29	syl13anc 1328	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
31	25, 28, 30	mpbi2and 956	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
32	1, 2, 4	latlej1 17060	. . . . 5 $/\$ $/\$ $.\/$
33	32	3adant3r3 1276	. . . 4 $/\$ $/\$ $/\$ $.\/$
34	1, 2, 4	latlej1 17060	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
35	3, 6, 7, 34	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
36	1, 2, 3, 10, 6, 9, 33, 35	lattrd 17058	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$
37	1, 2, 4	latlej2 17061	. . . . . 6 $/\$ $/\$ $.\/$
38	37	3adant3r3 1276	. . . . 5 $/\$ $/\$ $/\$ $.\/$
39	1, 2, 3, 17, 6, 9, 38, 35	lattrd 17058	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$
40	1, 2, 4	latlej2 17061	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$
41	3, 6, 7, 40	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$
42	1, 2, 4	latjle12 17062	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
43	3, 17, 7, 9, 42	syl13anc 1328	. . . 4 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
44	39, 41, 43	mpbi2and 956	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
45	1, 2, 4	latjle12 17062	. . . 4 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
46	3, 10, 12, 9, 45	syl13anc 1328	. . 3 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
47	36, 44, 46	mpbi2and 956	. 2 $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
48	1, 2, 3, 9, 14, 31, 47	latasymd 17057	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

cjn 16944

clat 17045

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

This theorem is referenced by: latj12 17096 latj32 17097 latj4 17101 latmass 17188 latmassOLD 34516 hlatjass 34656 cvrexchlem 34705 cvrat3 34728 2atmat 34847 4atlem3 34882 4atlem3a 34883 4atlem4a 34885 4atlem4d 34888 4at2 34900 2lplnja 34905 pmapjlln1 35141 dalawlem3 35159 dalawlem12 35168 cdleme30a 35666 trlcolem 36014 cdlemh1 36103 cdlemkid1 36210 doca2N 36415 djajN 36426

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